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+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with (generalised) nonlinear least squares</h6></li>
+    <li><a class="dropdown-item" href="../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a></li>
+    <li><a class="dropdown-item" href="../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a></li>
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+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with hierarchical models (nonlinear mixed-effects models)</h6></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2022_dmta_pathway.html">Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2023_mesotrione_parent.html">Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</a></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2022_cyan_pathway.html">Testing hierarchical pathway kinetics with residue data on cyantraniliprole</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/dimethenamid_2018.html">Comparison of saemix and nlme evaluations of dimethenamid data from 2018</a></li>
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+    <li><a class="dropdown-item" href="../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a></li>
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+      <h1>Example evaluation of FOCUS Example Dataset D</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change 31 January 2019
+(rebuilt 2025-02-14)</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/FOCUS_D.rmd" class="external-link"><code>vignettes/FOCUS_D.rmd</code></a></small>
+      <div class="d-none name"><code>FOCUS_D.rmd</code></div>
+    </div>
+
+    
+    
+<p>This is just a very simple vignette showing how to fit a degradation
+model for a parent compound with one transformation product using
+<code>mkin</code>. After loading the library we look at the data. We
+have observed concentrations in the column named <code>value</code> at
+the times specified in column <code>time</code> for the two observed
+variables named <code>parent</code> and <code>m1</code>.</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://pkgdown.jrwb.de/mkin/">mkin</a></span>, quietly <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">FOCUS_2006_D</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">##      name time  value</span></span>
+<span><span class="co">## 1  parent    0  99.46</span></span>
+<span><span class="co">## 2  parent    0 102.04</span></span>
+<span><span class="co">## 3  parent    1  93.50</span></span>
+<span><span class="co">## 4  parent    1  92.50</span></span>
+<span><span class="co">## 5  parent    3  63.23</span></span>
+<span><span class="co">## 6  parent    3  68.99</span></span>
+<span><span class="co">## 7  parent    7  52.32</span></span>
+<span><span class="co">## 8  parent    7  55.13</span></span>
+<span><span class="co">## 9  parent   14  27.27</span></span>
+<span><span class="co">## 10 parent   14  26.64</span></span>
+<span><span class="co">## 11 parent   21  11.50</span></span>
+<span><span class="co">## 12 parent   21  11.64</span></span>
+<span><span class="co">## 13 parent   35   2.85</span></span>
+<span><span class="co">## 14 parent   35   2.91</span></span>
+<span><span class="co">## 15 parent   50   0.69</span></span>
+<span><span class="co">## 16 parent   50   0.63</span></span>
+<span><span class="co">## 17 parent   75   0.05</span></span>
+<span><span class="co">## 18 parent   75   0.06</span></span>
+<span><span class="co">## 19 parent  100     NA</span></span>
+<span><span class="co">## 20 parent  100     NA</span></span>
+<span><span class="co">## 21 parent  120     NA</span></span>
+<span><span class="co">## 22 parent  120     NA</span></span>
+<span><span class="co">## 23     m1    0   0.00</span></span>
+<span><span class="co">## 24     m1    0   0.00</span></span>
+<span><span class="co">## 25     m1    1   4.84</span></span>
+<span><span class="co">## 26     m1    1   5.64</span></span>
+<span><span class="co">## 27     m1    3  12.91</span></span>
+<span><span class="co">## 28     m1    3  12.96</span></span>
+<span><span class="co">## 29     m1    7  22.97</span></span>
+<span><span class="co">## 30     m1    7  24.47</span></span>
+<span><span class="co">## 31     m1   14  41.69</span></span>
+<span><span class="co">## 32     m1   14  33.21</span></span>
+<span><span class="co">## 33     m1   21  44.37</span></span>
+<span><span class="co">## 34     m1   21  46.44</span></span>
+<span><span class="co">## 35     m1   35  41.22</span></span>
+<span><span class="co">## 36     m1   35  37.95</span></span>
+<span><span class="co">## 37     m1   50  41.19</span></span>
+<span><span class="co">## 38     m1   50  40.01</span></span>
+<span><span class="co">## 39     m1   75  40.09</span></span>
+<span><span class="co">## 40     m1   75  33.85</span></span>
+<span><span class="co">## 41     m1  100  31.04</span></span>
+<span><span class="co">## 42     m1  100  33.13</span></span>
+<span><span class="co">## 43     m1  120  25.15</span></span>
+<span><span class="co">## 44     m1  120  33.31</span></span></code></pre>
+<p>Next we specify the degradation model: The parent compound degrades
+with simple first-order kinetics (SFO) to one metabolite named m1, which
+also degrades with SFO kinetics.</p>
+<p>The call to mkinmod returns a degradation model. The differential
+equations represented in R code can be found in the character vector
+<code>$diffs</code> of the <code>mkinmod</code> object. If a C compiler
+(gcc) is installed and functional, the differential equation model will
+be compiled from auto-generated C code.</p>
+<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">SFO_SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fu"><a href="../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"m1"</span><span class="op">)</span>, m1 <span class="op">=</span> <span class="fu"><a href="../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">SFO_SFO</span><span class="op">$</span><span class="va">diffs</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">##                                                    parent </span></span>
+<span><span class="co">##                          "d_parent = - k_parent * parent" </span></span>
+<span><span class="co">##                                                        m1 </span></span>
+<span><span class="co">## "d_m1 = + f_parent_to_m1 * k_parent * parent - k_m1 * m1"</span></span></code></pre>
+<p>We do the fitting without progress report
+(<code>quiet = TRUE</code>).</p>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">fit</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">SFO_SFO</span>, <span class="va">FOCUS_2006_D</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit(SFO_SFO, FOCUS_2006_D, quiet = TRUE): Observations with</span></span>
+<span><span class="co">## value of zero were removed from the data</span></span></code></pre>
+<p>A plot of the fit including a residual plot for both observed
+variables is obtained using the <code>plot_sep</code> method for
+<code>mkinfit</code> objects, which shows separate graphs for all
+compounds and their residuals.</p>
+<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">fit</span>, lpos <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"topright"</span>, <span class="st">"bottomright"</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_D_files/figure-html/plot-1.png" width="768"></p>
+<p>Confidence intervals for the parameter estimates are obtained using
+the <code>mkinparplot</code> function.</p>
+<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../reference/mkinparplot.html">mkinparplot</a></span><span class="op">(</span><span class="va">fit</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_D_files/figure-html/plot_2-1.png" width="768"></p>
+<p>A comprehensive report of the results is obtained using the
+<code>summary</code> method for <code>mkinfit</code> objects.</p>
+<div class="sourceCode" id="cb11"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">fit</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## mkin version used for fitting:    1.2.10 </span></span>
+<span><span class="co">## R version used for fitting:       4.4.2 </span></span>
+<span><span class="co">## Date of fit:     Fri Feb 14 08:59:09 2025 </span></span>
+<span><span class="co">## Date of summary: Fri Feb 14 08:59:09 2025 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Equations:</span></span>
+<span><span class="co">## d_parent/dt = - k_parent * parent</span></span>
+<span><span class="co">## d_m1/dt = + f_parent_to_m1 * k_parent * parent - k_m1 * m1</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Model predictions using solution type analytical </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fitted using 401 model solutions performed in 0.053 s</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model: Constant variance </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model algorithm: OLS </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for parameters to be optimised:</span></span>
+<span><span class="co">##                   value   type</span></span>
+<span><span class="co">## parent_0       100.7500  state</span></span>
+<span><span class="co">## k_parent         0.1000 deparm</span></span>
+<span><span class="co">## k_m1             0.1001 deparm</span></span>
+<span><span class="co">## f_parent_to_m1   0.5000 deparm</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for the transformed parameters actually optimised:</span></span>
+<span><span class="co">##                      value lower upper</span></span>
+<span><span class="co">## parent_0        100.750000  -Inf   Inf</span></span>
+<span><span class="co">## log_k_parent     -2.302585  -Inf   Inf</span></span>
+<span><span class="co">## log_k_m1         -2.301586  -Inf   Inf</span></span>
+<span><span class="co">## f_parent_qlogis   0.000000  -Inf   Inf</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fixed parameter values:</span></span>
+<span><span class="co">##      value  type</span></span>
+<span><span class="co">## m1_0     0 state</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Warning(s): </span></span>
+<span><span class="co">## Observations with value of zero were removed from the data</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Results:</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">##        AIC      BIC    logLik</span></span>
+<span><span class="co">##   204.4486 212.6365 -97.22429</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Optimised, transformed parameters with symmetric confidence intervals:</span></span>
+<span><span class="co">##                 Estimate Std. Error   Lower    Upper</span></span>
+<span><span class="co">## parent_0        99.60000    1.57000 96.4000 102.8000</span></span>
+<span><span class="co">## log_k_parent    -2.31600    0.04087 -2.3990  -2.2330</span></span>
+<span><span class="co">## log_k_m1        -5.24700    0.13320 -5.5180  -4.9770</span></span>
+<span><span class="co">## f_parent_qlogis  0.05792    0.08926 -0.1237   0.2395</span></span>
+<span><span class="co">## sigma            3.12600    0.35850  2.3960   3.8550</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameter correlation:</span></span>
+<span><span class="co">##                   parent_0 log_k_parent   log_k_m1 f_parent_qlogis      sigma</span></span>
+<span><span class="co">## parent_0         1.000e+00    5.174e-01 -1.688e-01      -5.471e-01 -1.172e-06</span></span>
+<span><span class="co">## log_k_parent     5.174e-01    1.000e+00 -3.263e-01      -5.426e-01 -8.479e-07</span></span>
+<span><span class="co">## log_k_m1        -1.688e-01   -3.263e-01  1.000e+00       7.478e-01  8.211e-07</span></span>
+<span><span class="co">## f_parent_qlogis -5.471e-01   -5.426e-01  7.478e-01       1.000e+00  1.305e-06</span></span>
+<span><span class="co">## sigma           -1.172e-06   -8.479e-07  8.211e-07       1.305e-06  1.000e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Backtransformed parameters:</span></span>
+<span><span class="co">## Confidence intervals for internally transformed parameters are asymmetric.</span></span>
+<span><span class="co">## t-test (unrealistically) based on the assumption of normal distribution</span></span>
+<span><span class="co">## for estimators of untransformed parameters.</span></span>
+<span><span class="co">##                 Estimate t value    Pr(&gt;t)     Lower     Upper</span></span>
+<span><span class="co">## parent_0       99.600000  63.430 2.298e-36 96.400000 1.028e+02</span></span>
+<span><span class="co">## k_parent        0.098700  24.470 4.955e-23  0.090820 1.073e-01</span></span>
+<span><span class="co">## k_m1            0.005261   7.510 6.165e-09  0.004012 6.898e-03</span></span>
+<span><span class="co">## f_parent_to_m1  0.514500  23.070 3.104e-22  0.469100 5.596e-01</span></span>
+<span><span class="co">## sigma           3.126000   8.718 2.235e-10  2.396000 3.855e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## FOCUS Chi2 error levels in percent:</span></span>
+<span><span class="co">##          err.min n.optim df</span></span>
+<span><span class="co">## All data   6.398       4 15</span></span>
+<span><span class="co">## parent     6.459       2  7</span></span>
+<span><span class="co">## m1         4.690       2  8</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Resulting formation fractions:</span></span>
+<span><span class="co">##                 ff</span></span>
+<span><span class="co">## parent_m1   0.5145</span></span>
+<span><span class="co">## parent_sink 0.4855</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Estimated disappearance times:</span></span>
+<span><span class="co">##           DT50   DT90</span></span>
+<span><span class="co">## parent   7.023  23.33</span></span>
+<span><span class="co">## m1     131.761 437.70</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Data:</span></span>
+<span><span class="co">##  time variable observed predicted   residual</span></span>
+<span><span class="co">##     0   parent    99.46  99.59848 -1.385e-01</span></span>
+<span><span class="co">##     0   parent   102.04  99.59848  2.442e+00</span></span>
+<span><span class="co">##     1   parent    93.50  90.23787  3.262e+00</span></span>
+<span><span class="co">##     1   parent    92.50  90.23787  2.262e+00</span></span>
+<span><span class="co">##     3   parent    63.23  74.07319 -1.084e+01</span></span>
+<span><span class="co">##     3   parent    68.99  74.07319 -5.083e+00</span></span>
+<span><span class="co">##     7   parent    52.32  49.91207  2.408e+00</span></span>
+<span><span class="co">##     7   parent    55.13  49.91207  5.218e+00</span></span>
+<span><span class="co">##    14   parent    27.27  25.01258  2.257e+00</span></span>
+<span><span class="co">##    14   parent    26.64  25.01258  1.627e+00</span></span>
+<span><span class="co">##    21   parent    11.50  12.53462 -1.035e+00</span></span>
+<span><span class="co">##    21   parent    11.64  12.53462 -8.946e-01</span></span>
+<span><span class="co">##    35   parent     2.85   3.14787 -2.979e-01</span></span>
+<span><span class="co">##    35   parent     2.91   3.14787 -2.379e-01</span></span>
+<span><span class="co">##    50   parent     0.69   0.71624 -2.624e-02</span></span>
+<span><span class="co">##    50   parent     0.63   0.71624 -8.624e-02</span></span>
+<span><span class="co">##    75   parent     0.05   0.06074 -1.074e-02</span></span>
+<span><span class="co">##    75   parent     0.06   0.06074 -7.382e-04</span></span>
+<span><span class="co">##     1       m1     4.84   4.80296  3.704e-02</span></span>
+<span><span class="co">##     1       m1     5.64   4.80296  8.370e-01</span></span>
+<span><span class="co">##     3       m1    12.91  13.02400 -1.140e-01</span></span>
+<span><span class="co">##     3       m1    12.96  13.02400 -6.400e-02</span></span>
+<span><span class="co">##     7       m1    22.97  25.04476 -2.075e+00</span></span>
+<span><span class="co">##     7       m1    24.47  25.04476 -5.748e-01</span></span>
+<span><span class="co">##    14       m1    41.69  36.69003  5.000e+00</span></span>
+<span><span class="co">##    14       m1    33.21  36.69003 -3.480e+00</span></span>
+<span><span class="co">##    21       m1    44.37  41.65310  2.717e+00</span></span>
+<span><span class="co">##    21       m1    46.44  41.65310  4.787e+00</span></span>
+<span><span class="co">##    35       m1    41.22  43.31313 -2.093e+00</span></span>
+<span><span class="co">##    35       m1    37.95  43.31313 -5.363e+00</span></span>
+<span><span class="co">##    50       m1    41.19  41.21832 -2.832e-02</span></span>
+<span><span class="co">##    50       m1    40.01  41.21832 -1.208e+00</span></span>
+<span><span class="co">##    75       m1    40.09  36.44704  3.643e+00</span></span>
+<span><span class="co">##    75       m1    33.85  36.44704 -2.597e+00</span></span>
+<span><span class="co">##   100       m1    31.04  31.98162 -9.416e-01</span></span>
+<span><span class="co">##   100       m1    33.13  31.98162  1.148e+00</span></span>
+<span><span class="co">##   120       m1    25.15  28.78984 -3.640e+00</span></span>
+<span><span class="co">##   120       m1    33.31  28.78984  4.520e+00</span></span></code></pre>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
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+
+      <h1>Example evaluation of FOCUS Laboratory Data L1 to L3</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change 18 May 2023
+(rebuilt 2025-02-14)</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/FOCUS_L.rmd" class="external-link"><code>vignettes/FOCUS_L.rmd</code></a></small>
+      <div class="d-none name"><code>FOCUS_L.rmd</code></div>
+    </div>
+
+    
+    
+<div class="section level2">
+<h2 id="laboratory-data-l1">Laboratory Data L1<a class="anchor" aria-label="anchor" href="#laboratory-data-l1"></a>
+</h2>
+<p>The following code defines example dataset L1 from the FOCUS kinetics
+report, p. 284:</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="st"><a href="https://pkgdown.jrwb.de/mkin/">"mkin"</a></span>, quietly <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="va">FOCUS_2006_L1</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html" class="external-link">data.frame</a></span><span class="op">(</span></span>
+<span>  t <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/rep.html" class="external-link">rep</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">1</span>, <span class="fl">2</span>, <span class="fl">3</span>, <span class="fl">5</span>, <span class="fl">7</span>, <span class="fl">14</span>, <span class="fl">21</span>, <span class="fl">30</span><span class="op">)</span>, each <span class="op">=</span> <span class="fl">2</span><span class="op">)</span>,</span>
+<span>  parent <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">88.3</span>, <span class="fl">91.4</span>, <span class="fl">85.6</span>, <span class="fl">84.5</span>, <span class="fl">78.9</span>, <span class="fl">77.6</span>,</span>
+<span>             <span class="fl">72.0</span>, <span class="fl">71.9</span>, <span class="fl">50.3</span>, <span class="fl">59.4</span>, <span class="fl">47.0</span>, <span class="fl">45.1</span>,</span>
+<span>             <span class="fl">27.7</span>, <span class="fl">27.3</span>, <span class="fl">10.0</span>, <span class="fl">10.4</span>, <span class="fl">2.9</span>, <span class="fl">4.0</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="va">FOCUS_2006_L1_mkin</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkin_wide_to_long.html">mkin_wide_to_long</a></span><span class="op">(</span><span class="va">FOCUS_2006_L1</span><span class="op">)</span></span></code></pre></div>
+<p>Here we use the assumptions of simple first order (SFO), the case of
+declining rate constant over time (FOMC) and the case of two different
+phases of the kinetics (DFOP). For a more detailed discussion of the
+models, please see the FOCUS kinetics report.</p>
+<p>Since mkin version 0.9-32 (July 2014), we can use shorthand notation
+like <code>"SFO"</code> for parent only degradation models. The
+following two lines fit the model and produce the summary report of the
+model fit. This covers the numerical analysis given in the FOCUS
+report.</p>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.L1.SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="va">FOCUS_2006_L1_mkin</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">m.L1.SFO</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## mkin version used for fitting:    1.2.10 </span></span>
+<span><span class="co">## R version used for fitting:       4.4.2 </span></span>
+<span><span class="co">## Date of fit:     Fri Feb 14 08:59:11 2025 </span></span>
+<span><span class="co">## Date of summary: Fri Feb 14 08:59:11 2025 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Equations:</span></span>
+<span><span class="co">## d_parent/dt = - k_parent * parent</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Model predictions using solution type analytical </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fitted using 133 model solutions performed in 0.01 s</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model: Constant variance </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model algorithm: OLS </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for parameters to be optimised:</span></span>
+<span><span class="co">##          value   type</span></span>
+<span><span class="co">## parent_0 89.85  state</span></span>
+<span><span class="co">## k_parent  0.10 deparm</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for the transformed parameters actually optimised:</span></span>
+<span><span class="co">##                  value lower upper</span></span>
+<span><span class="co">## parent_0     89.850000  -Inf   Inf</span></span>
+<span><span class="co">## log_k_parent -2.302585  -Inf   Inf</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fixed parameter values:</span></span>
+<span><span class="co">## None</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Results:</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">##        AIC     BIC    logLik</span></span>
+<span><span class="co">##   93.88778 96.5589 -43.94389</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Optimised, transformed parameters with symmetric confidence intervals:</span></span>
+<span><span class="co">##              Estimate Std. Error  Lower  Upper</span></span>
+<span><span class="co">## parent_0       92.470    1.28200 89.740 95.200</span></span>
+<span><span class="co">## log_k_parent   -2.347    0.03763 -2.428 -2.267</span></span>
+<span><span class="co">## sigma           2.780    0.46330  1.792  3.767</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameter correlation:</span></span>
+<span><span class="co">##                parent_0 log_k_parent      sigma</span></span>
+<span><span class="co">## parent_0      1.000e+00    6.186e-01 -1.516e-09</span></span>
+<span><span class="co">## log_k_parent  6.186e-01    1.000e+00 -3.124e-09</span></span>
+<span><span class="co">## sigma        -1.516e-09   -3.124e-09  1.000e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Backtransformed parameters:</span></span>
+<span><span class="co">## Confidence intervals for internally transformed parameters are asymmetric.</span></span>
+<span><span class="co">## t-test (unrealistically) based on the assumption of normal distribution</span></span>
+<span><span class="co">## for estimators of untransformed parameters.</span></span>
+<span><span class="co">##          Estimate t value    Pr(&gt;t)    Lower   Upper</span></span>
+<span><span class="co">## parent_0 92.47000   72.13 8.824e-21 89.74000 95.2000</span></span>
+<span><span class="co">## k_parent  0.09561   26.57 2.487e-14  0.08824  0.1036</span></span>
+<span><span class="co">## sigma     2.78000    6.00 1.216e-05  1.79200  3.7670</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## FOCUS Chi2 error levels in percent:</span></span>
+<span><span class="co">##          err.min n.optim df</span></span>
+<span><span class="co">## All data   3.424       2  7</span></span>
+<span><span class="co">## parent     3.424       2  7</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Estimated disappearance times:</span></span>
+<span><span class="co">##         DT50  DT90</span></span>
+<span><span class="co">## parent 7.249 24.08</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Data:</span></span>
+<span><span class="co">##  time variable observed predicted residual</span></span>
+<span><span class="co">##     0   parent     88.3    92.471  -4.1710</span></span>
+<span><span class="co">##     0   parent     91.4    92.471  -1.0710</span></span>
+<span><span class="co">##     1   parent     85.6    84.039   1.5610</span></span>
+<span><span class="co">##     1   parent     84.5    84.039   0.4610</span></span>
+<span><span class="co">##     2   parent     78.9    76.376   2.5241</span></span>
+<span><span class="co">##     2   parent     77.6    76.376   1.2241</span></span>
+<span><span class="co">##     3   parent     72.0    69.412   2.5884</span></span>
+<span><span class="co">##     3   parent     71.9    69.412   2.4884</span></span>
+<span><span class="co">##     5   parent     50.3    57.330  -7.0301</span></span>
+<span><span class="co">##     5   parent     59.4    57.330   2.0699</span></span>
+<span><span class="co">##     7   parent     47.0    47.352  -0.3515</span></span>
+<span><span class="co">##     7   parent     45.1    47.352  -2.2515</span></span>
+<span><span class="co">##    14   parent     27.7    24.247   3.4528</span></span>
+<span><span class="co">##    14   parent     27.3    24.247   3.0528</span></span>
+<span><span class="co">##    21   parent     10.0    12.416  -2.4163</span></span>
+<span><span class="co">##    21   parent     10.4    12.416  -2.0163</span></span>
+<span><span class="co">##    30   parent      2.9     5.251  -2.3513</span></span>
+<span><span class="co">##    30   parent      4.0     5.251  -1.2513</span></span></code></pre>
+<p>A plot of the fit is obtained with the plot function for mkinfit
+objects.</p>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">m.L1.SFO</span>, show_errmin <span class="op">=</span> <span class="cn">TRUE</span>, main <span class="op">=</span> <span class="st">"FOCUS L1 - SFO"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-4-1.png" width="576"></p>
+<p>The residual plot can be easily obtained by</p>
+<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../reference/mkinresplot.html">mkinresplot</a></span><span class="op">(</span><span class="va">m.L1.SFO</span>, ylab <span class="op">=</span> <span class="st">"Observed"</span>, xlab <span class="op">=</span> <span class="st">"Time"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-5-1.png" width="576"></p>
+<p>For comparison, the FOMC model is fitted as well, and the
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level is checked.</p>
+<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.L1.FOMC</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="st">"FOMC"</span>, <span class="va">FOCUS_2006_L1_mkin</span>, quiet<span class="op">=</span><span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet = TRUE): Optimisation did not converge:</span></span>
+<span><span class="co">## false convergence (8)</span></span></code></pre>
+<div class="sourceCode" id="cb8"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">m.L1.FOMC</span>, show_errmin <span class="op">=</span> <span class="cn">TRUE</span>, main <span class="op">=</span> <span class="st">"FOCUS L1 - FOMC"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-6-1.png" width="576"></p>
+<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">m.L1.FOMC</span>, data <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in sqrt(diag(covar)): NaNs produced</span></span></code></pre>
+<pre><code><span><span class="co">## Warning in cov2cor(ans$covar): diag(V) had non-positive or NA entries; the</span></span>
+<span><span class="co">## non-finite result may be dubious</span></span></code></pre>
+<pre><code><span><span class="co">## mkin version used for fitting:    1.2.10 </span></span>
+<span><span class="co">## R version used for fitting:       4.4.2 </span></span>
+<span><span class="co">## Date of fit:     Fri Feb 14 08:59:11 2025 </span></span>
+<span><span class="co">## Date of summary: Fri Feb 14 08:59:11 2025 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Equations:</span></span>
+<span><span class="co">## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Model predictions using solution type analytical </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fitted using 342 model solutions performed in 0.023 s</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model: Constant variance </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model algorithm: OLS </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for parameters to be optimised:</span></span>
+<span><span class="co">##          value   type</span></span>
+<span><span class="co">## parent_0 89.85  state</span></span>
+<span><span class="co">## alpha     1.00 deparm</span></span>
+<span><span class="co">## beta     10.00 deparm</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for the transformed parameters actually optimised:</span></span>
+<span><span class="co">##               value lower upper</span></span>
+<span><span class="co">## parent_0  89.850000  -Inf   Inf</span></span>
+<span><span class="co">## log_alpha  0.000000  -Inf   Inf</span></span>
+<span><span class="co">## log_beta   2.302585  -Inf   Inf</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fixed parameter values:</span></span>
+<span><span class="co">## None</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Warning(s): </span></span>
+<span><span class="co">## Optimisation did not converge:</span></span>
+<span><span class="co">## false convergence (8)</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Results:</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">##        AIC      BIC    logLik</span></span>
+<span><span class="co">##   95.88782 99.44931 -43.94391</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Optimised, transformed parameters with symmetric confidence intervals:</span></span>
+<span><span class="co">##           Estimate Std. Error  Lower  Upper</span></span>
+<span><span class="co">## parent_0     92.47     1.2820 89.720 95.220</span></span>
+<span><span class="co">## log_alpha    13.20        NaN    NaN    NaN</span></span>
+<span><span class="co">## log_beta     15.54        NaN    NaN    NaN</span></span>
+<span><span class="co">## sigma         2.78     0.4607  1.792  3.768</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameter correlation:</span></span>
+<span><span class="co">##           parent_0 log_alpha log_beta    sigma</span></span>
+<span><span class="co">## parent_0  1.000000       NaN      NaN 0.000603</span></span>
+<span><span class="co">## log_alpha      NaN         1      NaN      NaN</span></span>
+<span><span class="co">## log_beta       NaN       NaN        1      NaN</span></span>
+<span><span class="co">## sigma     0.000603       NaN      NaN 1.000000</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Backtransformed parameters:</span></span>
+<span><span class="co">## Confidence intervals for internally transformed parameters are asymmetric.</span></span>
+<span><span class="co">## t-test (unrealistically) based on the assumption of normal distribution</span></span>
+<span><span class="co">## for estimators of untransformed parameters.</span></span>
+<span><span class="co">##           Estimate t value Pr(&gt;t)  Lower  Upper</span></span>
+<span><span class="co">## parent_0 9.247e+01      NA     NA 89.720 95.220</span></span>
+<span><span class="co">## alpha    5.386e+05      NA     NA     NA     NA</span></span>
+<span><span class="co">## beta     5.633e+06      NA     NA     NA     NA</span></span>
+<span><span class="co">## sigma    2.780e+00      NA     NA  1.792  3.768</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## FOCUS Chi2 error levels in percent:</span></span>
+<span><span class="co">##          err.min n.optim df</span></span>
+<span><span class="co">## All data   3.619       3  6</span></span>
+<span><span class="co">## parent     3.619       3  6</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Estimated disappearance times:</span></span>
+<span><span class="co">##         DT50  DT90 DT50back</span></span>
+<span><span class="co">## parent 7.249 24.08    7.249</span></span></code></pre>
+<p>We get a warning that the default optimisation algorithm
+<code>Port</code> did not converge, which is an indication that the
+model is overparameterised, <em>i.e.</em> contains too many parameters
+that are ill-defined as a consequence.</p>
+<p>And in fact, due to the higher number of parameters, and the lower
+number of degrees of freedom of the fit, the
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level is actually higher for the FOMC model (3.6%) than for the
+SFO model (3.4%). Additionally, the parameters <code>log_alpha</code>
+and <code>log_beta</code> internally fitted in the model have excessive
+confidence intervals, that span more than 25 orders of magnitude (!)
+when backtransformed to the scale of <code>alpha</code> and
+<code>beta</code>. Also, the t-test for significant difference from zero
+does not indicate such a significant difference, with p-values greater
+than 0.1, and finally, the parameter correlation of
+<code>log_alpha</code> and <code>log_beta</code> is 1.000, clearly
+indicating that the model is overparameterised.</p>
+<p>The
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error levels reported in Appendix 3 and Appendix 7 to the FOCUS kinetics
+report are rounded to integer percentages and partly deviate by one
+percentage point from the results calculated by mkin. The reason for
+this is not known. However, mkin gives the same
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error levels as the kinfit package and the calculation routines of the
+kinfit package have been extensively compared to the results obtained by
+the KinGUI software, as documented in the kinfit package vignette.
+KinGUI was the first widely used standard package in this field. Also,
+the calculation of
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error levels was compared with KinGUII, CAKE and DegKin manager in a
+project sponsored by the German Umweltbundesamt <span class="citation">(Ranke 2014)</span>.</p>
+</div>
+<div class="section level2">
+<h2 id="laboratory-data-l2">Laboratory Data L2<a class="anchor" aria-label="anchor" href="#laboratory-data-l2"></a>
+</h2>
+<p>The following code defines example dataset L2 from the FOCUS kinetics
+report, p. 287:</p>
+<div class="sourceCode" id="cb13"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">FOCUS_2006_L2</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html" class="external-link">data.frame</a></span><span class="op">(</span></span>
+<span>  t <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/rep.html" class="external-link">rep</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">1</span>, <span class="fl">3</span>, <span class="fl">7</span>, <span class="fl">14</span>, <span class="fl">28</span><span class="op">)</span>, each <span class="op">=</span> <span class="fl">2</span><span class="op">)</span>,</span>
+<span>  parent <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">96.1</span>, <span class="fl">91.8</span>, <span class="fl">41.4</span>, <span class="fl">38.7</span>,</span>
+<span>             <span class="fl">19.3</span>, <span class="fl">22.3</span>, <span class="fl">4.6</span>, <span class="fl">4.6</span>,</span>
+<span>             <span class="fl">2.6</span>, <span class="fl">1.2</span>, <span class="fl">0.3</span>, <span class="fl">0.6</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="va">FOCUS_2006_L2_mkin</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkin_wide_to_long.html">mkin_wide_to_long</a></span><span class="op">(</span><span class="va">FOCUS_2006_L2</span><span class="op">)</span></span></code></pre></div>
+<div class="section level3">
+<h3 id="sfo-fit-for-l2">SFO fit for L2<a class="anchor" aria-label="anchor" href="#sfo-fit-for-l2"></a>
+</h3>
+<p>Again, the SFO model is fitted and the result is plotted. The
+residual plot can be obtained simply by adding the argument
+<code>show_residuals</code> to the plot command.</p>
+<div class="sourceCode" id="cb14"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.L2.SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="va">FOCUS_2006_L2_mkin</span>, quiet<span class="op">=</span><span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">m.L2.SFO</span>, show_residuals <span class="op">=</span> <span class="cn">TRUE</span>, show_errmin <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>     main <span class="op">=</span> <span class="st">"FOCUS L2 - SFO"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-8-1.png" width="672"></p>
+<p>The
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level of 14% suggests that the model does not fit very well. This
+is also obvious from the plots of the fit, in which we have included the
+residual plot.</p>
+<p>In the FOCUS kinetics report, it is stated that there is no apparent
+systematic error observed from the residual plot up to the measured DT90
+(approximately at day 5), and there is an underestimation beyond that
+point.</p>
+<p>We may add that it is difficult to judge the random nature of the
+residuals just from the three samplings at days 0, 1 and 3. Also, it is
+not clear <em>a priori</em> why a consistent underestimation after the
+approximate DT90 should be irrelevant. However, this can be rationalised
+by the fact that the FOCUS fate models generally only implement SFO
+kinetics.</p>
+</div>
+<div class="section level3">
+<h3 id="fomc-fit-for-l2">FOMC fit for L2<a class="anchor" aria-label="anchor" href="#fomc-fit-for-l2"></a>
+</h3>
+<p>For comparison, the FOMC model is fitted as well, and the
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level is checked.</p>
+<div class="sourceCode" id="cb15"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.L2.FOMC</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="st">"FOMC"</span>, <span class="va">FOCUS_2006_L2_mkin</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">m.L2.FOMC</span>, show_residuals <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>     main <span class="op">=</span> <span class="st">"FOCUS L2 - FOMC"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-9-1.png" width="672"></p>
+<div class="sourceCode" id="cb16"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">m.L2.FOMC</span>, data <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## mkin version used for fitting:    1.2.10 </span></span>
+<span><span class="co">## R version used for fitting:       4.4.2 </span></span>
+<span><span class="co">## Date of fit:     Fri Feb 14 08:59:12 2025 </span></span>
+<span><span class="co">## Date of summary: Fri Feb 14 08:59:12 2025 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Equations:</span></span>
+<span><span class="co">## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Model predictions using solution type analytical </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fitted using 239 model solutions performed in 0.015 s</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model: Constant variance </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model algorithm: OLS </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for parameters to be optimised:</span></span>
+<span><span class="co">##          value   type</span></span>
+<span><span class="co">## parent_0 93.95  state</span></span>
+<span><span class="co">## alpha     1.00 deparm</span></span>
+<span><span class="co">## beta     10.00 deparm</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for the transformed parameters actually optimised:</span></span>
+<span><span class="co">##               value lower upper</span></span>
+<span><span class="co">## parent_0  93.950000  -Inf   Inf</span></span>
+<span><span class="co">## log_alpha  0.000000  -Inf   Inf</span></span>
+<span><span class="co">## log_beta   2.302585  -Inf   Inf</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fixed parameter values:</span></span>
+<span><span class="co">## None</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Results:</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">##        AIC      BIC    logLik</span></span>
+<span><span class="co">##   61.78966 63.72928 -26.89483</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Optimised, transformed parameters with symmetric confidence intervals:</span></span>
+<span><span class="co">##           Estimate Std. Error    Lower   Upper</span></span>
+<span><span class="co">## parent_0   93.7700     1.6130 90.05000 97.4900</span></span>
+<span><span class="co">## log_alpha   0.3180     0.1559 -0.04149  0.6776</span></span>
+<span><span class="co">## log_beta    0.2102     0.2493 -0.36460  0.7850</span></span>
+<span><span class="co">## sigma       2.2760     0.4645  1.20500  3.3470</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameter correlation:</span></span>
+<span><span class="co">##             parent_0  log_alpha   log_beta      sigma</span></span>
+<span><span class="co">## parent_0   1.000e+00 -1.151e-01 -2.085e-01 -7.436e-09</span></span>
+<span><span class="co">## log_alpha -1.151e-01  1.000e+00  9.741e-01 -1.617e-07</span></span>
+<span><span class="co">## log_beta  -2.085e-01  9.741e-01  1.000e+00 -1.386e-07</span></span>
+<span><span class="co">## sigma     -7.436e-09 -1.617e-07 -1.386e-07  1.000e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Backtransformed parameters:</span></span>
+<span><span class="co">## Confidence intervals for internally transformed parameters are asymmetric.</span></span>
+<span><span class="co">## t-test (unrealistically) based on the assumption of normal distribution</span></span>
+<span><span class="co">## for estimators of untransformed parameters.</span></span>
+<span><span class="co">##          Estimate t value    Pr(&gt;t)   Lower  Upper</span></span>
+<span><span class="co">## parent_0   93.770  58.120 4.267e-12 90.0500 97.490</span></span>
+<span><span class="co">## alpha       1.374   6.414 1.030e-04  0.9594  1.969</span></span>
+<span><span class="co">## beta        1.234   4.012 1.942e-03  0.6945  2.192</span></span>
+<span><span class="co">## sigma       2.276   4.899 5.977e-04  1.2050  3.347</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## FOCUS Chi2 error levels in percent:</span></span>
+<span><span class="co">##          err.min n.optim df</span></span>
+<span><span class="co">## All data   6.205       3  3</span></span>
+<span><span class="co">## parent     6.205       3  3</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Estimated disappearance times:</span></span>
+<span><span class="co">##          DT50  DT90 DT50back</span></span>
+<span><span class="co">## parent 0.8092 5.356    1.612</span></span></code></pre>
+<p>The error level at which the
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+test passes is much lower in this case. Therefore, the FOMC model
+provides a better description of the data, as less experimental error
+has to be assumed in order to explain the data.</p>
+</div>
+<div class="section level3">
+<h3 id="dfop-fit-for-l2">DFOP fit for L2<a class="anchor" aria-label="anchor" href="#dfop-fit-for-l2"></a>
+</h3>
+<p>Fitting the four parameter DFOP model further reduces the
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level.</p>
+<div class="sourceCode" id="cb18"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.L2.DFOP</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="st">"DFOP"</span>, <span class="va">FOCUS_2006_L2_mkin</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">m.L2.DFOP</span>, show_residuals <span class="op">=</span> <span class="cn">TRUE</span>, show_errmin <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>     main <span class="op">=</span> <span class="st">"FOCUS L2 - DFOP"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-10-1.png" width="672"></p>
+<div class="sourceCode" id="cb19"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">m.L2.DFOP</span>, data <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## mkin version used for fitting:    1.2.10 </span></span>
+<span><span class="co">## R version used for fitting:       4.4.2 </span></span>
+<span><span class="co">## Date of fit:     Fri Feb 14 08:59:12 2025 </span></span>
+<span><span class="co">## Date of summary: Fri Feb 14 08:59:12 2025 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Equations:</span></span>
+<span><span class="co">## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *</span></span>
+<span><span class="co">##            time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))</span></span>
+<span><span class="co">##            * parent</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Model predictions using solution type analytical </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fitted using 581 model solutions performed in 0.042 s</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model: Constant variance </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model algorithm: OLS </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for parameters to be optimised:</span></span>
+<span><span class="co">##          value   type</span></span>
+<span><span class="co">## parent_0 93.95  state</span></span>
+<span><span class="co">## k1        0.10 deparm</span></span>
+<span><span class="co">## k2        0.01 deparm</span></span>
+<span><span class="co">## g         0.50 deparm</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for the transformed parameters actually optimised:</span></span>
+<span><span class="co">##              value lower upper</span></span>
+<span><span class="co">## parent_0 93.950000  -Inf   Inf</span></span>
+<span><span class="co">## log_k1   -2.302585  -Inf   Inf</span></span>
+<span><span class="co">## log_k2   -4.605170  -Inf   Inf</span></span>
+<span><span class="co">## g_qlogis  0.000000  -Inf   Inf</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fixed parameter values:</span></span>
+<span><span class="co">## None</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Results:</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">##        AIC      BIC    logLik</span></span>
+<span><span class="co">##   52.36695 54.79148 -21.18347</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Optimised, transformed parameters with symmetric confidence intervals:</span></span>
+<span><span class="co">##          Estimate Std. Error      Lower     Upper</span></span>
+<span><span class="co">## parent_0   93.950  9.998e-01    91.5900   96.3100</span></span>
+<span><span class="co">## log_k1      3.113  1.849e+03 -4369.0000 4375.0000</span></span>
+<span><span class="co">## log_k2     -1.088  6.285e-02    -1.2370   -0.9394</span></span>
+<span><span class="co">## g_qlogis   -0.399  9.946e-02    -0.6342   -0.1638</span></span>
+<span><span class="co">## sigma       1.414  2.886e-01     0.7314    2.0960</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameter correlation:</span></span>
+<span><span class="co">##            parent_0     log_k1     log_k2   g_qlogis      sigma</span></span>
+<span><span class="co">## parent_0  1.000e+00  6.763e-07 -8.944e-10  2.665e-01 -1.083e-09</span></span>
+<span><span class="co">## log_k1    6.763e-07  1.000e+00  1.112e-04 -2.187e-04 -1.027e-05</span></span>
+<span><span class="co">## log_k2   -8.944e-10  1.112e-04  1.000e+00 -7.903e-01  9.464e-09</span></span>
+<span><span class="co">## g_qlogis  2.665e-01 -2.187e-04 -7.903e-01  1.000e+00 -1.532e-08</span></span>
+<span><span class="co">## sigma    -1.083e-09 -1.027e-05  9.464e-09 -1.532e-08  1.000e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Backtransformed parameters:</span></span>
+<span><span class="co">## Confidence intervals for internally transformed parameters are asymmetric.</span></span>
+<span><span class="co">## t-test (unrealistically) based on the assumption of normal distribution</span></span>
+<span><span class="co">## for estimators of untransformed parameters.</span></span>
+<span><span class="co">##          Estimate   t value    Pr(&gt;t)   Lower   Upper</span></span>
+<span><span class="co">## parent_0  93.9500 9.397e+01 2.036e-12 91.5900 96.3100</span></span>
+<span><span class="co">## k1        22.4900 5.533e-04 4.998e-01  0.0000     Inf</span></span>
+<span><span class="co">## k2         0.3369 1.591e+01 4.697e-07  0.2904  0.3909</span></span>
+<span><span class="co">## g          0.4016 1.680e+01 3.238e-07  0.3466  0.4591</span></span>
+<span><span class="co">## sigma      1.4140 4.899e+00 8.776e-04  0.7314  2.0960</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## FOCUS Chi2 error levels in percent:</span></span>
+<span><span class="co">##          err.min n.optim df</span></span>
+<span><span class="co">## All data    2.53       4  2</span></span>
+<span><span class="co">## parent      2.53       4  2</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Estimated disappearance times:</span></span>
+<span><span class="co">##          DT50  DT90 DT50back DT50_k1 DT50_k2</span></span>
+<span><span class="co">## parent 0.5335 5.311    1.599 0.03083   2.058</span></span></code></pre>
+<p>Here, the DFOP model is clearly the best-fit model for dataset L2
+based on the chi^2 error level criterion.</p>
+</div>
+</div>
+<div class="section level2">
+<h2 id="laboratory-data-l3">Laboratory Data L3<a class="anchor" aria-label="anchor" href="#laboratory-data-l3"></a>
+</h2>
+<p>The following code defines example dataset L3 from the FOCUS kinetics
+report, p. 290.</p>
+<div class="sourceCode" id="cb21"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">FOCUS_2006_L3</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html" class="external-link">data.frame</a></span><span class="op">(</span></span>
+<span>  t <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">3</span>, <span class="fl">7</span>, <span class="fl">14</span>, <span class="fl">30</span>, <span class="fl">60</span>, <span class="fl">91</span>, <span class="fl">120</span><span class="op">)</span>,</span>
+<span>  parent <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">97.8</span>, <span class="fl">60</span>, <span class="fl">51</span>, <span class="fl">43</span>, <span class="fl">35</span>, <span class="fl">22</span>, <span class="fl">15</span>, <span class="fl">12</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="va">FOCUS_2006_L3_mkin</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkin_wide_to_long.html">mkin_wide_to_long</a></span><span class="op">(</span><span class="va">FOCUS_2006_L3</span><span class="op">)</span></span></code></pre></div>
+<div class="section level3">
+<h3 id="fit-multiple-models">Fit multiple models<a class="anchor" aria-label="anchor" href="#fit-multiple-models"></a>
+</h3>
+<p>As of mkin version 0.9-39 (June 2015), we can fit several models to
+one or more datasets in one call to the function <code>mmkin</code>. The
+datasets have to be passed in a list, in this case a named list holding
+only the L3 dataset prepared above.</p>
+<div class="sourceCode" id="cb22"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="co"># Only use one core here, not to offend the CRAN checks</span></span>
+<span><span class="va">mm.L3</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"FOMC"</span>, <span class="st">"DFOP"</span><span class="op">)</span>, cores <span class="op">=</span> <span class="fl">1</span>,</span>
+<span>               <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="st">"FOCUS L3"</span> <span class="op">=</span> <span class="va">FOCUS_2006_L3_mkin</span><span class="op">)</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">mm.L3</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-12-1.png" width="700"></p>
+<p>The
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level of 21% as well as the plot suggest that the SFO model does
+not fit very well. The FOMC model performs better, with an error level
+at which the
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+test passes of 7%. Fitting the four parameter DFOP model further reduces
+the
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level considerably.</p>
+</div>
+<div class="section level3">
+<h3 id="accessing-mmkin-objects">Accessing mmkin objects<a class="anchor" aria-label="anchor" href="#accessing-mmkin-objects"></a>
+</h3>
+<p>The objects returned by mmkin are arranged like a matrix, with models
+as a row index and datasets as a column index.</p>
+<p>We can extract the summary and plot for <em>e.g.</em> the DFOP fit,
+using square brackets for indexing which will result in the use of the
+summary and plot functions working on mkinfit objects.</p>
+<div class="sourceCode" id="cb23"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">mm.L3</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="fl">1</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## mkin version used for fitting:    1.2.10 </span></span>
+<span><span class="co">## R version used for fitting:       4.4.2 </span></span>
+<span><span class="co">## Date of fit:     Fri Feb 14 08:59:12 2025 </span></span>
+<span><span class="co">## Date of summary: Fri Feb 14 08:59:12 2025 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Equations:</span></span>
+<span><span class="co">## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *</span></span>
+<span><span class="co">##            time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))</span></span>
+<span><span class="co">##            * parent</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Model predictions using solution type analytical </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fitted using 376 model solutions performed in 0.024 s</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model: Constant variance </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model algorithm: OLS </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for parameters to be optimised:</span></span>
+<span><span class="co">##          value   type</span></span>
+<span><span class="co">## parent_0 97.80  state</span></span>
+<span><span class="co">## k1        0.10 deparm</span></span>
+<span><span class="co">## k2        0.01 deparm</span></span>
+<span><span class="co">## g         0.50 deparm</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for the transformed parameters actually optimised:</span></span>
+<span><span class="co">##              value lower upper</span></span>
+<span><span class="co">## parent_0 97.800000  -Inf   Inf</span></span>
+<span><span class="co">## log_k1   -2.302585  -Inf   Inf</span></span>
+<span><span class="co">## log_k2   -4.605170  -Inf   Inf</span></span>
+<span><span class="co">## g_qlogis  0.000000  -Inf   Inf</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fixed parameter values:</span></span>
+<span><span class="co">## None</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Results:</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">##        AIC      BIC    logLik</span></span>
+<span><span class="co">##   32.97732 33.37453 -11.48866</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Optimised, transformed parameters with symmetric confidence intervals:</span></span>
+<span><span class="co">##          Estimate Std. Error   Lower      Upper</span></span>
+<span><span class="co">## parent_0  97.7500    1.01900 94.5000 101.000000</span></span>
+<span><span class="co">## log_k1    -0.6612    0.10050 -0.9812  -0.341300</span></span>
+<span><span class="co">## log_k2    -4.2860    0.04322 -4.4230  -4.148000</span></span>
+<span><span class="co">## g_qlogis  -0.1739    0.05270 -0.3416  -0.006142</span></span>
+<span><span class="co">## sigma      1.0170    0.25430  0.2079   1.827000</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameter correlation:</span></span>
+<span><span class="co">##            parent_0     log_k1     log_k2   g_qlogis      sigma</span></span>
+<span><span class="co">## parent_0  1.000e+00  1.732e-01  2.282e-02  4.009e-01 -9.696e-08</span></span>
+<span><span class="co">## log_k1    1.732e-01  1.000e+00  4.945e-01 -5.809e-01  7.148e-07</span></span>
+<span><span class="co">## log_k2    2.282e-02  4.945e-01  1.000e+00 -6.812e-01  1.022e-06</span></span>
+<span><span class="co">## g_qlogis  4.009e-01 -5.809e-01 -6.812e-01  1.000e+00 -7.930e-07</span></span>
+<span><span class="co">## sigma    -9.696e-08  7.148e-07  1.022e-06 -7.930e-07  1.000e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Backtransformed parameters:</span></span>
+<span><span class="co">## Confidence intervals for internally transformed parameters are asymmetric.</span></span>
+<span><span class="co">## t-test (unrealistically) based on the assumption of normal distribution</span></span>
+<span><span class="co">## for estimators of untransformed parameters.</span></span>
+<span><span class="co">##          Estimate t value    Pr(&gt;t)    Lower     Upper</span></span>
+<span><span class="co">## parent_0 97.75000  95.960 1.248e-06 94.50000 101.00000</span></span>
+<span><span class="co">## k1        0.51620   9.947 1.081e-03  0.37490   0.71090</span></span>
+<span><span class="co">## k2        0.01376  23.140 8.840e-05  0.01199   0.01579</span></span>
+<span><span class="co">## g         0.45660  34.920 2.581e-05  0.41540   0.49850</span></span>
+<span><span class="co">## sigma     1.01700   4.000 1.400e-02  0.20790   1.82700</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## FOCUS Chi2 error levels in percent:</span></span>
+<span><span class="co">##          err.min n.optim df</span></span>
+<span><span class="co">## All data   2.225       4  4</span></span>
+<span><span class="co">## parent     2.225       4  4</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Estimated disappearance times:</span></span>
+<span><span class="co">##         DT50 DT90 DT50back DT50_k1 DT50_k2</span></span>
+<span><span class="co">## parent 7.464  123    37.03   1.343   50.37</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Data:</span></span>
+<span><span class="co">##  time variable observed predicted residual</span></span>
+<span><span class="co">##     0   parent     97.8     97.75  0.05396</span></span>
+<span><span class="co">##     3   parent     60.0     60.45 -0.44933</span></span>
+<span><span class="co">##     7   parent     51.0     49.44  1.56338</span></span>
+<span><span class="co">##    14   parent     43.0     43.84 -0.83632</span></span>
+<span><span class="co">##    30   parent     35.0     35.15 -0.14707</span></span>
+<span><span class="co">##    60   parent     22.0     23.26 -1.25919</span></span>
+<span><span class="co">##    91   parent     15.0     15.18 -0.18181</span></span>
+<span><span class="co">##   120   parent     12.0     10.19  1.81395</span></span></code></pre>
+<div class="sourceCode" id="cb25"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">mm.L3</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="fl">1</span><span class="op">]</span><span class="op">]</span>, show_errmin <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-13-1.png" width="700"></p>
+<p>Here, a look to the model plot, the confidence intervals of the
+parameters and the correlation matrix suggest that the parameter
+estimates are reliable, and the DFOP model can be used as the best-fit
+model based on the
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level criterion for laboratory data L3.</p>
+<p>This is also an example where the standard t-test for the parameter
+<code>g_ilr</code> is misleading, as it tests for a significant
+difference from zero. In this case, zero appears to be the correct value
+for this parameter, and the confidence interval for the backtransformed
+parameter <code>g</code> is quite narrow.</p>
+</div>
+</div>
+<div class="section level2">
+<h2 id="laboratory-data-l4">Laboratory Data L4<a class="anchor" aria-label="anchor" href="#laboratory-data-l4"></a>
+</h2>
+<p>The following code defines example dataset L4 from the FOCUS kinetics
+report, p. 293:</p>
+<div class="sourceCode" id="cb26"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">FOCUS_2006_L4</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html" class="external-link">data.frame</a></span><span class="op">(</span></span>
+<span>  t <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">3</span>, <span class="fl">7</span>, <span class="fl">14</span>, <span class="fl">30</span>, <span class="fl">60</span>, <span class="fl">91</span>, <span class="fl">120</span><span class="op">)</span>,</span>
+<span>  parent <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">96.6</span>, <span class="fl">96.3</span>, <span class="fl">94.3</span>, <span class="fl">88.8</span>, <span class="fl">74.9</span>, <span class="fl">59.9</span>, <span class="fl">53.5</span>, <span class="fl">49.0</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="va">FOCUS_2006_L4_mkin</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkin_wide_to_long.html">mkin_wide_to_long</a></span><span class="op">(</span><span class="va">FOCUS_2006_L4</span><span class="op">)</span></span></code></pre></div>
+<p>Fits of the SFO and FOMC models, plots and summaries are produced
+below:</p>
+<div class="sourceCode" id="cb27"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="co"># Only use one core here, not to offend the CRAN checks</span></span>
+<span><span class="va">mm.L4</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"FOMC"</span><span class="op">)</span>, cores <span class="op">=</span> <span class="fl">1</span>,</span>
+<span>               <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="st">"FOCUS L4"</span> <span class="op">=</span> <span class="va">FOCUS_2006_L4_mkin</span><span class="op">)</span>,</span>
+<span>               quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">mm.L4</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-15-1.png" width="700"></p>
+<p>The
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level of 3.3% as well as the plot suggest that the SFO model fits
+very well. The error level at which the
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+test passes is slightly lower for the FOMC model. However, the
+difference appears negligible.</p>
+<div class="sourceCode" id="cb28"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">mm.L4</span><span class="op">[[</span><span class="st">"SFO"</span>, <span class="fl">1</span><span class="op">]</span><span class="op">]</span>, data <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## mkin version used for fitting:    1.2.10 </span></span>
+<span><span class="co">## R version used for fitting:       4.4.2 </span></span>
+<span><span class="co">## Date of fit:     Fri Feb 14 08:59:13 2025 </span></span>
+<span><span class="co">## Date of summary: Fri Feb 14 08:59:13 2025 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Equations:</span></span>
+<span><span class="co">## d_parent/dt = - k_parent * parent</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Model predictions using solution type analytical </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fitted using 142 model solutions performed in 0.01 s</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model: Constant variance </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model algorithm: OLS </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for parameters to be optimised:</span></span>
+<span><span class="co">##          value   type</span></span>
+<span><span class="co">## parent_0  96.6  state</span></span>
+<span><span class="co">## k_parent   0.1 deparm</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for the transformed parameters actually optimised:</span></span>
+<span><span class="co">##                  value lower upper</span></span>
+<span><span class="co">## parent_0     96.600000  -Inf   Inf</span></span>
+<span><span class="co">## log_k_parent -2.302585  -Inf   Inf</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fixed parameter values:</span></span>
+<span><span class="co">## None</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Results:</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">##        AIC      BIC    logLik</span></span>
+<span><span class="co">##   47.12133 47.35966 -20.56067</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Optimised, transformed parameters with symmetric confidence intervals:</span></span>
+<span><span class="co">##              Estimate Std. Error  Lower   Upper</span></span>
+<span><span class="co">## parent_0       96.440    1.69900 92.070 100.800</span></span>
+<span><span class="co">## log_k_parent   -5.030    0.07059 -5.211  -4.848</span></span>
+<span><span class="co">## sigma           3.162    0.79050  1.130   5.194</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameter correlation:</span></span>
+<span><span class="co">##               parent_0 log_k_parent     sigma</span></span>
+<span><span class="co">## parent_0     1.000e+00    5.938e-01 3.430e-07</span></span>
+<span><span class="co">## log_k_parent 5.938e-01    1.000e+00 5.885e-07</span></span>
+<span><span class="co">## sigma        3.430e-07    5.885e-07 1.000e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Backtransformed parameters:</span></span>
+<span><span class="co">## Confidence intervals for internally transformed parameters are asymmetric.</span></span>
+<span><span class="co">## t-test (unrealistically) based on the assumption of normal distribution</span></span>
+<span><span class="co">## for estimators of untransformed parameters.</span></span>
+<span><span class="co">##           Estimate t value    Pr(&gt;t)     Lower     Upper</span></span>
+<span><span class="co">## parent_0 96.440000   56.77 1.604e-08 92.070000 1.008e+02</span></span>
+<span><span class="co">## k_parent  0.006541   14.17 1.578e-05  0.005455 7.842e-03</span></span>
+<span><span class="co">## sigma     3.162000    4.00 5.162e-03  1.130000 5.194e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## FOCUS Chi2 error levels in percent:</span></span>
+<span><span class="co">##          err.min n.optim df</span></span>
+<span><span class="co">## All data   3.287       2  6</span></span>
+<span><span class="co">## parent     3.287       2  6</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Estimated disappearance times:</span></span>
+<span><span class="co">##        DT50 DT90</span></span>
+<span><span class="co">## parent  106  352</span></span></code></pre>
+<div class="sourceCode" id="cb30"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">mm.L4</span><span class="op">[[</span><span class="st">"FOMC"</span>, <span class="fl">1</span><span class="op">]</span><span class="op">]</span>, data <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## mkin version used for fitting:    1.2.10 </span></span>
+<span><span class="co">## R version used for fitting:       4.4.2 </span></span>
+<span><span class="co">## Date of fit:     Fri Feb 14 08:59:13 2025 </span></span>
+<span><span class="co">## Date of summary: Fri Feb 14 08:59:13 2025 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Equations:</span></span>
+<span><span class="co">## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Model predictions using solution type analytical </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fitted using 224 model solutions performed in 0.013 s</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model: Constant variance </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Error model algorithm: OLS </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for parameters to be optimised:</span></span>
+<span><span class="co">##          value   type</span></span>
+<span><span class="co">## parent_0  96.6  state</span></span>
+<span><span class="co">## alpha      1.0 deparm</span></span>
+<span><span class="co">## beta      10.0 deparm</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Starting values for the transformed parameters actually optimised:</span></span>
+<span><span class="co">##               value lower upper</span></span>
+<span><span class="co">## parent_0  96.600000  -Inf   Inf</span></span>
+<span><span class="co">## log_alpha  0.000000  -Inf   Inf</span></span>
+<span><span class="co">## log_beta   2.302585  -Inf   Inf</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Fixed parameter values:</span></span>
+<span><span class="co">## None</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Results:</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">##        AIC      BIC    logLik</span></span>
+<span><span class="co">##   40.37255 40.69032 -16.18628</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Optimised, transformed parameters with symmetric confidence intervals:</span></span>
+<span><span class="co">##           Estimate Std. Error   Lower    Upper</span></span>
+<span><span class="co">## parent_0   99.1400     1.2670 95.6300 102.7000</span></span>
+<span><span class="co">## log_alpha  -0.3506     0.2616 -1.0770   0.3756</span></span>
+<span><span class="co">## log_beta    4.1740     0.3938  3.0810   5.2670</span></span>
+<span><span class="co">## sigma       1.8300     0.4575  0.5598   3.1000</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameter correlation:</span></span>
+<span><span class="co">##             parent_0  log_alpha   log_beta      sigma</span></span>
+<span><span class="co">## parent_0   1.000e+00 -4.696e-01 -5.543e-01 -2.447e-07</span></span>
+<span><span class="co">## log_alpha -4.696e-01  1.000e+00  9.889e-01  2.198e-08</span></span>
+<span><span class="co">## log_beta  -5.543e-01  9.889e-01  1.000e+00  4.923e-08</span></span>
+<span><span class="co">## sigma     -2.447e-07  2.198e-08  4.923e-08  1.000e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Backtransformed parameters:</span></span>
+<span><span class="co">## Confidence intervals for internally transformed parameters are asymmetric.</span></span>
+<span><span class="co">## t-test (unrealistically) based on the assumption of normal distribution</span></span>
+<span><span class="co">## for estimators of untransformed parameters.</span></span>
+<span><span class="co">##          Estimate t value    Pr(&gt;t)   Lower   Upper</span></span>
+<span><span class="co">## parent_0  99.1400  78.250 7.993e-08 95.6300 102.700</span></span>
+<span><span class="co">## alpha      0.7042   3.823 9.365e-03  0.3407   1.456</span></span>
+<span><span class="co">## beta      64.9800   2.540 3.201e-02 21.7800 193.900</span></span>
+<span><span class="co">## sigma      1.8300   4.000 8.065e-03  0.5598   3.100</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## FOCUS Chi2 error levels in percent:</span></span>
+<span><span class="co">##          err.min n.optim df</span></span>
+<span><span class="co">## All data   2.029       3  5</span></span>
+<span><span class="co">## parent     2.029       3  5</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Estimated disappearance times:</span></span>
+<span><span class="co">##         DT50 DT90 DT50back</span></span>
+<span><span class="co">## parent 108.9 1644    494.9</span></span></code></pre>
+</div>
+<div class="section level2">
+<h2 class="unnumbered" id="references">References<a class="anchor" aria-label="anchor" href="#references"></a>
+</h2>
+<div id="refs" class="references csl-bib-body hanging-indent">
+<div id="ref-ranke2014" class="csl-entry">
+Ranke, Johannes. 2014. <span>“<span class="nocase">Prüfung und
+Validierung von Modellierungssoftware als Alternative zu ModelMaker
+4.0</span>.”</span> Umweltbundesamt Projektnummer 27452.
+</div>
+</div>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
+</div>
+
+    </footer>
+</div>
+
+
+
+
+
+  </body>
+</html>
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@@ -0,0 +1,114 @@
+<!DOCTYPE html>
+<!-- Generated by pkgdown: do not edit by hand --><html lang="en"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><meta charset="utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge"><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"><title>Articles • mkin</title><script src="../deps/jquery-3.6.0/jquery-3.6.0.min.js"></script><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"><link href="../deps/bootstrap-5.3.1/bootstrap.min.css" rel="stylesheet"><script src="../deps/bootstrap-5.3.1/bootstrap.bundle.min.js"></script><link href="../deps/font-awesome-6.5.2/css/all.min.css" rel="stylesheet"><link href="../deps/font-awesome-6.5.2/css/v4-shims.min.css" rel="stylesheet"><script src="../deps/headroom-0.11.0/headroom.min.js"></script><script src="../deps/headroom-0.11.0/jQuery.headroom.min.js"></script><script src="../deps/bootstrap-toc-1.0.1/bootstrap-toc.min.js"></script><script src="../deps/clipboard.js-2.0.11/clipboard.min.js"></script><script src="../deps/search-1.0.0/autocomplete.jquery.min.js"></script><script src="../deps/search-1.0.0/fuse.min.js"></script><script src="../deps/search-1.0.0/mark.min.js"></script><!-- pkgdown --><script src="../pkgdown.js"></script><meta property="og:title" content="Articles"><meta name="robots" content="noindex"></head><body>
+    <a href="#main" class="visually-hidden-focusable">Skip to contents</a>
+
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+    <nav class="navbar navbar-expand-lg fixed-top bg-light" data-bs-theme="default" aria-label="Site navigation"><div class="container">
+
+    <a class="navbar-brand me-2" href="../index.html">mkin</a>
+
+    <small class="nav-text text-info me-auto" data-bs-toggle="tooltip" data-bs-placement="bottom" title="In-development version">1.2.10</small>
+
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+    <button class="navbar-toggler" type="button" data-bs-toggle="collapse" data-bs-target="#navbar" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
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+      <ul class="navbar-nav me-auto"><li class="nav-item"><a class="nav-link" href="../reference/index.html">Reference</a></li>
+<li class="nav-item dropdown">
+  <button class="nav-link dropdown-toggle" type="button" id="dropdown-articles" data-bs-toggle="dropdown" aria-expanded="false" aria-haspopup="true">Articles</button>
+  <ul class="dropdown-menu" aria-labelledby="dropdown-articles"><li><a class="dropdown-item" href="../articles/mkin.html">Introduction to mkin</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with (generalised) nonlinear least squares</h6></li>
+    <li><a class="dropdown-item" href="../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a></li>
+    <li><a class="dropdown-item" href="../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/FOCUS_Z.html">Example evaluation of FOCUS Example Dataset Z</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with hierarchical models (nonlinear mixed-effects models)</h6></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2022_dmta_pathway.html">Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2023_mesotrione_parent.html">Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</a></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2022_cyan_pathway.html">Testing hierarchical pathway kinetics with residue data on cyantraniliprole</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/dimethenamid_2018.html">Comparison of saemix and nlme evaluations of dimethenamid data from 2018</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/multistart.html">Short demo of the multistart method</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Performance</h6></li>
+    <li><a class="dropdown-item" href="../articles/web_only/compiled_models.html">Performance benefit by using compiled model definitions in mkin</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/benchmarks.html">Benchmark timings for mkin</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/saem_benchmarks.html">Benchmark timings for saem.mmkin</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Miscellaneous</h6></li>
+    <li><a class="dropdown-item" href="../articles/twa.html">Calculation of time weighted average concentrations with mkin</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a></li>
+  </ul></li>
+<li class="nav-item"><a class="nav-link" href="../coverage/coverage.html">Test coverage</a></li>
+<li class="nav-item"><a class="nav-link" href="../news/index.html">News</a></li>
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+      <h1>Articles</h1>
+    </div>
+
+    <div class="section ">
+      <h3>All vignettes</h3>
+      <div class="section-desc"></div>
+
+      <dl><dt><a href="prebuilt/2022_cyan_pathway.html">Testing hierarchical pathway kinetics with residue data on cyantraniliprole</a></dt>
+        <dd>
+        </dd><dt><a href="prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a></dt>
+        <dd>
+        </dd><dt><a href="prebuilt/2022_dmta_pathway.html">Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P</a></dt>
+        <dd>
+        </dd><dt><a href="prebuilt/2023_mesotrione_parent.html">Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</a></dt>
+        <dd>
+        </dd><dt><a href="web_only/benchmarks.html">Benchmark timings for mkin</a></dt>
+        <dd>
+        </dd><dt><a href="web_only/compiled_models.html">Performance benefit by using compiled model definitions in mkin</a></dt>
+        <dd>
+        </dd><dt><a href="web_only/dimethenamid_2018.html">Example evaluations of the dimethenamid data from 2018</a></dt>
+        <dd>
+        </dd><dt><a href="FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a></dt>
+        <dd>
+        </dd><dt><a href="FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a></dt>
+        <dd>
+        </dd><dt><a href="web_only/FOCUS_Z.html">Example evaluation of FOCUS dataset Z</a></dt>
+        <dd>
+        </dd><dt><a href="mkin.html">Short introduction to mkin</a></dt>
+        <dd>
+        </dd><dt><a href="web_only/multistart.html">Short demo of the multistart method</a></dt>
+        <dd>
+        </dd><dt><a href="web_only/NAFTA_examples.html">Evaluation of example datasets from Attachment 1 to the US EPA SOP for the NAFTA guidance</a></dt>
+        <dd>
+        </dd><dt><a href="web_only/saem_benchmarks.html">Benchmark timings for saem.mmkin</a></dt>
+        <dd>
+        </dd><dt><a href="twa.html">Calculation of time weighted average concentrations with mkin</a></dt>
+        <dd>
+      </dd></dl></div>
+  </main></div>
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diff --git a/docs/dev/articles/mkin.html b/docs/dev/articles/mkin.html
new file mode 100644
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+++ b/docs/dev/articles/mkin.html
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+<li><a class="dropdown-item" href="../articles/mkin.html">Introduction to mkin</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with (generalised) nonlinear least squares</h6></li>
+    <li><a class="dropdown-item" href="../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a></li>
+    <li><a class="dropdown-item" href="../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/FOCUS_Z.html">Example evaluation of FOCUS Example Dataset Z</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with hierarchical models (nonlinear mixed-effects models)</h6></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2022_dmta_pathway.html">Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2023_mesotrione_parent.html">Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</a></li>
+    <li><a class="dropdown-item" href="../articles/prebuilt/2022_cyan_pathway.html">Testing hierarchical pathway kinetics with residue data on cyantraniliprole</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/dimethenamid_2018.html">Comparison of saemix and nlme evaluations of dimethenamid data from 2018</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/multistart.html">Short demo of the multistart method</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Performance</h6></li>
+    <li><a class="dropdown-item" href="../articles/web_only/compiled_models.html">Performance benefit by using compiled model definitions in mkin</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/benchmarks.html">Benchmark timings for mkin</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/saem_benchmarks.html">Benchmark timings for saem.mmkin</a></li>
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+    <li><h6 class="dropdown-header" data-toc-skip>Miscellaneous</h6></li>
+    <li><a class="dropdown-item" href="../articles/twa.html">Calculation of time weighted average concentrations with mkin</a></li>
+    <li><a class="dropdown-item" href="../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a></li>
+  </ul>
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+  <main id="main" class="col-md-9"><div class="page-header">
+
+      <h1>Short introduction to mkin</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change 18 May 2023
+(rebuilt 2025-02-14)</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/mkin.rmd" class="external-link"><code>vignettes/mkin.rmd</code></a></small>
+      <div class="d-none name"><code>mkin.rmd</code></div>
+    </div>
+
+    
+    
+<p><a href="https://www.jrwb.de" class="external-link">Wissenschaftlicher Berater, Kronacher
+Str. 12, 79639 Grenzach-Wyhlen, Germany</a><br> Privatdozent at the
+University of Freiburg</p>
+<div class="section level2">
+<h2 id="abstract">Abstract<a class="anchor" aria-label="anchor" href="#abstract"></a>
+</h2>
+<p>In the regulatory evaluation of chemical substances like plant
+protection products (pesticides), biocides and other chemicals,
+degradation data play an important role. For the evaluation of pesticide
+degradation experiments, detailed guidance has been developed, based on
+nonlinear optimisation. The <code>R</code> add-on package
+<code>mkin</code> implements fitting some of the models recommended in
+this guidance from within R and calculates some statistical measures for
+data series within one or more compartments, for parent and
+metabolites.</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="st"><a href="https://pkgdown.jrwb.de/mkin/">"mkin"</a></span>, quietly <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="co"># Define the kinetic model</span></span>
+<span><span class="va">m_SFO_SFO_SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fu"><a href="../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M1"</span><span class="op">)</span>,</span>
+<span>                         M1 <span class="op">=</span> <span class="fu"><a href="../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M2"</span><span class="op">)</span>,</span>
+<span>                         M2 <span class="op">=</span> <span class="fu"><a href="../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>                         use_of_ff <span class="op">=</span> <span class="st">"max"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span></span>
+<span></span>
+<span><span class="co"># Produce model predictions using some arbitrary parameters</span></span>
+<span><span class="va">sampling_times</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">1</span>, <span class="fl">3</span>, <span class="fl">7</span>, <span class="fl">14</span>, <span class="fl">28</span>, <span class="fl">60</span>, <span class="fl">90</span>, <span class="fl">120</span><span class="op">)</span></span>
+<span><span class="va">d_SFO_SFO_SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkinpredict.html">mkinpredict</a></span><span class="op">(</span><span class="va">m_SFO_SFO_SFO</span>,</span>
+<span>  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>k_parent <span class="op">=</span> <span class="fl">0.03</span>,</span>
+<span>    f_parent_to_M1 <span class="op">=</span> <span class="fl">0.5</span>, k_M1 <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/Log.html" class="external-link">log</a></span><span class="op">(</span><span class="fl">2</span><span class="op">)</span><span class="op">/</span><span class="fl">100</span>,</span>
+<span>    f_M1_to_M2 <span class="op">=</span> <span class="fl">0.9</span>, k_M2 <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/Log.html" class="external-link">log</a></span><span class="op">(</span><span class="fl">2</span><span class="op">)</span><span class="op">/</span><span class="fl">50</span><span class="op">)</span>,</span>
+<span>  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fl">100</span>, M1 <span class="op">=</span> <span class="fl">0</span>, M2 <span class="op">=</span> <span class="fl">0</span><span class="op">)</span>,</span>
+<span>  <span class="va">sampling_times</span><span class="op">)</span></span>
+<span></span>
+<span><span class="co"># Generate a dataset by adding normally distributed errors with</span></span>
+<span><span class="co"># standard deviation 3, for two replicates at each sampling time</span></span>
+<span><span class="va">d_SFO_SFO_SFO_err</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/add_err.html">add_err</a></span><span class="op">(</span><span class="va">d_SFO_SFO_SFO</span>, reps <span class="op">=</span> <span class="fl">2</span>,</span>
+<span>                             sdfunc <span class="op">=</span> <span class="kw">function</span><span class="op">(</span><span class="va">x</span><span class="op">)</span> <span class="fl">3</span>,</span>
+<span>                             n <span class="op">=</span> <span class="fl">1</span>, seed <span class="op">=</span> <span class="fl">123456789</span> <span class="op">)</span></span>
+<span></span>
+<span><span class="co"># Fit the model to the dataset</span></span>
+<span><span class="va">f_SFO_SFO_SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">m_SFO_SFO_SFO</span>, <span class="va">d_SFO_SFO_SFO_err</span><span class="op">[[</span><span class="fl">1</span><span class="op">]</span><span class="op">]</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span></span>
+<span><span class="co"># Plot the results separately for parent and metabolites</span></span>
+<span><span class="fu"><a href="../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">f_SFO_SFO_SFO</span>, lpos <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"topright"</span>, <span class="st">"bottomright"</span>, <span class="st">"bottomright"</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<p><img src="mkin_files/figure-html/unnamed-chunk-2-1.png" width="768"></p>
+</div>
+<div class="section level2">
+<h2 id="background">Background<a class="anchor" aria-label="anchor" href="#background"></a>
+</h2>
+<p>The <code>mkin</code> package <span class="citation">(J. Ranke
+2021)</span> implements the approach to degradation kinetics recommended
+in the kinetics report provided by the FOrum for Co-ordination of
+pesticide fate models and their USe <span class="citation">(FOCUS Work
+Group on Degradation Kinetics 2006, 2014)</span>. It covers data series
+describing the decline of one compound, data series with transformation
+products (commonly termed metabolites) and data series for more than one
+compartment. It is possible to include back reactions. Therefore,
+equilibrium reactions and equilibrium partitioning can be specified,
+although this often leads to an overparameterisation of the model.</p>
+<p>When the first <code>mkin</code> code was published in 2010, the most
+commonly used tools for fitting more complex kinetic degradation models
+to experimental data were KinGUI <span class="citation">(Schäfer et al.
+2007)</span>, a MATLAB based tool with a graphical user interface that
+was specifically tailored to the task and included some output as
+proposed by the FOCUS Kinetics Workgroup, and ModelMaker, a general
+purpose compartment based tool providing infrastructure for fitting
+dynamic simulation models based on differential equations to data.</p>
+<p>The ‘mkin’ code was first uploaded to the BerliOS development
+platform. When this was taken down, the version control history was
+imported into the R-Forge site (see <em>e.g.</em> <a href="https://cgit.jrwb.de/mkin/commit/?id=30cbb4092f6d2d3beff5800603374a0d009ad770" class="external-link">the
+initial commit on 11 May 2010</a>), where the code is still being
+updated.</p>
+<p>At that time, the R package <code>FME</code> (Flexible Modelling
+Environment) <span class="citation">(Soetaert and Petzoldt 2010)</span>
+was already available, and provided a good basis for developing a
+package specifically tailored to the task. The remaining challenge was
+to make it as easy as possible for the users (including the author of
+this vignette) to specify the system of differential equations and to
+include the output requested by the FOCUS guidance, such as the
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level as defined in this guidance.</p>
+<p>Also, <code>mkin</code> introduced using analytical solutions for
+parent only kinetics for improved optimization speed. Later, Eigenvalue
+based solutions were introduced to <code>mkin</code> for the case of
+linear differential equations (<em>i.e.</em> where the FOMC or DFOP
+models were not used for the parent compound), greatly improving the
+optimization speed for these cases. This, has become somehow obsolete,
+as the use of compiled code described below gives even faster execution
+times.</p>
+<p>The possibility to specify back-reactions and a biphasic model
+(SFORB) for metabolites were present in <code>mkin</code> from the very
+beginning.</p>
+<div class="section level3">
+<h3 id="derived-software-tools">Derived software tools<a class="anchor" aria-label="anchor" href="#derived-software-tools"></a>
+</h3>
+<p>Soon after the publication of <code>mkin</code>, two derived tools
+were published, namely KinGUII (developed at Bayer Crop Science) and
+CAKE (commissioned to Tessella by Syngenta), which added a graphical
+user interface (GUI), and added fitting by iteratively reweighted least
+squares (IRLS) and characterisation of likely parameter distributions by
+Markov Chain Monte Carlo (MCMC) sampling.</p>
+<p>CAKE focuses on a smooth use experience, sacrificing some flexibility
+in the model definition, originally allowing only two primary
+metabolites in parallel. The current version 3.4 of CAKE released in May
+2020 uses a scheme for up to six metabolites in a flexible arrangement
+and supports biphasic modelling of metabolites, but does not support
+back-reactions (non-instantaneous equilibria).</p>
+<p>KinGUI offers an even more flexible widget for specifying complex
+kinetic models. Back-reactions (non-instantaneous equilibria) were
+supported early on, but until 2014, only simple first-order models could
+be specified for transformation products. Starting with KinGUII version
+2.1, biphasic modelling of metabolites was also available in
+KinGUII.</p>
+<p>A further graphical user interface (GUI) that has recently been
+brought to a decent degree of maturity is the browser based GUI named
+<code>gmkin</code>. Please see its <a href="https://pkgdown.jrwb.de/gmkin/" class="external-link">documentation page</a> and <a href="https://pkgdown.jrwb.de/gmkin/articles/gmkin_manual.html" class="external-link">manual</a>
+for further information.</p>
+<p>A comparison of scope, usability and numerical results obtained with
+these tools has been recently been published by <span class="citation">Johannes Ranke, Wöltjen, and Meinecke
+(2018)</span>.</p>
+</div>
+</div>
+<div class="section level2">
+<h2 id="unique-features">Unique features<a class="anchor" aria-label="anchor" href="#unique-features"></a>
+</h2>
+<p>Currently, the main unique features available in <code>mkin</code>
+are</p>
+<ul>
+<li>the <a href="https://pkgdown.jrwb.de/mkin/articles/web_only/compiled_models.html">speed
+increase</a> by using compiled code when a compiler is present,</li>
+<li>parallel model fitting on multicore machines using the <a href="https://pkgdown.jrwb.de/mkin/reference/mmkin.html"><code>mmkin</code>
+function</a>,</li>
+<li>the estimation of parameter confidence intervals based on
+transformed parameters (see below) and</li>
+<li>the possibility to use the <a href="https://pkgdown.jrwb.de/mkin/reference/sigma_twocomp.html">two-component
+error model</a>
+</li>
+</ul>
+<p>The iteratively reweighted least squares fitting of different
+variances for each variable as introduced by <span class="citation">Gao
+et al. (2011)</span> has been available in mkin since <a href="https://pkgdown.jrwb.de/mkin/news/index.html#mkin-0-9-22-2013-10-26">version
+0.9-22</a>. With <a href="https://pkgdown.jrwb.de/mkin/news/index.html#mkin-0-9-49-5-2019-07-04">release
+0.9.49.5</a>, the IRLS algorithm has been complemented by direct or
+step-wise maximisation of the likelihood function, which makes it
+possible not only to fit the variance by variable error model but also a
+<a href="https://pkgdown.jrwb.de/mkin/reference/sigma_twocomp.html">two-component
+error model</a> inspired by error models developed in analytical
+chemistry <span class="citation">(Johannes Ranke and Meinecke
+2019)</span>.</p>
+</div>
+<div class="section level2">
+<h2 id="internal-parameter-transformations">Internal parameter transformations<a class="anchor" aria-label="anchor" href="#internal-parameter-transformations"></a>
+</h2>
+<p>For rate constants, the log transformation is used, as proposed by
+Bates and Watts <span class="citation">(1988, 77, 149)</span>.
+Approximate intervals are constructed for the transformed rate constants
+<span class="citation">(compare Bates and Watts 1988, 135)</span>,
+<em>i.e.</em> for their logarithms. Confidence intervals for the rate
+constants are then obtained using the appropriate backtransformation
+using the exponential function.</p>
+<p>In the first version of <code>mkin</code> allowing for specifying
+models using formation fractions, a home-made reparameterisation was
+used in order to ensure that the sum of formation fractions would not
+exceed unity.</p>
+<p>This method is still used in the current version of KinGUII (v2.1
+from April 2014), with a modification that allows for fixing the pathway
+to sink to zero. CAKE uses penalties in the objective function in order
+to enforce this constraint.</p>
+<p>In 2012, an alternative reparameterisation of the formation fractions
+was proposed together with René Lehmann <span class="citation">(J. Ranke
+and Lehmann 2012)</span>, based on isometric logratio transformation
+(ILR). The aim was to improve the validity of the linear approximation
+of the objective function during the parameter estimation procedure as
+well as in the subsequent calculation of parameter confidence intervals.
+In the current version of mkin, a logit transformation is used for
+parameters that are bound between 0 and 1, such as the g parameter of
+the DFOP model.</p>
+<div class="section level3">
+<h3 id="confidence-intervals-based-on-transformed-parameters">Confidence intervals based on transformed parameters<a class="anchor" aria-label="anchor" href="#confidence-intervals-based-on-transformed-parameters"></a>
+</h3>
+<p>In the first attempt at providing improved parameter confidence
+intervals introduced to <code>mkin</code> in 2013, confidence intervals
+obtained from FME on the transformed parameters were simply all
+backtransformed one by one to yield asymmetric confidence intervals for
+the backtransformed parameters.</p>
+<p>However, while there is a 1:1 relation between the rate constants in
+the model and the transformed parameters fitted in the model, the
+parameters obtained by the isometric logratio transformation are
+calculated from the set of formation fractions that quantify the paths
+to each of the compounds formed from a specific parent compound, and no
+such 1:1 relation exists.</p>
+<p>Therefore, parameter confidence intervals for formation fractions
+obtained with this method only appear valid for the case of a single
+transformation product, where currently the logit transformation is used
+for the formation fraction.</p>
+<p>The confidence intervals obtained by backtransformation for the cases
+where a 1:1 relation between transformed and original parameter exist
+are considered by the author of this vignette to be more accurate than
+those obtained using a re-estimation of the Hessian matrix after
+backtransformation, as implemented in the FME package.</p>
+</div>
+<div class="section level3">
+<h3 id="parameter-t-test-based-on-untransformed-parameters">Parameter t-test based on untransformed parameters<a class="anchor" aria-label="anchor" href="#parameter-t-test-based-on-untransformed-parameters"></a>
+</h3>
+<p>The standard output of many nonlinear regression software packages
+includes the results from a test for significant difference from zero
+for all parameters. Such a test is also recommended to check the
+validity of rate constants in the FOCUS guidance <span class="citation">(FOCUS Work Group on Degradation Kinetics 2014,
+96ff)</span>.</p>
+<p>It has been argued that the precondition for this test, <em>i.e.</em>
+normal distribution of the estimator for the parameters, is not
+fulfilled in the case of nonlinear regression <span class="citation">(J.
+Ranke and Lehmann 2015)</span>. However, this test is commonly used by
+industry, consultants and national authorities in order to decide on the
+reliability of parameter estimates, based on the FOCUS guidance
+mentioned above. Therefore, the results of this one-sided t-test are
+included in the summary output from <code>mkin</code>.</p>
+<p>As it is not reasonable to test for significant difference of the
+transformed parameters (<em>e.g.</em>
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>g</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>k</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">log(k)</annotation></semantics></math>)
+from zero, the t-test is calculated based on the model definition before
+parameter transformation, <em>i.e.</em> in a similar way as in packages
+that do not apply such an internal parameter transformation. A note is
+included in the <code>mkin</code> output, pointing to the fact that the
+t-test is based on the unjustified assumption of normal distribution of
+the parameter estimators.</p>
+</div>
+</div>
+<div class="section level2">
+<h2 id="references">References<a class="anchor" aria-label="anchor" href="#references"></a>
+</h2>
+<!-- vim: set foldmethod=syntax: -->
+<div id="refs" class="references csl-bib-body hanging-indent">
+<div id="ref-bates1988" class="csl-entry">
+Bates, D., and D. Watts. 1988. <em>Nonlinear Regression and Its
+Applications</em>. Wiley-Interscience.
+</div>
+<div id="ref-FOCUS2006" class="csl-entry">
+FOCUS Work Group on Degradation Kinetics. 2006. <em>Guidance Document on
+Estimating Persistence and Degradation Kinetics from Environmental Fate
+Studies on Pesticides in EU Registration. Report of the FOCUS Work Group
+on Degradation Kinetics</em>. <a href="http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics" class="external-link">http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics</a>.
+</div>
+<div id="ref-FOCUSkinetics2014" class="csl-entry">
+———. 2014. <em>Generic Guidance for Estimating Persistence and
+Degradation Kinetics from Environmental Fate Studies on Pesticides in EU
+Registration</em>. 1.1 ed. <a href="http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics" class="external-link">http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics</a>.
+</div>
+<div id="ref-gao11" class="csl-entry">
+Gao, Z., J. W. Green, J. Vanderborght, and W. Schmitt. 2011.
+<span>“Improving Uncertainty Analysis in Kinetic Evaluations Using
+Iteratively Reweighted Least Squares.”</span> Journal. <em>Environmental
+Science and Technology</em> 45: 4429–37.
+</div>
+<div id="ref-pkg:mkin" class="csl-entry">
+Ranke, J. 2021. <em>‘<span class="nocase">mkin</span>‘:
+<span>K</span>inetic Evaluation of Chemical Degradation Data</em>. <a href="https://CRAN.R-project.org/package=mkin" class="external-link">https://CRAN.R-project.org/package=mkin</a>.
+</div>
+<div id="ref-ranke2012" class="csl-entry">
+Ranke, J., and R. Lehmann. 2012. <span>“Parameter Reliability in Kinetic
+Evaluation of Environmental Metabolism Data - Assessment and the
+Influence of Model Specification.”</span> In <em>SETAC World 20-24
+May</em>. Berlin. <a href="https://jrwb.de/posters/Poster_SETAC_2012_Kinetic_parameter_uncertainty_model_parameterization_Lehmann_Ranke.pdf" class="external-link">https://jrwb.de/posters/Poster_SETAC_2012_Kinetic_parameter_uncertainty_model_parameterization_Lehmann_Ranke.pdf</a>.
+</div>
+<div id="ref-ranke2015" class="csl-entry">
+———. 2015. <span>“To t-Test or Not to t-Test, That Is the
+Question.”</span> In <em>XV Symposium on Pesticide Chemistry 2-4
+September 2015</em>. Piacenza. <a href="https://jrwb.de/posters/piacenza_2015.pdf" class="external-link">https://jrwb.de/posters/piacenza_2015.pdf</a>.
+</div>
+<div id="ref-ranke2019" class="csl-entry">
+Ranke, Johannes, and Stefan Meinecke. 2019. <span>“Error Models for the
+Kinetic Evaluation of Chemical Degradation Data.”</span>
+<em>Environments</em> 6 (12). <a href="https://doi.org/10.3390/environments6120124" class="external-link">https://doi.org/10.3390/environments6120124</a>.
+</div>
+<div id="ref-ranke2018" class="csl-entry">
+Ranke, Johannes, Janina Wöltjen, and Stefan Meinecke. 2018.
+<span>“Comparison of Software Tools for Kinetic Evaluation of Chemical
+Degradation Data.”</span> <em>Environmental Sciences Europe</em> 30 (1):
+17. <a href="https://doi.org/10.1186/s12302-018-0145-1" class="external-link">https://doi.org/10.1186/s12302-018-0145-1</a>.
+</div>
+<div id="ref-schaefer2007" class="csl-entry">
+Schäfer, D., B. Mikolasch, P. Rainbird, and B. Harvey. 2007.
+<span>“<span>KinGUI</span>: A New Kinetic Software Tool for Evaluations
+According to <span>FOCUS</span> Degradation Kinetics.”</span> In
+<em>Proceedings of the XIII Symposium Pesticide Chemistry</em>, edited
+by Del Re A. A. M., Capri E., Fragoulis G., and Trevisan M., 916–23.
+Piacenza.
+</div>
+<div id="ref-soetaert2010" class="csl-entry">
+Soetaert, Karline, and Thomas Petzoldt. 2010. <span>“Inverse Modelling,
+Sensitivity and Monte Carlo Analysis in <span>R</span> Using Package
+<span>FME</span>.”</span> <em>Journal of Statistical Software</em> 33
+(3): 1–28. <a href="https://doi.org/10.18637/jss.v033.i03" class="external-link">https://doi.org/10.18637/jss.v033.i03</a>.
+</div>
+</div>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
+</div>
+
+    </footer>
+</div>
+
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+
+
+
+  </body>
+</html>
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+
+      <h1>Testing hierarchical pathway kinetics with residue data on cyantraniliprole</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change on 13 February
+2023, last compiled on 14 Februar 2025</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/prebuilt/2022_cyan_pathway.rmd" class="external-link"><code>vignettes/prebuilt/2022_cyan_pathway.rmd</code></a></small>
+      <div class="d-none name"><code>2022_cyan_pathway.rmd</code></div>
+    </div>
+
+    
+    
+<div class="section level2">
+<h2 id="introduction">Introduction<a class="anchor" aria-label="anchor" href="#introduction"></a>
+</h2>
+<p>The purpose of this document is to test demonstrate how nonlinear
+hierarchical models (NLHM) based on the parent degradation models SFO,
+FOMC, DFOP and HS, with serial formation of two or more metabolites can
+be fitted with the mkin package.</p>
+<p>It was assembled in the course of work package 1.2 of Project Number
+173340 (Application of nonlinear hierarchical models to the kinetic
+evaluation of chemical degradation data) of the German Environment
+Agency carried out in 2022 and 2023.</p>
+<p>The mkin package is used in version 1.2.10 which is currently under
+development. The newly introduced functionality that is used here is a
+simplification of excluding random effects for a set of fits based on a
+related set of fits with a reduced model, and the documentation of the
+starting parameters of the fit, so that all starting parameters of
+<code>saem</code> fits are now listed in the summary. The
+<code>saemix</code> package is used as a backend for fitting the NLHM,
+but is also loaded to make the convergence plot function available.</p>
+<p>This document is processed with the <code>knitr</code> package, which
+also provides the <code>kable</code> function that is used to improve
+the display of tabular data in R markdown documents. For parallel
+processing, the <code>parallel</code> package is used.</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://pkgdown.jrwb.de/mkin/">mkin</a></span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://yihui.org/knitr/" class="external-link">knitr</a></span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">saemix</span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">parallel</span><span class="op">)</span></span>
+<span><span class="va">n_cores</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/detectCores.html" class="external-link">detectCores</a></span><span class="op">(</span><span class="op">)</span></span>
+<span></span>
+<span><span class="co"># We need to start a new cluster after defining a compiled model that is</span></span>
+<span><span class="co"># saved as a DLL to the user directory, therefore we define a function</span></span>
+<span><span class="co"># This is used again after defining the pathway model</span></span>
+<span><span class="va">start_cluster</span> <span class="op">&lt;-</span> <span class="kw">function</span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="kw">if</span> <span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/Sys.info.html" class="external-link">Sys.info</a></span><span class="op">(</span><span class="op">)</span><span class="op">[</span><span class="st">"sysname"</span><span class="op">]</span> <span class="op">==</span> <span class="st">"Windows"</span><span class="op">)</span> <span class="op">{</span></span>
+<span>    <span class="va">ret</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">makePSOCKcluster</a></span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
+<span>  <span class="op">}</span> <span class="kw">else</span> <span class="op">{</span></span>
+<span>    <span class="va">ret</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">makeForkCluster</a></span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
+<span>  <span class="op">}</span></span>
+<span>  <span class="kw"><a href="https://rdrr.io/r/base/function.html" class="external-link">return</a></span><span class="op">(</span><span class="va">ret</span><span class="op">)</span></span>
+<span><span class="op">}</span></span>
+<span><span class="va">cl</span> <span class="op">&lt;-</span> <span class="fu">start_cluster</span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span></code></pre></div>
+<div class="section level3">
+<h3 id="test-data">Test data<a class="anchor" aria-label="anchor" href="#test-data"></a>
+</h3>
+<p>The example data are taken from the final addendum to the DAR from
+2014 and are distributed with the mkin package. Residue data and time
+step normalisation factors are read in using the function
+<code>read_spreadsheet</code> from the mkin package. This function also
+performs the time step normalisation.</p>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">data_file</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.file.html" class="external-link">system.file</a></span><span class="op">(</span></span>
+<span>  <span class="st">"testdata"</span>, <span class="st">"cyantraniliprole_soil_efsa_2014.xlsx"</span>,</span>
+<span>  package <span class="op">=</span> <span class="st">"mkin"</span><span class="op">)</span></span>
+<span><span class="va">cyan_ds</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/read_spreadsheet.html">read_spreadsheet</a></span><span class="op">(</span><span class="va">data_file</span>, parent_only <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span></span></code></pre></div>
+<p>The following tables show the covariate data and the 5 datasets that
+were read in from the spreadsheet file.</p>
+<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">pH</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/attr.html" class="external-link">attr</a></span><span class="op">(</span><span class="va">cyan_ds</span>, <span class="st">"covariates"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="va">pH</span>, caption <span class="op">=</span> <span class="st">"Covariate data"</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<caption>Covariate data</caption>
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">pH</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">Nambsheim</td>
+<td align="right">7.90</td>
+</tr>
+<tr class="even">
+<td align="left">Tama</td>
+<td align="right">6.20</td>
+</tr>
+<tr class="odd">
+<td align="left">Gross-Umstadt</td>
+<td align="right">7.04</td>
+</tr>
+<tr class="even">
+<td align="left">Sassafras</td>
+<td align="right">4.62</td>
+</tr>
+<tr class="odd">
+<td align="left">Lleida</td>
+<td align="right">8.05</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw">for</span> <span class="op">(</span><span class="va">ds_name</span> <span class="kw">in</span> <span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">cyan_ds</span><span class="op">)</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span></span>
+<span>    <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="fu"><a href="../../reference/mkin_long_to_wide.html">mkin_long_to_wide</a></span><span class="op">(</span><span class="va">cyan_ds</span><span class="op">[[</span><span class="va">ds_name</span><span class="op">]</span><span class="op">]</span><span class="op">)</span>,</span>
+<span>      caption <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/paste.html" class="external-link">paste</a></span><span class="op">(</span><span class="st">"Dataset"</span>, <span class="va">ds_name</span><span class="op">)</span>,</span>
+<span>      booktabs <span class="op">=</span> <span class="cn">TRUE</span>, row.names <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span><span class="op">)</span></span>
+<span>    <span class="fu"><a href="https://rdrr.io/r/base/cat.html" class="external-link">cat</a></span><span class="op">(</span><span class="st">"\n\\clearpage\n"</span><span class="op">)</span></span>
+<span><span class="op">}</span></span></code></pre></div>
+<table class="table">
+<caption>Dataset Nambsheim</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">cyan</th>
+<th align="right">JCZ38</th>
+<th align="right">J9C38</th>
+<th align="right">JSE76</th>
+<th align="right">J9Z38</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">105.79</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">3.210424</td>
+<td align="right">77.26</td>
+<td align="right">7.92</td>
+<td align="right">11.94</td>
+<td align="right">5.58</td>
+<td align="right">9.12</td>
+</tr>
+<tr class="odd">
+<td align="right">7.490988</td>
+<td align="right">57.13</td>
+<td align="right">15.46</td>
+<td align="right">16.58</td>
+<td align="right">12.59</td>
+<td align="right">11.74</td>
+</tr>
+<tr class="even">
+<td align="right">17.122259</td>
+<td align="right">37.74</td>
+<td align="right">15.98</td>
+<td align="right">13.36</td>
+<td align="right">26.05</td>
+<td align="right">10.77</td>
+</tr>
+<tr class="odd">
+<td align="right">23.543105</td>
+<td align="right">31.47</td>
+<td align="right">6.05</td>
+<td align="right">14.49</td>
+<td align="right">34.71</td>
+<td align="right">4.96</td>
+</tr>
+<tr class="even">
+<td align="right">43.875788</td>
+<td align="right">16.74</td>
+<td align="right">6.07</td>
+<td align="right">7.57</td>
+<td align="right">40.38</td>
+<td align="right">6.52</td>
+</tr>
+<tr class="odd">
+<td align="right">67.418893</td>
+<td align="right">8.85</td>
+<td align="right">10.34</td>
+<td align="right">6.39</td>
+<td align="right">30.71</td>
+<td align="right">8.90</td>
+</tr>
+<tr class="even">
+<td align="right">107.014116</td>
+<td align="right">5.19</td>
+<td align="right">9.61</td>
+<td align="right">1.95</td>
+<td align="right">20.41</td>
+<td align="right">12.93</td>
+</tr>
+<tr class="odd">
+<td align="right">129.487080</td>
+<td align="right">3.45</td>
+<td align="right">6.18</td>
+<td align="right">1.36</td>
+<td align="right">21.78</td>
+<td align="right">6.99</td>
+</tr>
+<tr class="even">
+<td align="right">195.835832</td>
+<td align="right">2.15</td>
+<td align="right">9.13</td>
+<td align="right">0.95</td>
+<td align="right">16.29</td>
+<td align="right">7.69</td>
+</tr>
+<tr class="odd">
+<td align="right">254.693596</td>
+<td align="right">1.92</td>
+<td align="right">6.92</td>
+<td align="right">0.20</td>
+<td align="right">13.57</td>
+<td align="right">7.16</td>
+</tr>
+<tr class="even">
+<td align="right">321.042348</td>
+<td align="right">2.26</td>
+<td align="right">7.02</td>
+<td align="right">NA</td>
+<td align="right">11.12</td>
+<td align="right">8.66</td>
+</tr>
+<tr class="odd">
+<td align="right">383.110535</td>
+<td align="right">NA</td>
+<td align="right">5.05</td>
+<td align="right">NA</td>
+<td align="right">10.64</td>
+<td align="right">5.56</td>
+</tr>
+<tr class="even">
+<td align="right">0.000000</td>
+<td align="right">105.57</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">3.210424</td>
+<td align="right">78.88</td>
+<td align="right">12.77</td>
+<td align="right">11.94</td>
+<td align="right">5.47</td>
+<td align="right">9.12</td>
+</tr>
+<tr class="even">
+<td align="right">7.490988</td>
+<td align="right">59.94</td>
+<td align="right">15.27</td>
+<td align="right">16.58</td>
+<td align="right">13.60</td>
+<td align="right">11.74</td>
+</tr>
+<tr class="odd">
+<td align="right">17.122259</td>
+<td align="right">39.67</td>
+<td align="right">14.26</td>
+<td align="right">13.36</td>
+<td align="right">29.44</td>
+<td align="right">10.77</td>
+</tr>
+<tr class="even">
+<td align="right">23.543105</td>
+<td align="right">30.21</td>
+<td align="right">16.07</td>
+<td align="right">14.49</td>
+<td align="right">35.90</td>
+<td align="right">4.96</td>
+</tr>
+<tr class="odd">
+<td align="right">43.875788</td>
+<td align="right">18.06</td>
+<td align="right">9.44</td>
+<td align="right">7.57</td>
+<td align="right">42.30</td>
+<td align="right">6.52</td>
+</tr>
+<tr class="even">
+<td align="right">67.418893</td>
+<td align="right">8.54</td>
+<td align="right">5.78</td>
+<td align="right">6.39</td>
+<td align="right">34.70</td>
+<td align="right">8.90</td>
+</tr>
+<tr class="odd">
+<td align="right">107.014116</td>
+<td align="right">7.26</td>
+<td align="right">4.54</td>
+<td align="right">1.95</td>
+<td align="right">23.33</td>
+<td align="right">12.93</td>
+</tr>
+<tr class="even">
+<td align="right">129.487080</td>
+<td align="right">3.60</td>
+<td align="right">4.22</td>
+<td align="right">1.36</td>
+<td align="right">23.56</td>
+<td align="right">6.99</td>
+</tr>
+<tr class="odd">
+<td align="right">195.835832</td>
+<td align="right">2.84</td>
+<td align="right">3.05</td>
+<td align="right">0.95</td>
+<td align="right">16.21</td>
+<td align="right">7.69</td>
+</tr>
+<tr class="even">
+<td align="right">254.693596</td>
+<td align="right">2.00</td>
+<td align="right">2.90</td>
+<td align="right">0.20</td>
+<td align="right">15.53</td>
+<td align="right">7.16</td>
+</tr>
+<tr class="odd">
+<td align="right">321.042348</td>
+<td align="right">1.79</td>
+<td align="right">0.94</td>
+<td align="right">NA</td>
+<td align="right">9.80</td>
+<td align="right">8.66</td>
+</tr>
+<tr class="even">
+<td align="right">383.110535</td>
+<td align="right">NA</td>
+<td align="right">1.82</td>
+<td align="right">NA</td>
+<td align="right">9.49</td>
+<td align="right">5.56</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Tama</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">cyan</th>
+<th align="right">JCZ38</th>
+<th align="right">J9Z38</th>
+<th align="right">JSE76</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">106.14</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">2.400833</td>
+<td align="right">93.47</td>
+<td align="right">6.46</td>
+<td align="right">2.85</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">5.601943</td>
+<td align="right">88.39</td>
+<td align="right">10.86</td>
+<td align="right">4.65</td>
+<td align="right">3.85</td>
+</tr>
+<tr class="even">
+<td align="right">12.804442</td>
+<td align="right">72.29</td>
+<td align="right">11.97</td>
+<td align="right">4.91</td>
+<td align="right">11.24</td>
+</tr>
+<tr class="odd">
+<td align="right">17.606108</td>
+<td align="right">65.79</td>
+<td align="right">13.11</td>
+<td align="right">6.63</td>
+<td align="right">13.79</td>
+</tr>
+<tr class="even">
+<td align="right">32.811382</td>
+<td align="right">53.16</td>
+<td align="right">11.24</td>
+<td align="right">8.90</td>
+<td align="right">23.40</td>
+</tr>
+<tr class="odd">
+<td align="right">50.417490</td>
+<td align="right">44.01</td>
+<td align="right">11.34</td>
+<td align="right">9.98</td>
+<td align="right">29.56</td>
+</tr>
+<tr class="even">
+<td align="right">80.027761</td>
+<td align="right">33.23</td>
+<td align="right">8.82</td>
+<td align="right">11.31</td>
+<td align="right">35.63</td>
+</tr>
+<tr class="odd">
+<td align="right">96.833591</td>
+<td align="right">40.68</td>
+<td align="right">5.94</td>
+<td align="right">8.32</td>
+<td align="right">29.09</td>
+</tr>
+<tr class="even">
+<td align="right">146.450803</td>
+<td align="right">20.65</td>
+<td align="right">4.49</td>
+<td align="right">8.72</td>
+<td align="right">36.88</td>
+</tr>
+<tr class="odd">
+<td align="right">190.466072</td>
+<td align="right">17.71</td>
+<td align="right">4.66</td>
+<td align="right">11.10</td>
+<td align="right">40.97</td>
+</tr>
+<tr class="even">
+<td align="right">240.083284</td>
+<td align="right">14.86</td>
+<td align="right">2.27</td>
+<td align="right">11.62</td>
+<td align="right">40.11</td>
+</tr>
+<tr class="odd">
+<td align="right">286.499386</td>
+<td align="right">12.02</td>
+<td align="right">NA</td>
+<td align="right">10.73</td>
+<td align="right">42.58</td>
+</tr>
+<tr class="even">
+<td align="right">0.000000</td>
+<td align="right">109.11</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">2.400833</td>
+<td align="right">96.84</td>
+<td align="right">5.52</td>
+<td align="right">2.04</td>
+<td align="right">2.02</td>
+</tr>
+<tr class="even">
+<td align="right">5.601943</td>
+<td align="right">85.29</td>
+<td align="right">9.65</td>
+<td align="right">2.99</td>
+<td align="right">4.39</td>
+</tr>
+<tr class="odd">
+<td align="right">12.804442</td>
+<td align="right">73.68</td>
+<td align="right">12.48</td>
+<td align="right">5.05</td>
+<td align="right">11.47</td>
+</tr>
+<tr class="even">
+<td align="right">17.606108</td>
+<td align="right">64.89</td>
+<td align="right">12.44</td>
+<td align="right">6.29</td>
+<td align="right">15.00</td>
+</tr>
+<tr class="odd">
+<td align="right">32.811382</td>
+<td align="right">52.27</td>
+<td align="right">10.86</td>
+<td align="right">7.65</td>
+<td align="right">23.30</td>
+</tr>
+<tr class="even">
+<td align="right">50.417490</td>
+<td align="right">42.61</td>
+<td align="right">10.54</td>
+<td align="right">9.37</td>
+<td align="right">31.06</td>
+</tr>
+<tr class="odd">
+<td align="right">80.027761</td>
+<td align="right">34.29</td>
+<td align="right">10.02</td>
+<td align="right">9.04</td>
+<td align="right">37.87</td>
+</tr>
+<tr class="even">
+<td align="right">96.833591</td>
+<td align="right">30.50</td>
+<td align="right">6.34</td>
+<td align="right">8.14</td>
+<td align="right">33.97</td>
+</tr>
+<tr class="odd">
+<td align="right">146.450803</td>
+<td align="right">19.21</td>
+<td align="right">6.29</td>
+<td align="right">8.52</td>
+<td align="right">26.15</td>
+</tr>
+<tr class="even">
+<td align="right">190.466072</td>
+<td align="right">17.55</td>
+<td align="right">5.81</td>
+<td align="right">9.89</td>
+<td align="right">32.08</td>
+</tr>
+<tr class="odd">
+<td align="right">240.083284</td>
+<td align="right">13.22</td>
+<td align="right">5.99</td>
+<td align="right">10.79</td>
+<td align="right">40.66</td>
+</tr>
+<tr class="even">
+<td align="right">286.499386</td>
+<td align="right">11.09</td>
+<td align="right">6.05</td>
+<td align="right">8.82</td>
+<td align="right">42.90</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Gross-Umstadt</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">cyan</th>
+<th align="right">JCZ38</th>
+<th align="right">J9Z38</th>
+<th align="right">JSE76</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.0000000</td>
+<td align="right">103.03</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">2.1014681</td>
+<td align="right">87.85</td>
+<td align="right">4.79</td>
+<td align="right">3.26</td>
+<td align="right">0.62</td>
+</tr>
+<tr class="odd">
+<td align="right">4.9034255</td>
+<td align="right">77.35</td>
+<td align="right">8.05</td>
+<td align="right">9.89</td>
+<td align="right">1.32</td>
+</tr>
+<tr class="even">
+<td align="right">10.5073404</td>
+<td align="right">69.33</td>
+<td align="right">9.74</td>
+<td align="right">12.32</td>
+<td align="right">4.74</td>
+</tr>
+<tr class="odd">
+<td align="right">21.0146807</td>
+<td align="right">55.65</td>
+<td align="right">14.57</td>
+<td align="right">13.59</td>
+<td align="right">9.84</td>
+</tr>
+<tr class="even">
+<td align="right">31.5220211</td>
+<td align="right">49.03</td>
+<td align="right">14.66</td>
+<td align="right">16.71</td>
+<td align="right">12.32</td>
+</tr>
+<tr class="odd">
+<td align="right">42.0293615</td>
+<td align="right">41.86</td>
+<td align="right">15.97</td>
+<td align="right">13.64</td>
+<td align="right">15.53</td>
+</tr>
+<tr class="even">
+<td align="right">63.0440422</td>
+<td align="right">34.88</td>
+<td align="right">18.20</td>
+<td align="right">14.12</td>
+<td align="right">22.02</td>
+</tr>
+<tr class="odd">
+<td align="right">84.0587230</td>
+<td align="right">28.26</td>
+<td align="right">15.64</td>
+<td align="right">14.06</td>
+<td align="right">25.60</td>
+</tr>
+<tr class="even">
+<td align="right">0.0000000</td>
+<td align="right">104.05</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">2.1014681</td>
+<td align="right">85.25</td>
+<td align="right">2.68</td>
+<td align="right">7.32</td>
+<td align="right">0.69</td>
+</tr>
+<tr class="even">
+<td align="right">4.9034255</td>
+<td align="right">77.22</td>
+<td align="right">7.28</td>
+<td align="right">8.37</td>
+<td align="right">1.45</td>
+</tr>
+<tr class="odd">
+<td align="right">10.5073404</td>
+<td align="right">65.23</td>
+<td align="right">10.73</td>
+<td align="right">10.93</td>
+<td align="right">4.74</td>
+</tr>
+<tr class="even">
+<td align="right">21.0146807</td>
+<td align="right">57.78</td>
+<td align="right">12.29</td>
+<td align="right">14.80</td>
+<td align="right">9.05</td>
+</tr>
+<tr class="odd">
+<td align="right">31.5220211</td>
+<td align="right">54.83</td>
+<td align="right">14.05</td>
+<td align="right">12.01</td>
+<td align="right">11.05</td>
+</tr>
+<tr class="even">
+<td align="right">42.0293615</td>
+<td align="right">45.17</td>
+<td align="right">12.12</td>
+<td align="right">17.89</td>
+<td align="right">15.71</td>
+</tr>
+<tr class="odd">
+<td align="right">63.0440422</td>
+<td align="right">34.83</td>
+<td align="right">12.90</td>
+<td align="right">15.86</td>
+<td align="right">22.52</td>
+</tr>
+<tr class="even">
+<td align="right">84.0587230</td>
+<td align="right">26.59</td>
+<td align="right">14.28</td>
+<td align="right">14.91</td>
+<td align="right">28.48</td>
+</tr>
+<tr class="odd">
+<td align="right">0.0000000</td>
+<td align="right">104.62</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">0.8145225</td>
+<td align="right">97.21</td>
+<td align="right">NA</td>
+<td align="right">4.00</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">1.9005525</td>
+<td align="right">89.64</td>
+<td align="right">3.59</td>
+<td align="right">5.24</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">4.0726125</td>
+<td align="right">87.90</td>
+<td align="right">4.10</td>
+<td align="right">9.58</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">8.1452251</td>
+<td align="right">86.90</td>
+<td align="right">5.96</td>
+<td align="right">9.45</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">12.2178376</td>
+<td align="right">74.74</td>
+<td align="right">7.83</td>
+<td align="right">15.03</td>
+<td align="right">5.33</td>
+</tr>
+<tr class="odd">
+<td align="right">16.2904502</td>
+<td align="right">74.13</td>
+<td align="right">8.84</td>
+<td align="right">14.41</td>
+<td align="right">5.10</td>
+</tr>
+<tr class="even">
+<td align="right">24.4356753</td>
+<td align="right">65.26</td>
+<td align="right">11.84</td>
+<td align="right">18.33</td>
+<td align="right">6.71</td>
+</tr>
+<tr class="odd">
+<td align="right">32.5809004</td>
+<td align="right">57.70</td>
+<td align="right">12.74</td>
+<td align="right">19.93</td>
+<td align="right">9.74</td>
+</tr>
+<tr class="even">
+<td align="right">0.0000000</td>
+<td align="right">101.94</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">0.8145225</td>
+<td align="right">99.94</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">1.9005525</td>
+<td align="right">94.87</td>
+<td align="right">NA</td>
+<td align="right">4.56</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">4.0726125</td>
+<td align="right">86.96</td>
+<td align="right">6.75</td>
+<td align="right">6.90</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">8.1452251</td>
+<td align="right">80.51</td>
+<td align="right">10.68</td>
+<td align="right">7.43</td>
+<td align="right">2.58</td>
+</tr>
+<tr class="odd">
+<td align="right">12.2178376</td>
+<td align="right">78.38</td>
+<td align="right">10.35</td>
+<td align="right">9.46</td>
+<td align="right">3.69</td>
+</tr>
+<tr class="even">
+<td align="right">16.2904502</td>
+<td align="right">70.05</td>
+<td align="right">13.73</td>
+<td align="right">9.27</td>
+<td align="right">7.18</td>
+</tr>
+<tr class="odd">
+<td align="right">24.4356753</td>
+<td align="right">61.28</td>
+<td align="right">12.57</td>
+<td align="right">13.28</td>
+<td align="right">13.19</td>
+</tr>
+<tr class="even">
+<td align="right">32.5809004</td>
+<td align="right">52.85</td>
+<td align="right">12.67</td>
+<td align="right">12.95</td>
+<td align="right">13.69</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Sassafras</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">cyan</th>
+<th align="right">JCZ38</th>
+<th align="right">J9Z38</th>
+<th align="right">JSE76</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">102.17</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">2.216719</td>
+<td align="right">95.49</td>
+<td align="right">1.11</td>
+<td align="right">0.10</td>
+<td align="right">0.83</td>
+</tr>
+<tr class="odd">
+<td align="right">5.172343</td>
+<td align="right">83.35</td>
+<td align="right">6.43</td>
+<td align="right">2.89</td>
+<td align="right">3.30</td>
+</tr>
+<tr class="even">
+<td align="right">11.083593</td>
+<td align="right">78.18</td>
+<td align="right">10.00</td>
+<td align="right">5.59</td>
+<td align="right">0.81</td>
+</tr>
+<tr class="odd">
+<td align="right">22.167186</td>
+<td align="right">70.44</td>
+<td align="right">17.21</td>
+<td align="right">4.23</td>
+<td align="right">1.09</td>
+</tr>
+<tr class="even">
+<td align="right">33.250779</td>
+<td align="right">68.00</td>
+<td align="right">20.45</td>
+<td align="right">5.86</td>
+<td align="right">1.17</td>
+</tr>
+<tr class="odd">
+<td align="right">44.334371</td>
+<td align="right">59.64</td>
+<td align="right">24.64</td>
+<td align="right">3.17</td>
+<td align="right">2.72</td>
+</tr>
+<tr class="even">
+<td align="right">66.501557</td>
+<td align="right">50.73</td>
+<td align="right">27.50</td>
+<td align="right">6.19</td>
+<td align="right">1.27</td>
+</tr>
+<tr class="odd">
+<td align="right">88.668742</td>
+<td align="right">45.65</td>
+<td align="right">32.77</td>
+<td align="right">5.69</td>
+<td align="right">4.54</td>
+</tr>
+<tr class="even">
+<td align="right">0.000000</td>
+<td align="right">100.43</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">2.216719</td>
+<td align="right">95.34</td>
+<td align="right">3.21</td>
+<td align="right">0.14</td>
+<td align="right">0.46</td>
+</tr>
+<tr class="even">
+<td align="right">5.172343</td>
+<td align="right">84.38</td>
+<td align="right">5.73</td>
+<td align="right">4.75</td>
+<td align="right">0.62</td>
+</tr>
+<tr class="odd">
+<td align="right">11.083593</td>
+<td align="right">78.50</td>
+<td align="right">11.89</td>
+<td align="right">3.99</td>
+<td align="right">0.73</td>
+</tr>
+<tr class="even">
+<td align="right">22.167186</td>
+<td align="right">71.17</td>
+<td align="right">17.28</td>
+<td align="right">4.39</td>
+<td align="right">0.66</td>
+</tr>
+<tr class="odd">
+<td align="right">33.250779</td>
+<td align="right">59.41</td>
+<td align="right">18.73</td>
+<td align="right">11.85</td>
+<td align="right">2.65</td>
+</tr>
+<tr class="even">
+<td align="right">44.334371</td>
+<td align="right">64.57</td>
+<td align="right">22.93</td>
+<td align="right">5.13</td>
+<td align="right">2.01</td>
+</tr>
+<tr class="odd">
+<td align="right">66.501557</td>
+<td align="right">49.08</td>
+<td align="right">33.39</td>
+<td align="right">5.67</td>
+<td align="right">3.63</td>
+</tr>
+<tr class="even">
+<td align="right">88.668742</td>
+<td align="right">40.41</td>
+<td align="right">39.60</td>
+<td align="right">5.93</td>
+<td align="right">6.17</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Lleida</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">cyan</th>
+<th align="right">JCZ38</th>
+<th align="right">J9Z38</th>
+<th align="right">JSE76</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">102.71</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">2.821051</td>
+<td align="right">79.11</td>
+<td align="right">5.70</td>
+<td align="right">8.07</td>
+<td align="right">0.97</td>
+</tr>
+<tr class="odd">
+<td align="right">6.582451</td>
+<td align="right">70.03</td>
+<td align="right">7.17</td>
+<td align="right">11.31</td>
+<td align="right">4.72</td>
+</tr>
+<tr class="even">
+<td align="right">14.105253</td>
+<td align="right">50.93</td>
+<td align="right">10.25</td>
+<td align="right">14.84</td>
+<td align="right">9.95</td>
+</tr>
+<tr class="odd">
+<td align="right">28.210505</td>
+<td align="right">33.43</td>
+<td align="right">10.40</td>
+<td align="right">14.82</td>
+<td align="right">24.06</td>
+</tr>
+<tr class="even">
+<td align="right">42.315758</td>
+<td align="right">24.69</td>
+<td align="right">9.75</td>
+<td align="right">16.38</td>
+<td align="right">29.38</td>
+</tr>
+<tr class="odd">
+<td align="right">56.421010</td>
+<td align="right">22.99</td>
+<td align="right">10.06</td>
+<td align="right">15.51</td>
+<td align="right">29.25</td>
+</tr>
+<tr class="even">
+<td align="right">84.631516</td>
+<td align="right">14.63</td>
+<td align="right">5.63</td>
+<td align="right">14.74</td>
+<td align="right">31.04</td>
+</tr>
+<tr class="odd">
+<td align="right">112.842021</td>
+<td align="right">12.43</td>
+<td align="right">4.17</td>
+<td align="right">13.53</td>
+<td align="right">33.28</td>
+</tr>
+<tr class="even">
+<td align="right">0.000000</td>
+<td align="right">99.31</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">2.821051</td>
+<td align="right">82.07</td>
+<td align="right">6.55</td>
+<td align="right">5.60</td>
+<td align="right">1.12</td>
+</tr>
+<tr class="even">
+<td align="right">6.582451</td>
+<td align="right">70.65</td>
+<td align="right">7.61</td>
+<td align="right">8.01</td>
+<td align="right">3.21</td>
+</tr>
+<tr class="odd">
+<td align="right">14.105253</td>
+<td align="right">53.52</td>
+<td align="right">11.48</td>
+<td align="right">10.82</td>
+<td align="right">12.24</td>
+</tr>
+<tr class="even">
+<td align="right">28.210505</td>
+<td align="right">35.60</td>
+<td align="right">11.19</td>
+<td align="right">15.43</td>
+<td align="right">23.53</td>
+</tr>
+<tr class="odd">
+<td align="right">42.315758</td>
+<td align="right">34.26</td>
+<td align="right">11.09</td>
+<td align="right">13.26</td>
+<td align="right">27.42</td>
+</tr>
+<tr class="even">
+<td align="right">56.421010</td>
+<td align="right">21.79</td>
+<td align="right">4.80</td>
+<td align="right">18.30</td>
+<td align="right">30.20</td>
+</tr>
+<tr class="odd">
+<td align="right">84.631516</td>
+<td align="right">14.06</td>
+<td align="right">6.30</td>
+<td align="right">16.35</td>
+<td align="right">32.32</td>
+</tr>
+<tr class="even">
+<td align="right">112.842021</td>
+<td align="right">11.51</td>
+<td align="right">5.57</td>
+<td align="right">12.64</td>
+<td align="right">32.51</td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="section level2">
+<h2 id="parent-only-evaluations">Parent only evaluations<a class="anchor" aria-label="anchor" href="#parent-only-evaluations"></a>
+</h2>
+<p>As the pathway fits have very long run times, evaluations of the
+parent data are performed first, in order to determine for each
+hierarchical parent degradation model which random effects on the
+degradation model parameters are ill-defined.</p>
+<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">cyan_sep_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"FOMC"</span>, <span class="st">"DFOP"</span>, <span class="st">"SFORB"</span>, <span class="st">"HS"</span><span class="op">)</span>,</span>
+<span>  <span class="va">cyan_ds</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>, cores <span class="op">=</span> <span class="va">n_cores</span><span class="op">)</span></span>
+<span><span class="va">cyan_sep_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">cyan_sep_const</span>, error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span></span>
+<span><span class="va">cyan_saem_full</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mhmkin.html">mhmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">cyan_sep_const</span>, <span class="va">cyan_sep_tc</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">cyan_saem_full</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">HS</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>All fits converged successfully.</p>
+<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">cyan_saem_full</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">sd(cyan_0)</td>
+<td align="left">sd(cyan_0)</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">sd(log_beta)</td>
+<td align="left">sd(cyan_0)</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">sd(cyan_0)</td>
+<td align="left">sd(cyan_0), sd(log_k1)</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB</td>
+<td align="left">sd(cyan_free_0)</td>
+<td align="left">sd(cyan_free_0), sd(log_k_cyan_free_bound)</td>
+</tr>
+<tr class="odd">
+<td align="left">HS</td>
+<td align="left">sd(cyan_0)</td>
+<td align="left">sd(cyan_0)</td>
+</tr>
+</tbody>
+</table>
+<p>In almost all models, the random effect for the initial concentration
+of the parent compound is ill-defined. For the biexponential models DFOP
+and SFORB, the random effect of one additional parameter is ill-defined
+when the two-component error model is used.</p>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">cyan_saem_full</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO const</td>
+<td align="right">5</td>
+<td align="right">833.9</td>
+<td align="right">832.0</td>
+<td align="right">-412.0</td>
+</tr>
+<tr class="even">
+<td align="left">SFO tc</td>
+<td align="right">6</td>
+<td align="right">831.6</td>
+<td align="right">829.3</td>
+<td align="right">-409.8</td>
+</tr>
+<tr class="odd">
+<td align="left">FOMC const</td>
+<td align="right">7</td>
+<td align="right">709.1</td>
+<td align="right">706.4</td>
+<td align="right">-347.6</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC tc</td>
+<td align="right">8</td>
+<td align="right">689.2</td>
+<td align="right">686.1</td>
+<td align="right">-336.6</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP const</td>
+<td align="right">9</td>
+<td align="right">703.0</td>
+<td align="right">699.5</td>
+<td align="right">-342.5</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB const</td>
+<td align="right">9</td>
+<td align="right">701.3</td>
+<td align="right">697.8</td>
+<td align="right">-341.7</td>
+</tr>
+<tr class="odd">
+<td align="left">HS const</td>
+<td align="right">9</td>
+<td align="right">718.6</td>
+<td align="right">715.1</td>
+<td align="right">-350.3</td>
+</tr>
+<tr class="even">
+<td align="left">DFOP tc</td>
+<td align="right">10</td>
+<td align="right">703.1</td>
+<td align="right">699.2</td>
+<td align="right">-341.6</td>
+</tr>
+<tr class="odd">
+<td align="left">SFORB tc</td>
+<td align="right">10</td>
+<td align="right">700.0</td>
+<td align="right">696.1</td>
+<td align="right">-340.0</td>
+</tr>
+<tr class="even">
+<td align="left">HS tc</td>
+<td align="right">10</td>
+<td align="right">716.7</td>
+<td align="right">712.8</td>
+<td align="right">-348.3</td>
+</tr>
+</tbody>
+</table>
+<p>Model comparison based on AIC and BIC indicates that the
+two-component error model is preferable for all parent models with the
+exception of DFOP. The lowest AIC and BIC values are are obtained with
+the FOMC model, followed by SFORB and DFOP.</p>
+<div class="sourceCode" id="cb8"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">stopCluster</a></span><span class="op">(</span><span class="va">cl</span><span class="op">)</span></span></code></pre></div>
+</div>
+<div class="section level2">
+<h2 id="pathway-fits">Pathway fits<a class="anchor" aria-label="anchor" href="#pathway-fits"></a>
+</h2>
+<div class="section level3">
+<h3 id="evaluations-with-pathway-established-previously">Evaluations with pathway established previously<a class="anchor" aria-label="anchor" href="#evaluations-with-pathway-established-previously"></a>
+</h3>
+<p>To test the technical feasibility of coupling the relevant parent
+degradation models with different transformation pathway models, a list
+of <code>mkinmod</code> models is set up below. As in the EU evaluation,
+parallel formation of metabolites JCZ38 and J9Z38 and secondary
+formation of metabolite JSE76 from JCZ38 is used.</p>
+<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw">if</span> <span class="op">(</span><span class="op">!</span><span class="fu"><a href="https://rdrr.io/r/base/files2.html" class="external-link">dir.exists</a></span><span class="op">(</span><span class="st">"cyan_dlls"</span><span class="op">)</span><span class="op">)</span> <span class="fu"><a href="https://rdrr.io/r/base/files2.html" class="external-link">dir.create</a></span><span class="op">(</span><span class="st">"cyan_dlls"</span><span class="op">)</span></span>
+<span><span class="va">cyan_path_1</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span></span>
+<span>  sfo_path_1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    cyan <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"JCZ38"</span>, <span class="st">"J9Z38"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>    JCZ38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"JSE76"</span><span class="op">)</span>,</span>
+<span>    J9Z38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>    JSE76 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>    name <span class="op">=</span> <span class="st">"sfo_path_1"</span>, dll_dir <span class="op">=</span> <span class="st">"cyan_dlls"</span>, overwrite <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span>,</span>
+<span>  fomc_path_1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    cyan <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"FOMC"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"JCZ38"</span>, <span class="st">"J9Z38"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>    JCZ38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"JSE76"</span><span class="op">)</span>,</span>
+<span>    J9Z38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>    JSE76 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>    name <span class="op">=</span> <span class="st">"fomc_path_1"</span>, dll_dir <span class="op">=</span> <span class="st">"cyan_dlls"</span>, overwrite <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span>,</span>
+<span>  dfop_path_1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    cyan <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"DFOP"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"JCZ38"</span>, <span class="st">"J9Z38"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>    JCZ38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"JSE76"</span><span class="op">)</span>,</span>
+<span>    J9Z38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>    JSE76 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>    name <span class="op">=</span> <span class="st">"dfop_path_1"</span>, dll_dir <span class="op">=</span> <span class="st">"cyan_dlls"</span>, overwrite <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span>,</span>
+<span>  sforb_path_1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    cyan <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFORB"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"JCZ38"</span>, <span class="st">"J9Z38"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>    JCZ38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"JSE76"</span><span class="op">)</span>,</span>
+<span>    J9Z38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>    JSE76 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>    name <span class="op">=</span> <span class="st">"sforb_path_1"</span>, dll_dir <span class="op">=</span> <span class="st">"cyan_dlls"</span>, overwrite <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span>,</span>
+<span>  hs_path_1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    cyan <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"HS"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"JCZ38"</span>, <span class="st">"J9Z38"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>    JCZ38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"JSE76"</span><span class="op">)</span>,</span>
+<span>    J9Z38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>    JSE76 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>    name <span class="op">=</span> <span class="st">"hs_path_1"</span>, dll_dir <span class="op">=</span> <span class="st">"cyan_dlls"</span>, overwrite <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="op">)</span></span>
+<span><span class="va">cl_path_1</span> <span class="op">&lt;-</span> <span class="fu">start_cluster</span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span></code></pre></div>
+<p>To obtain suitable starting values for the NLHM fits, separate
+pathway fits are performed for all datasets.</p>
+<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_sep_1_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span></span>
+<span>  <span class="va">cyan_path_1</span>,</span>
+<span>  <span class="va">cyan_ds</span>,</span>
+<span>  error_model <span class="op">=</span> <span class="st">"const"</span>,</span>
+<span>  cluster <span class="op">=</span> <span class="va">cl_path_1</span>,</span>
+<span>  quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_1_const</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">Nambsheim</th>
+<th align="left">Tama</th>
+<th align="left">Gross-Umstadt</th>
+<th align="left">Sassafras</th>
+<th align="left">Lleida</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">fomc_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">hs_path_1</td>
+<td align="left">C</td>
+<td align="left">C</td>
+<td align="left">C</td>
+<td align="left">C</td>
+<td align="left">C</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb11"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_sep_1_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_sep_1_const</span>, error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_1_tc</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">Nambsheim</th>
+<th align="left">Tama</th>
+<th align="left">Gross-Umstadt</th>
+<th align="left">Sassafras</th>
+<th align="left">Lleida</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">fomc_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">hs_path_1</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+</tr>
+</tbody>
+</table>
+<p>Most separate fits converged successfully. The biggest convergence
+problems are seen when using the HS model with constant variance.</p>
+<p>For the hierarchical pathway fits, those random effects that could
+not be quantified in the corresponding parent data analyses are
+excluded.</p>
+<p>In the code below, the output of the <code>illparms</code> function
+for the parent only fits is used as an argument
+<code>no_random_effect</code> to the <code>mhmkin</code> function. The
+possibility to do so was introduced in mkin version <code>1.2.2</code>
+which is currently under development.</p>
+<div class="sourceCode" id="cb12"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_saem_1</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mhmkin.html">mhmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">f_sep_1_const</span>, <span class="va">f_sep_1_tc</span><span class="op">)</span>,</span>
+<span>  no_random_effect <span class="op">=</span> <span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">cyan_saem_full</span><span class="op">)</span>,</span>
+<span>  cluster <span class="op">=</span> <span class="va">cl_path_1</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb13"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_path_1</td>
+<td align="left">FO</td>
+<td align="left">Fth, FO</td>
+</tr>
+<tr class="even">
+<td align="left">fomc_path_1</td>
+<td align="left">OK</td>
+<td align="left">Fth, FO</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1</td>
+<td align="left">Fth, FO</td>
+<td align="left">Fth, FO</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1</td>
+<td align="left">Fth, FO</td>
+<td align="left">Fth, FO</td>
+</tr>
+<tr class="odd">
+<td align="left">hs_path_1</td>
+<td align="left">FO</td>
+<td align="left">E</td>
+</tr>
+</tbody>
+</table>
+<p>The status information from the individual fits shows that all fits
+completed successfully. The matrix entries Fth and FO indicate that the
+Fisher Information Matrix could not be inverted for the fixed effects
+(theta) and the random effects (Omega), respectively. For the affected
+fits, ill-defined parameters cannot be determined using the
+<code>illparms</code> function, because it relies on the Fisher
+Information Matrix.</p>
+<div class="sourceCode" id="cb14"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<colgroup>
+<col width="18%">
+<col width="77%">
+<col width="4%">
+</colgroup>
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_path_1</td>
+<td align="left">NA</td>
+<td align="left">NA</td>
+</tr>
+<tr class="even">
+<td align="left">fomc_path_1</td>
+<td align="left">sd(log_k_J9Z38), sd(f_cyan_ilr_2),
+sd(f_JCZ38_qlogis)</td>
+<td align="left">NA</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1</td>
+<td align="left">NA</td>
+<td align="left">NA</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1</td>
+<td align="left">NA</td>
+<td align="left">NA</td>
+</tr>
+<tr class="odd">
+<td align="left">hs_path_1</td>
+<td align="left">NA</td>
+<td align="left">E</td>
+</tr>
+</tbody>
+</table>
+<p>The model comparisons below suggest that the pathway fits using DFOP
+or SFORB for the parent compound provide the best fit.</p>
+<div class="sourceCode" id="cb15"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">[</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_path_1 const</td>
+<td align="right">16</td>
+<td align="right">2693.0</td>
+<td align="right">2686.8</td>
+<td align="right">-1330.5</td>
+</tr>
+<tr class="even">
+<td align="left">fomc_path_1 const</td>
+<td align="right">18</td>
+<td align="right">2427.9</td>
+<td align="right">2420.9</td>
+<td align="right">-1196.0</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1 const</td>
+<td align="right">20</td>
+<td align="right">2403.2</td>
+<td align="right">2395.4</td>
+<td align="right">-1181.6</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1 const</td>
+<td align="right">20</td>
+<td align="right">2401.4</td>
+<td align="right">2393.6</td>
+<td align="right">-1180.7</td>
+</tr>
+<tr class="odd">
+<td align="left">hs_path_1 const</td>
+<td align="right">20</td>
+<td align="right">2427.2</td>
+<td align="right">2419.4</td>
+<td align="right">-1193.6</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb16"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">[</span><span class="fl">1</span><span class="op">:</span><span class="fl">4</span>, <span class="op">]</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_path_1 const</td>
+<td align="right">16</td>
+<td align="right">2693.0</td>
+<td align="right">2686.8</td>
+<td align="right">-1330.5</td>
+</tr>
+<tr class="even">
+<td align="left">sfo_path_1 tc</td>
+<td align="right">17</td>
+<td align="right">2657.6</td>
+<td align="right">2651.0</td>
+<td align="right">-1311.8</td>
+</tr>
+<tr class="odd">
+<td align="left">fomc_path_1 const</td>
+<td align="right">18</td>
+<td align="right">2427.9</td>
+<td align="right">2420.9</td>
+<td align="right">-1196.0</td>
+</tr>
+<tr class="even">
+<td align="left">fomc_path_1 tc</td>
+<td align="right">19</td>
+<td align="right">2423.6</td>
+<td align="right">2416.2</td>
+<td align="right">-1192.8</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1 const</td>
+<td align="right">20</td>
+<td align="right">2403.2</td>
+<td align="right">2395.4</td>
+<td align="right">-1181.6</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1 const</td>
+<td align="right">20</td>
+<td align="right">2401.4</td>
+<td align="right">2393.6</td>
+<td align="right">-1180.7</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1 tc</td>
+<td align="right">20</td>
+<td align="right">2398.0</td>
+<td align="right">2390.1</td>
+<td align="right">-1179.0</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1 tc</td>
+<td align="right">20</td>
+<td align="right">2399.9</td>
+<td align="right">2392.1</td>
+<td align="right">-1180.0</td>
+</tr>
+</tbody>
+</table>
+<p>For these two parent model, successful fits are shown below. Plots of
+the fits with the other parent models are shown in the Appendix.</p>
+<div class="sourceCode" id="cb17"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">[[</span><span class="st">"dfop_path_1"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_cyan_pathway_files/figure-html/unnamed-chunk-7-1.png" alt="DFOP pathway fit with two-component error" width="700"><p class="caption">
+DFOP pathway fit with two-component error
+</p>
+</div>
+<div class="sourceCode" id="cb18"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">[[</span><span class="st">"sforb_path_1"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_cyan_pathway_files/figure-html/unnamed-chunk-8-1.png" alt="SFORB pathway fit with two-component error" width="700"><p class="caption">
+SFORB pathway fit with two-component error
+</p>
+</div>
+<p>A closer graphical analysis of these Figures shows that the residues
+of transformation product JCZ38 in the soils Tama and Nambsheim observed
+at later time points are strongly and systematically underestimated.</p>
+<div class="sourceCode" id="cb19"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">stopCluster</a></span><span class="op">(</span><span class="va">cl_path_1</span><span class="op">)</span></span></code></pre></div>
+</div>
+<div class="section level3">
+<h3 id="alternative-pathway-fits">Alternative pathway fits<a class="anchor" aria-label="anchor" href="#alternative-pathway-fits"></a>
+</h3>
+<p>To improve the fit for JCZ38, a back-reaction from JSE76 to JCZ38 was
+introduced in an alternative version of the transformation pathway, in
+analogy to the back-reaction from K5A78 to K5A77. Both pairs of
+transformation products are pairs of an organic acid with its
+corresponding amide (Addendum 2014, p. 109). As FOMC provided the best
+fit for the parent, and the biexponential models DFOP and SFORB provided
+the best initial pathway fits, these three parent models are used in the
+alternative pathway fits.</p>
+<div class="sourceCode" id="cb20"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">cyan_path_2</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span></span>
+<span>  fomc_path_2 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    cyan <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"FOMC"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"JCZ38"</span>, <span class="st">"J9Z38"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>    JCZ38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"JSE76"</span><span class="op">)</span>,</span>
+<span>    J9Z38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>    JSE76 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"JCZ38"</span><span class="op">)</span>,</span>
+<span>    name <span class="op">=</span> <span class="st">"fomc_path_2"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>    dll_dir <span class="op">=</span> <span class="st">"cyan_dlls"</span>,</span>
+<span>    overwrite <span class="op">=</span> <span class="cn">TRUE</span></span>
+<span>  <span class="op">)</span>,</span>
+<span>  dfop_path_2 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    cyan <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"DFOP"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"JCZ38"</span>, <span class="st">"J9Z38"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>    JCZ38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"JSE76"</span><span class="op">)</span>,</span>
+<span>    J9Z38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>    JSE76 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"JCZ38"</span><span class="op">)</span>,</span>
+<span>    name <span class="op">=</span> <span class="st">"dfop_path_2"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>    dll_dir <span class="op">=</span> <span class="st">"cyan_dlls"</span>,</span>
+<span>    overwrite <span class="op">=</span> <span class="cn">TRUE</span></span>
+<span>  <span class="op">)</span>,</span>
+<span>  sforb_path_2 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    cyan <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFORB"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"JCZ38"</span>, <span class="st">"J9Z38"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>    JCZ38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"JSE76"</span><span class="op">)</span>,</span>
+<span>    J9Z38 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>    JSE76 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"JCZ38"</span><span class="op">)</span>,</span>
+<span>    name <span class="op">=</span> <span class="st">"sforb_path_2"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>    dll_dir <span class="op">=</span> <span class="st">"cyan_dlls"</span>,</span>
+<span>    overwrite <span class="op">=</span> <span class="cn">TRUE</span></span>
+<span>  <span class="op">)</span></span>
+<span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">cl_path_2</span> <span class="op">&lt;-</span> <span class="fu">start_cluster</span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
+<span><span class="va">f_sep_2_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span></span>
+<span>  <span class="va">cyan_path_2</span>,</span>
+<span>  <span class="va">cyan_ds</span>,</span>
+<span>  error_model <span class="op">=</span> <span class="st">"const"</span>,</span>
+<span>  cluster <span class="op">=</span> <span class="va">cl_path_2</span>,</span>
+<span>  quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span></span>
+<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_2_const</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">Nambsheim</th>
+<th align="left">Tama</th>
+<th align="left">Gross-Umstadt</th>
+<th align="left">Sassafras</th>
+<th align="left">Lleida</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">fomc_path_2</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">dfop_path_2</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">sforb_path_2</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>Using constant variance, separate fits converge with the exception of
+the fits to the Sassafras soil data.</p>
+<div class="sourceCode" id="cb21"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_sep_2_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_sep_2_const</span>, error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_2_tc</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">Nambsheim</th>
+<th align="left">Tama</th>
+<th align="left">Gross-Umstadt</th>
+<th align="left">Sassafras</th>
+<th align="left">Lleida</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">fomc_path_2</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">dfop_path_2</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">sforb_path_2</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>Using the two-component error model, all separate fits converge with
+the exception of the alternative pathway fit with DFOP used for the
+parent and the Sassafras dataset.</p>
+<div class="sourceCode" id="cb22"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_saem_2</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mhmkin.html">mhmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">f_sep_2_const</span>, <span class="va">f_sep_2_tc</span><span class="op">)</span>,</span>
+<span>  no_random_effect <span class="op">=</span> <span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">cyan_saem_full</span><span class="op">[</span><span class="fl">2</span><span class="op">:</span><span class="fl">4</span>, <span class="op">]</span><span class="op">)</span>,</span>
+<span>  cluster <span class="op">=</span> <span class="va">cl_path_2</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb23"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">fomc_path_2</td>
+<td align="left">E</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">dfop_path_2</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">sforb_path_2</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>The hierarchical fits for the alternative pathway completed
+successfully, with the exception of the model using FOMC for the parent
+compound and constant variance as the error model.</p>
+<div class="sourceCode" id="cb24"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<colgroup>
+<col width="14%">
+<col width="42%">
+<col width="42%">
+</colgroup>
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">fomc_path_2</td>
+<td align="left">E</td>
+<td align="left">sd(f_JSE76_qlogis)</td>
+</tr>
+<tr class="even">
+<td align="left">dfop_path_2</td>
+<td align="left">sd(f_JCZ38_qlogis), sd(f_JSE76_qlogis)</td>
+<td align="left">sd(f_JCZ38_qlogis), sd(f_JSE76_qlogis)</td>
+</tr>
+<tr class="odd">
+<td align="left">sforb_path_2</td>
+<td align="left">sd(f_JCZ38_qlogis), sd(f_JSE76_qlogis)</td>
+<td align="left">sd(f_JCZ38_qlogis), sd(f_JSE76_qlogis)</td>
+</tr>
+</tbody>
+</table>
+<p>In all biphasic fits (DFOP or SFORB for the parent compound), the
+random effects for the formation fractions for the pathways from JCZ38
+to JSE76, and for the reverse pathway from JSE76 to JCZ38 are
+ill-defined.</p>
+<div class="sourceCode" id="cb25"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">fomc_path_2 tc</td>
+<td align="right">21</td>
+<td align="right">2249.0</td>
+<td align="right">2240.8</td>
+<td align="right">-1103.5</td>
+</tr>
+<tr class="even">
+<td align="left">dfop_path_2 tc</td>
+<td align="right">22</td>
+<td align="right">2234.4</td>
+<td align="right">2225.8</td>
+<td align="right">-1095.2</td>
+</tr>
+<tr class="odd">
+<td align="left">sforb_path_2 tc</td>
+<td align="right">22</td>
+<td align="right">2239.7</td>
+<td align="right">2231.1</td>
+<td align="right">-1097.9</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb26"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[</span><span class="fl">2</span><span class="op">:</span><span class="fl">3</span>,<span class="op">]</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">dfop_path_2 const</td>
+<td align="right">22</td>
+<td align="right">2288.4</td>
+<td align="right">2279.8</td>
+<td align="right">-1122.2</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_2 const</td>
+<td align="right">22</td>
+<td align="right">2283.3</td>
+<td align="right">2274.7</td>
+<td align="right">-1119.7</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_2 tc</td>
+<td align="right">22</td>
+<td align="right">2234.4</td>
+<td align="right">2225.8</td>
+<td align="right">-1095.2</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_2 tc</td>
+<td align="right">22</td>
+<td align="right">2239.7</td>
+<td align="right">2231.1</td>
+<td align="right">-1097.9</td>
+</tr>
+</tbody>
+</table>
+<p>The variants using the biexponential models DFOP and SFORB for the
+parent compound and the two-component error model give the lowest AIC
+and BIC values and are plotted below. Compared with the original
+pathway, the AIC and BIC values indicate a large improvement. This is
+confirmed by the plots, which show that the metabolite JCZ38 is fitted
+much better with this model.</p>
+<div class="sourceCode" id="cb27"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"fomc_path_2"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_cyan_pathway_files/figure-html/unnamed-chunk-13-1.png" alt="FOMC pathway fit with two-component error, alternative pathway" width="700"><p class="caption">
+FOMC pathway fit with two-component error, alternative pathway
+</p>
+</div>
+<div class="sourceCode" id="cb28"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"dfop_path_2"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_cyan_pathway_files/figure-html/unnamed-chunk-14-1.png" alt="DFOP pathway fit with two-component error, alternative pathway" width="700"><p class="caption">
+DFOP pathway fit with two-component error, alternative pathway
+</p>
+</div>
+<div class="sourceCode" id="cb29"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"sforb_path_2"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_cyan_pathway_files/figure-html/unnamed-chunk-15-1.png" alt="SFORB pathway fit with two-component error, alternative pathway" width="700"><p class="caption">
+SFORB pathway fit with two-component error, alternative pathway
+</p>
+</div>
+</div>
+<div class="section level3">
+<h3 id="refinement-of-alternative-pathway-fits">Refinement of alternative pathway fits<a class="anchor" aria-label="anchor" href="#refinement-of-alternative-pathway-fits"></a>
+</h3>
+<p>All ill-defined random effects that were identified in the parent
+only fits and in the above pathway fits, are excluded for the final
+evaluations below. For this purpose, a list of character vectors is
+created below that can be indexed by row and column indices, and which
+contains the degradation parameter names for which random effects should
+be excluded for each of the hierarchical fits contained in
+<code>f_saem_2</code>.</p>
+<div class="sourceCode" id="cb30"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">no_ranef</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/matrix.html" class="external-link">matrix</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="op">)</span>, nrow <span class="op">=</span> <span class="fl">3</span>, ncol <span class="op">=</span> <span class="fl">2</span>, dimnames <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/dimnames.html" class="external-link">dimnames</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="va">no_ranef</span><span class="op">[[</span><span class="st">"fomc_path_2"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"log_beta"</span>, <span class="st">"f_JCZ38_qlogis"</span>, <span class="st">"f_JSE76_qlogis"</span><span class="op">)</span></span>
+<span><span class="va">no_ranef</span><span class="op">[[</span><span class="st">"fomc_path_2"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"cyan_0"</span>, <span class="st">"f_JCZ38_qlogis"</span>, <span class="st">"f_JSE76_qlogis"</span><span class="op">)</span></span>
+<span><span class="va">no_ranef</span><span class="op">[[</span><span class="st">"dfop_path_2"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"cyan_0"</span>, <span class="st">"f_JCZ38_qlogis"</span>, <span class="st">"f_JSE76_qlogis"</span><span class="op">)</span></span>
+<span><span class="va">no_ranef</span><span class="op">[[</span><span class="st">"dfop_path_2"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"cyan_0"</span>, <span class="st">"log_k1"</span>, <span class="st">"f_JCZ38_qlogis"</span>, <span class="st">"f_JSE76_qlogis"</span><span class="op">)</span></span>
+<span><span class="va">no_ranef</span><span class="op">[[</span><span class="st">"sforb_path_2"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"cyan_free_0"</span>,</span>
+<span>  <span class="st">"f_JCZ38_qlogis"</span>, <span class="st">"f_JSE76_qlogis"</span><span class="op">)</span></span>
+<span><span class="va">no_ranef</span><span class="op">[[</span><span class="st">"sforb_path_2"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"cyan_free_0"</span>, <span class="st">"log_k_cyan_free_bound"</span>,</span>
+<span>  <span class="st">"f_JCZ38_qlogis"</span>, <span class="st">"f_JSE76_qlogis"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/parallel/clusterApply.html" class="external-link">clusterExport</a></span><span class="op">(</span><span class="va">cl_path_2</span>, <span class="st">"no_ranef"</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">f_saem_3</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_saem_2</span>,</span>
+<span>  no_random_effect <span class="op">=</span> <span class="va">no_ranef</span>,</span>
+<span>  cluster <span class="op">=</span> <span class="va">cl_path_2</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb31"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_saem_3</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">fomc_path_2</td>
+<td align="left">E</td>
+<td align="left">Fth</td>
+</tr>
+<tr class="even">
+<td align="left">dfop_path_2</td>
+<td align="left">Fth</td>
+<td align="left">Fth</td>
+</tr>
+<tr class="odd">
+<td align="left">sforb_path_2</td>
+<td align="left">Fth</td>
+<td align="left">Fth</td>
+</tr>
+</tbody>
+</table>
+<p>With the exception of the FOMC pathway fit with constant variance,
+all updated fits completed successfully. However, the Fisher Information
+Matrix for the fixed effects (Fth) could not be inverted, so no
+confidence intervals for the optimised parameters are available.</p>
+<div class="sourceCode" id="cb32"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_3</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">fomc_path_2</td>
+<td align="left">E</td>
+<td align="left"></td>
+</tr>
+<tr class="even">
+<td align="left">dfop_path_2</td>
+<td align="left"></td>
+<td align="left"></td>
+</tr>
+<tr class="odd">
+<td align="left">sforb_path_2</td>
+<td align="left"></td>
+<td align="left"></td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb33"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_3</span><span class="op">[</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">fomc_path_2 tc</td>
+<td align="right">19</td>
+<td align="right">2249.1</td>
+<td align="right">2241.6</td>
+<td align="right">-1105.5</td>
+</tr>
+<tr class="even">
+<td align="left">dfop_path_2 tc</td>
+<td align="right">20</td>
+<td align="right">2237.3</td>
+<td align="right">2229.5</td>
+<td align="right">-1098.6</td>
+</tr>
+<tr class="odd">
+<td align="left">sforb_path_2 tc</td>
+<td align="right">20</td>
+<td align="right">2241.3</td>
+<td align="right">2233.5</td>
+<td align="right">-1100.7</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb34"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_3</span><span class="op">[</span><span class="fl">2</span><span class="op">:</span><span class="fl">3</span>,<span class="op">]</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">dfop_path_2 const</td>
+<td align="right">20</td>
+<td align="right">2282.2</td>
+<td align="right">2274.4</td>
+<td align="right">-1121.1</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_2 const</td>
+<td align="right">20</td>
+<td align="right">2279.7</td>
+<td align="right">2271.9</td>
+<td align="right">-1119.9</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_2 tc</td>
+<td align="right">20</td>
+<td align="right">2237.3</td>
+<td align="right">2229.5</td>
+<td align="right">-1098.6</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_2 tc</td>
+<td align="right">20</td>
+<td align="right">2241.3</td>
+<td align="right">2233.5</td>
+<td align="right">-1100.7</td>
+</tr>
+</tbody>
+</table>
+<p>While the AIC and BIC values of the best fit (DFOP pathway fit with
+two-component error) are lower than in the previous fits with the
+alternative pathway, the practical value of these refined evaluations is
+limited as no confidence intervals are obtained.</p>
+<div class="sourceCode" id="cb35"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">stopCluster</a></span><span class="op">(</span><span class="va">cl_path_2</span><span class="op">)</span></span></code></pre></div>
+</div>
+</div>
+<div class="section level2">
+<h2 id="conclusion">Conclusion<a class="anchor" aria-label="anchor" href="#conclusion"></a>
+</h2>
+<p>It was demonstrated that a relatively complex transformation pathway
+with parallel formation of two primary metabolites and one secondary
+metabolite can be fitted even if the data in the individual datasets are
+quite different and partly only cover the formation phase.</p>
+<p>The run times of the pathway fits were several hours, limiting the
+practical feasibility of iterative refinements based on ill-defined
+parameters and of alternative checks of parameter identifiability based
+on multistart runs.</p>
+</div>
+<div class="section level2">
+<h2 id="acknowledgements">Acknowledgements<a class="anchor" aria-label="anchor" href="#acknowledgements"></a>
+</h2>
+<p>The helpful comments by Janina Wöltjen of the German Environment
+Agency are gratefully acknowledged.</p>
+</div>
+<div class="section level2">
+<h2 id="appendix">Appendix<a class="anchor" aria-label="anchor" href="#appendix"></a>
+</h2>
+<div class="section level3">
+<h3 id="plots-of-fits-that-were-not-refined-further">Plots of fits that were not refined further<a class="anchor" aria-label="anchor" href="#plots-of-fits-that-were-not-refined-further"></a>
+</h3>
+<div class="sourceCode" id="cb36"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">[[</span><span class="st">"sfo_path_1"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_cyan_pathway_files/figure-html/unnamed-chunk-20-1.png" alt="SFO pathway fit with two-component error" width="700"><p class="caption">
+SFO pathway fit with two-component error
+</p>
+</div>
+<div class="sourceCode" id="cb37"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">[[</span><span class="st">"fomc_path_1"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_cyan_pathway_files/figure-html/unnamed-chunk-21-1.png" alt="FOMC pathway fit with two-component error" width="700"><p class="caption">
+FOMC pathway fit with two-component error
+</p>
+</div>
+<div class="sourceCode" id="cb38"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">[[</span><span class="st">"sforb_path_1"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_cyan_pathway_files/figure-html/unnamed-chunk-22-1.png" alt="HS pathway fit with two-component error" width="700"><p class="caption">
+HS pathway fit with two-component error
+</p>
+</div>
+</div>
+<div class="section level3">
+<h3 id="hierarchical-fit-listings">Hierarchical fit listings<a class="anchor" aria-label="anchor" href="#hierarchical-fit-listings"></a>
+</h3>
+<div class="section level4">
+<h4 id="pathway-1">Pathway 1<a class="anchor" aria-label="anchor" href="#pathway-1"></a>
+</h4>
+<caption>
+Hierarchical SFO path 1 fit with constant variance
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:42:26 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - k_cyan * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * k_cyan * cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * k_cyan * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 437.151 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0     log_k_cyan    log_k_JCZ38    log_k_J9Z38    log_k_JSE76 
+       95.3304        -3.8459        -3.1305        -5.0678        -5.3196 
+  f_cyan_ilr_1   f_cyan_ilr_2 f_JCZ38_qlogis 
+        0.8158        23.5335        11.8774 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_cyan log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_0          4.797     0.0000       0.000       0.000      0.0000
+log_k_cyan      0.000     0.9619       0.000       0.000      0.0000
+log_k_JCZ38     0.000     0.0000       2.139       0.000      0.0000
+log_k_J9Z38     0.000     0.0000       0.000       1.639      0.0000
+log_k_JSE76     0.000     0.0000       0.000       0.000      0.7894
+f_cyan_ilr_1    0.000     0.0000       0.000       0.000      0.0000
+f_cyan_ilr_2    0.000     0.0000       0.000       0.000      0.0000
+f_JCZ38_qlogis  0.000     0.0000       0.000       0.000      0.0000
+               f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis
+cyan_0               0.0000        0.000           0.00
+log_k_cyan           0.0000        0.000           0.00
+log_k_JCZ38          0.0000        0.000           0.00
+log_k_J9Z38          0.0000        0.000           0.00
+log_k_JSE76          0.0000        0.000           0.00
+f_cyan_ilr_1         0.7714        0.000           0.00
+f_cyan_ilr_2         0.0000        9.247           0.00
+f_JCZ38_qlogis       0.0000        0.000          16.61
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2693 2687  -1331
+
+Optimised parameters:
+                     est.      lower      upper
+cyan_0            95.1279  9.354e+01  9.671e+01
+log_k_cyan        -3.8527 -4.367e+00 -3.338e+00
+log_k_JCZ38       -3.0381 -4.187e+00 -1.889e+00
+log_k_J9Z38       -5.0095 -5.623e+00 -4.396e+00
+log_k_JSE76       -5.3357 -6.025e+00 -4.646e+00
+f_cyan_ilr_1       0.8050  5.174e-01  1.093e+00
+f_cyan_ilr_2      12.4820 -1.050e+06  1.051e+06
+f_JCZ38_qlogis     1.2912  3.561e-01  2.226e+00
+a.1                4.8393         NA         NA
+SD.log_k_cyan      0.5840         NA         NA
+SD.log_k_JCZ38     1.2740         NA         NA
+SD.log_k_J9Z38     0.3172         NA         NA
+SD.log_k_JSE76     0.5677         NA         NA
+SD.f_cyan_ilr_1    0.2623         NA         NA
+SD.f_cyan_ilr_2    1.3724         NA         NA
+SD.f_JCZ38_qlogis  0.1464         NA         NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_cyan     0.5840    NA    NA
+SD.log_k_JCZ38    1.2740    NA    NA
+SD.log_k_J9Z38    0.3172    NA    NA
+SD.log_k_JSE76    0.5677    NA    NA
+SD.f_cyan_ilr_1   0.2623    NA    NA
+SD.f_cyan_ilr_2   1.3724    NA    NA
+SD.f_JCZ38_qlogis 0.1464    NA    NA
+
+Variance model:
+     est. lower upper
+a.1 4.839    NA    NA
+
+Backtransformed parameters:
+                      est.     lower     upper
+cyan_0           95.127935 93.542456 96.713413
+k_cyan            0.021221  0.012687  0.035497
+k_JCZ38           0.047924  0.015189  0.151213
+k_J9Z38           0.006674  0.003612  0.012332
+k_JSE76           0.004817  0.002417  0.009601
+f_cyan_to_JCZ38   0.757402        NA        NA
+f_cyan_to_J9Z38   0.242597        NA        NA
+f_JCZ38_to_JSE76  0.784347  0.588098  0.902582
+
+Resulting formation fractions:
+                   ff
+cyan_JCZ38  7.574e-01
+cyan_J9Z38  2.426e-01
+cyan_sink   9.839e-08
+JCZ38_JSE76 7.843e-01
+JCZ38_sink  2.157e-01
+
+Estimated disappearance times:
+        DT50   DT90
+cyan   32.66 108.50
+JCZ38  14.46  48.05
+J9Z38 103.86 345.00
+JSE76 143.91 478.04
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical SFO path 1 fit with two-component error
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:42:06 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - k_cyan * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * k_cyan * cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * k_cyan * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 417.143 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0     log_k_cyan    log_k_JCZ38    log_k_J9Z38    log_k_JSE76 
+       96.0039        -3.8907        -3.1276        -5.0069        -4.9367 
+  f_cyan_ilr_1   f_cyan_ilr_2 f_JCZ38_qlogis 
+        0.7937        22.3422        17.8932 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_cyan log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_0          4.859      0.000        0.00        0.00      0.0000
+log_k_cyan      0.000      0.962        0.00        0.00      0.0000
+log_k_JCZ38     0.000      0.000        2.04        0.00      0.0000
+log_k_J9Z38     0.000      0.000        0.00        1.72      0.0000
+log_k_JSE76     0.000      0.000        0.00        0.00      0.9076
+f_cyan_ilr_1    0.000      0.000        0.00        0.00      0.0000
+f_cyan_ilr_2    0.000      0.000        0.00        0.00      0.0000
+f_JCZ38_qlogis  0.000      0.000        0.00        0.00      0.0000
+               f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis
+cyan_0               0.0000        0.000           0.00
+log_k_cyan           0.0000        0.000           0.00
+log_k_JCZ38          0.0000        0.000           0.00
+log_k_J9Z38          0.0000        0.000           0.00
+log_k_JSE76          0.0000        0.000           0.00
+f_cyan_ilr_1         0.7598        0.000           0.00
+f_cyan_ilr_2         0.0000        8.939           0.00
+f_JCZ38_qlogis       0.0000        0.000          14.49
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2658 2651  -1312
+
+Optimised parameters:
+                      est. lower upper
+cyan_0            94.81681    NA    NA
+log_k_cyan        -3.91558    NA    NA
+log_k_JCZ38       -3.12715    NA    NA
+log_k_J9Z38       -5.04840    NA    NA
+log_k_JSE76       -5.10443    NA    NA
+f_cyan_ilr_1       0.80760    NA    NA
+f_cyan_ilr_2      48.66960    NA    NA
+f_JCZ38_qlogis     3.03397    NA    NA
+a.1                3.93879    NA    NA
+b.1                0.08057    NA    NA
+SD.log_k_cyan      0.58921    NA    NA
+SD.log_k_JCZ38     1.29813    NA    NA
+SD.log_k_J9Z38     0.68372    NA    NA
+SD.log_k_JSE76     0.35128    NA    NA
+SD.f_cyan_ilr_1    0.38352    NA    NA
+SD.f_cyan_ilr_2    4.98884    NA    NA
+SD.f_JCZ38_qlogis  1.75636    NA    NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_cyan     0.5892    NA    NA
+SD.log_k_JCZ38    1.2981    NA    NA
+SD.log_k_J9Z38    0.6837    NA    NA
+SD.log_k_JSE76    0.3513    NA    NA
+SD.f_cyan_ilr_1   0.3835    NA    NA
+SD.f_cyan_ilr_2   4.9888    NA    NA
+SD.f_JCZ38_qlogis 1.7564    NA    NA
+
+Variance model:
+       est. lower upper
+a.1 3.93879    NA    NA
+b.1 0.08057    NA    NA
+
+Backtransformed parameters:
+                     est. lower upper
+cyan_0           94.81681    NA    NA
+k_cyan            0.01993    NA    NA
+k_JCZ38           0.04384    NA    NA
+k_J9Z38           0.00642    NA    NA
+k_JSE76           0.00607    NA    NA
+f_cyan_to_JCZ38   0.75807    NA    NA
+f_cyan_to_J9Z38   0.24193    NA    NA
+f_JCZ38_to_JSE76  0.95409    NA    NA
+
+Resulting formation fractions:
+                 ff
+cyan_JCZ38  0.75807
+cyan_J9Z38  0.24193
+cyan_sink   0.00000
+JCZ38_JSE76 0.95409
+JCZ38_sink  0.04591
+
+Estimated disappearance times:
+        DT50   DT90
+cyan   34.78 115.54
+JCZ38  15.81  52.52
+J9Z38 107.97 358.68
+JSE76 114.20 379.35
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical FOMC path 1 fit with constant variance
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:42:57 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - (alpha/beta) * 1/((time/beta) + 1) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 468.609 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      101.2314        -3.3680        -5.1108        -5.9416         0.7144 
+  f_cyan_ilr_2 f_JCZ38_qlogis      log_alpha       log_beta 
+        7.0229        14.9234        -0.1791         2.9811 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.416       0.000         0.0       0.000       0.0000
+log_k_JCZ38     0.000       2.439         0.0       0.000       0.0000
+log_k_J9Z38     0.000       0.000         1.7       0.000       0.0000
+log_k_JSE76     0.000       0.000         0.0       1.856       0.0000
+f_cyan_ilr_1    0.000       0.000         0.0       0.000       0.7164
+f_cyan_ilr_2    0.000       0.000         0.0       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000         0.0       0.000       0.0000
+log_alpha       0.000       0.000         0.0       0.000       0.0000
+log_beta        0.000       0.000         0.0       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis log_alpha log_beta
+cyan_0                 0.00           0.00    0.0000   0.0000
+log_k_JCZ38            0.00           0.00    0.0000   0.0000
+log_k_J9Z38            0.00           0.00    0.0000   0.0000
+log_k_JSE76            0.00           0.00    0.0000   0.0000
+f_cyan_ilr_1           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_2          11.57           0.00    0.0000   0.0000
+f_JCZ38_qlogis         0.00          18.81    0.0000   0.0000
+log_alpha              0.00           0.00    0.4144   0.0000
+log_beta               0.00           0.00    0.0000   0.5077
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2428 2421  -1196
+
+Optimised parameters:
+                      est.    lower    upper
+cyan_0            101.1664 98.51265 103.8202
+log_k_JCZ38        -3.3883 -4.78250  -1.9941
+log_k_J9Z38        -5.3087 -5.91564  -4.7017
+log_k_JSE76        -6.1313 -7.30061  -4.9619
+f_cyan_ilr_1        0.7456  0.43782   1.0534
+f_cyan_ilr_2        0.8181  0.24956   1.3866
+f_JCZ38_qlogis      2.0467  0.61165   3.4817
+log_alpha          -0.2391 -0.62806   0.1499
+log_beta            2.8739  2.67664   3.0711
+a.1                 3.4160  3.17960   3.6525
+SD.cyan_0           2.4355  0.40399   4.4671
+SD.log_k_JCZ38      1.5654  0.57311   2.5576
+SD.log_k_J9Z38      0.4645 -0.06533   0.9943
+SD.log_k_JSE76      0.9841  0.10738   1.8609
+SD.f_cyan_ilr_1     0.3285  0.10546   0.5515
+SD.f_cyan_ilr_2     0.2276 -0.38711   0.8424
+SD.f_JCZ38_qlogis   0.8340 -0.20970   1.8777
+SD.log_alpha        0.4250  0.16017   0.6898
+
+Correlation: 
+               cyan_0  l__JCZ3 l__J9Z3 l__JSE7 f_cy__1 f_cy__2 f_JCZ38 log_lph
+log_k_JCZ38    -0.0159                                                        
+log_k_J9Z38    -0.0546  0.0080                                                
+log_k_JSE76    -0.0337  0.0016  0.0074                                        
+f_cyan_ilr_1   -0.0095  0.0194 -0.1573  0.0003                                
+f_cyan_ilr_2   -0.2733  0.0799  0.3059  0.0263  0.0125                        
+f_JCZ38_qlogis  0.0755 -0.0783 -0.0516  0.1222 -0.1155 -0.5231                
+log_alpha      -0.0567  0.0120  0.0351  0.0189  0.0040  0.0829 -0.0502        
+log_beta       -0.2980  0.0461  0.1382  0.0758  0.0209  0.4079 -0.2053  0.2759
+
+Random effects:
+                    est.    lower  upper
+SD.cyan_0         2.4355  0.40399 4.4671
+SD.log_k_JCZ38    1.5654  0.57311 2.5576
+SD.log_k_J9Z38    0.4645 -0.06533 0.9943
+SD.log_k_JSE76    0.9841  0.10738 1.8609
+SD.f_cyan_ilr_1   0.3285  0.10546 0.5515
+SD.f_cyan_ilr_2   0.2276 -0.38711 0.8424
+SD.f_JCZ38_qlogis 0.8340 -0.20970 1.8777
+SD.log_alpha      0.4250  0.16017 0.6898
+
+Variance model:
+     est. lower upper
+a.1 3.416  3.18 3.652
+
+Backtransformed parameters:
+                      est.     lower     upper
+cyan_0           1.012e+02 9.851e+01 103.82023
+k_JCZ38          3.377e-02 8.375e-03   0.13614
+k_J9Z38          4.948e-03 2.697e-03   0.00908
+k_JSE76          2.174e-03 6.751e-04   0.00700
+f_cyan_to_JCZ38  6.389e-01        NA        NA
+f_cyan_to_J9Z38  2.226e-01        NA        NA
+f_JCZ38_to_JSE76 8.856e-01 6.483e-01   0.97016
+alpha            7.873e-01 5.336e-01   1.16166
+beta             1.771e+01 1.454e+01  21.56509
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.6389
+cyan_J9Z38  0.2226
+cyan_sink   0.1385
+JCZ38_JSE76 0.8856
+JCZ38_sink  0.1144
+
+Estimated disappearance times:
+        DT50    DT90 DT50back
+cyan   25.00  312.06    93.94
+JCZ38  20.53   68.19       NA
+J9Z38 140.07  465.32       NA
+JSE76 318.86 1059.22       NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical FOMC path 1 fit with two-component error
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:42:50 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - (alpha/beta) * 1/((time/beta) + 1) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 460.309 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+     101.13294       -3.32499       -5.09097       -5.93566        0.71359 
+  f_cyan_ilr_2 f_JCZ38_qlogis      log_alpha       log_beta 
+      10.30315       14.62272       -0.09633        3.10634 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.649       0.000        0.00        0.00       0.0000
+log_k_JCZ38     0.000       2.319        0.00        0.00       0.0000
+log_k_J9Z38     0.000       0.000        1.73        0.00       0.0000
+log_k_JSE76     0.000       0.000        0.00        1.86       0.0000
+f_cyan_ilr_1    0.000       0.000        0.00        0.00       0.7183
+f_cyan_ilr_2    0.000       0.000        0.00        0.00       0.0000
+f_JCZ38_qlogis  0.000       0.000        0.00        0.00       0.0000
+log_alpha       0.000       0.000        0.00        0.00       0.0000
+log_beta        0.000       0.000        0.00        0.00       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis log_alpha log_beta
+cyan_0                 0.00           0.00    0.0000   0.0000
+log_k_JCZ38            0.00           0.00    0.0000   0.0000
+log_k_J9Z38            0.00           0.00    0.0000   0.0000
+log_k_JSE76            0.00           0.00    0.0000   0.0000
+f_cyan_ilr_1           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_2          12.85           0.00    0.0000   0.0000
+f_JCZ38_qlogis         0.00          18.54    0.0000   0.0000
+log_alpha              0.00           0.00    0.3142   0.0000
+log_beta               0.00           0.00    0.0000   0.7333
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2424 2416  -1193
+
+Optimised parameters:
+                       est. lower upper
+cyan_0            100.65667    NA    NA
+log_k_JCZ38        -3.45782    NA    NA
+log_k_J9Z38        -5.23476    NA    NA
+log_k_JSE76        -5.71827    NA    NA
+f_cyan_ilr_1        0.68389    NA    NA
+f_cyan_ilr_2        0.61027    NA    NA
+f_JCZ38_qlogis    116.27482    NA    NA
+log_alpha          -0.14484    NA    NA
+log_beta            3.03220    NA    NA
+a.1                 3.11051    NA    NA
+b.1                 0.04508    NA    NA
+SD.log_k_JCZ38      1.39961    NA    NA
+SD.log_k_J9Z38      0.57920    NA    NA
+SD.log_k_JSE76      0.68364    NA    NA
+SD.f_cyan_ilr_1     0.31477    NA    NA
+SD.f_cyan_ilr_2     0.37716    NA    NA
+SD.f_JCZ38_qlogis   5.52695    NA    NA
+SD.log_alpha        0.22823    NA    NA
+SD.log_beta         0.39161    NA    NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_JCZ38    1.3996    NA    NA
+SD.log_k_J9Z38    0.5792    NA    NA
+SD.log_k_JSE76    0.6836    NA    NA
+SD.f_cyan_ilr_1   0.3148    NA    NA
+SD.f_cyan_ilr_2   0.3772    NA    NA
+SD.f_JCZ38_qlogis 5.5270    NA    NA
+SD.log_alpha      0.2282    NA    NA
+SD.log_beta       0.3916    NA    NA
+
+Variance model:
+       est. lower upper
+a.1 3.11051    NA    NA
+b.1 0.04508    NA    NA
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.007e+02    NA    NA
+k_JCZ38          3.150e-02    NA    NA
+k_J9Z38          5.328e-03    NA    NA
+k_JSE76          3.285e-03    NA    NA
+f_cyan_to_JCZ38  5.980e-01    NA    NA
+f_cyan_to_J9Z38  2.273e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+alpha            8.652e-01    NA    NA
+beta             2.074e+01    NA    NA
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.5980
+cyan_J9Z38  0.2273
+cyan_sink   0.1746
+JCZ38_JSE76 1.0000
+JCZ38_sink  0.0000
+
+Estimated disappearance times:
+        DT50  DT90 DT50back
+cyan   25.48 276.2    83.15
+JCZ38  22.01  73.1       NA
+J9Z38 130.09 432.2       NA
+JSE76 210.98 700.9       NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical DFOP path 1 fit with constant variance
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:43:33 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 504.014 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      102.0643        -3.4008        -5.0024        -5.8612         0.6855 
+  f_cyan_ilr_2 f_JCZ38_qlogis         log_k1         log_k2       g_qlogis 
+        1.2366        13.6901        -1.8641        -4.5063        -0.6468 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          4.466       0.000       0.000       0.000       0.0000
+log_k_JCZ38     0.000       2.382       0.000       0.000       0.0000
+log_k_J9Z38     0.000       0.000       1.595       0.000       0.0000
+log_k_JSE76     0.000       0.000       0.000       1.245       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000       0.000       0.6852
+f_cyan_ilr_2    0.000       0.000       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000       0.000       0.0000
+log_k1          0.000       0.000       0.000       0.000       0.0000
+log_k2          0.000       0.000       0.000       0.000       0.0000
+g_qlogis        0.000       0.000       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis log_k1 log_k2 g_qlogis
+cyan_0                 0.00           0.00 0.0000 0.0000    0.000
+log_k_JCZ38            0.00           0.00 0.0000 0.0000    0.000
+log_k_J9Z38            0.00           0.00 0.0000 0.0000    0.000
+log_k_JSE76            0.00           0.00 0.0000 0.0000    0.000
+f_cyan_ilr_1           0.00           0.00 0.0000 0.0000    0.000
+f_cyan_ilr_2           1.28           0.00 0.0000 0.0000    0.000
+f_JCZ38_qlogis         0.00          16.08 0.0000 0.0000    0.000
+log_k1                 0.00           0.00 0.9866 0.0000    0.000
+log_k2                 0.00           0.00 0.0000 0.5953    0.000
+g_qlogis               0.00           0.00 0.0000 0.0000    1.583
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2403 2395  -1182
+
+Optimised parameters:
+                      est. lower upper
+cyan_0            102.5565    NA    NA
+log_k_JCZ38        -3.4729    NA    NA
+log_k_J9Z38        -5.1533    NA    NA
+log_k_JSE76        -5.6669    NA    NA
+f_cyan_ilr_1        0.6665    NA    NA
+f_cyan_ilr_2        0.5191    NA    NA
+f_JCZ38_qlogis     37.0113    NA    NA
+log_k1             -1.8497    NA    NA
+log_k2             -4.4931    NA    NA
+g_qlogis           -0.6383    NA    NA
+a.1                 3.2397    NA    NA
+SD.log_k_JCZ38      1.4286    NA    NA
+SD.log_k_J9Z38      0.5312    NA    NA
+SD.log_k_JSE76      0.6627    NA    NA
+SD.f_cyan_ilr_1     0.3013    NA    NA
+SD.f_cyan_ilr_2     0.2980    NA    NA
+SD.f_JCZ38_qlogis   0.1637    NA    NA
+SD.log_k1           0.5069    NA    NA
+SD.log_k2           0.3828    NA    NA
+SD.g_qlogis         0.8641    NA    NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_JCZ38    1.4286    NA    NA
+SD.log_k_J9Z38    0.5312    NA    NA
+SD.log_k_JSE76    0.6627    NA    NA
+SD.f_cyan_ilr_1   0.3013    NA    NA
+SD.f_cyan_ilr_2   0.2980    NA    NA
+SD.f_JCZ38_qlogis 0.1637    NA    NA
+SD.log_k1         0.5069    NA    NA
+SD.log_k2         0.3828    NA    NA
+SD.g_qlogis       0.8641    NA    NA
+
+Variance model:
+    est. lower upper
+a.1 3.24    NA    NA
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.026e+02    NA    NA
+k_JCZ38          3.103e-02    NA    NA
+k_J9Z38          5.780e-03    NA    NA
+k_JSE76          3.459e-03    NA    NA
+f_cyan_to_JCZ38  5.813e-01    NA    NA
+f_cyan_to_J9Z38  2.265e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+k1               1.573e-01    NA    NA
+k2               1.119e-02    NA    NA
+g                3.456e-01    NA    NA
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.5813
+cyan_J9Z38  0.2265
+cyan_sink   0.1922
+JCZ38_JSE76 1.0000
+JCZ38_sink  0.0000
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   25.23 167.94    50.55   4.407   61.97
+JCZ38  22.34  74.22       NA      NA      NA
+J9Z38 119.92 398.36       NA      NA      NA
+JSE76 200.41 665.76       NA      NA      NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical DFOP path 1 fit with two-component error
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:46:07 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 658.043 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      101.3964        -3.3626        -4.9792        -5.8727         0.6814 
+  f_cyan_ilr_2 f_JCZ38_qlogis         log_k1         log_k2       g_qlogis 
+        6.8713        13.6901        -1.9222        -4.5035        -0.7172 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.317       0.000       0.000       0.000       0.0000
+log_k_JCZ38     0.000       2.272       0.000       0.000       0.0000
+log_k_J9Z38     0.000       0.000       1.633       0.000       0.0000
+log_k_JSE76     0.000       0.000       0.000       1.271       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000       0.000       0.6839
+f_cyan_ilr_2    0.000       0.000       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000       0.000       0.0000
+log_k1          0.000       0.000       0.000       0.000       0.0000
+log_k2          0.000       0.000       0.000       0.000       0.0000
+g_qlogis        0.000       0.000       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis log_k1 log_k2 g_qlogis
+cyan_0                 0.00           0.00 0.0000 0.0000    0.000
+log_k_JCZ38            0.00           0.00 0.0000 0.0000    0.000
+log_k_J9Z38            0.00           0.00 0.0000 0.0000    0.000
+log_k_JSE76            0.00           0.00 0.0000 0.0000    0.000
+f_cyan_ilr_1           0.00           0.00 0.0000 0.0000    0.000
+f_cyan_ilr_2          11.95           0.00 0.0000 0.0000    0.000
+f_JCZ38_qlogis         0.00          16.08 0.0000 0.0000    0.000
+log_k1                 0.00           0.00 0.9496 0.0000    0.000
+log_k2                 0.00           0.00 0.0000 0.5846    0.000
+g_qlogis               0.00           0.00 0.0000 0.0000    1.719
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2398 2390  -1179
+
+Optimised parameters:
+                       est. lower upper
+cyan_0            100.69709    NA    NA
+log_k_JCZ38        -3.46669    NA    NA
+log_k_J9Z38        -5.05076    NA    NA
+log_k_JSE76        -5.55558    NA    NA
+f_cyan_ilr_1        0.66045    NA    NA
+f_cyan_ilr_2        0.84275    NA    NA
+f_JCZ38_qlogis     64.22404    NA    NA
+log_k1             -2.17715    NA    NA
+log_k2             -4.55002    NA    NA
+g_qlogis           -0.55920    NA    NA
+a.1                 2.95785    NA    NA
+b.1                 0.04456    NA    NA
+SD.log_k_JCZ38      1.39881    NA    NA
+SD.log_k_J9Z38      0.67788    NA    NA
+SD.log_k_JSE76      0.52603    NA    NA
+SD.f_cyan_ilr_1     0.32490    NA    NA
+SD.f_cyan_ilr_2     0.53923    NA    NA
+SD.f_JCZ38_qlogis   2.75576    NA    NA
+SD.log_k2           0.30694    NA    NA
+SD.g_qlogis         0.83619    NA    NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_JCZ38    1.3988    NA    NA
+SD.log_k_J9Z38    0.6779    NA    NA
+SD.log_k_JSE76    0.5260    NA    NA
+SD.f_cyan_ilr_1   0.3249    NA    NA
+SD.f_cyan_ilr_2   0.5392    NA    NA
+SD.f_JCZ38_qlogis 2.7558    NA    NA
+SD.log_k2         0.3069    NA    NA
+SD.g_qlogis       0.8362    NA    NA
+
+Variance model:
+       est. lower upper
+a.1 2.95785    NA    NA
+b.1 0.04456    NA    NA
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.007e+02    NA    NA
+k_JCZ38          3.122e-02    NA    NA
+k_J9Z38          6.404e-03    NA    NA
+k_JSE76          3.866e-03    NA    NA
+f_cyan_to_JCZ38  6.187e-01    NA    NA
+f_cyan_to_J9Z38  2.431e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+k1               1.134e-01    NA    NA
+k2               1.057e-02    NA    NA
+g                3.637e-01    NA    NA
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.6187
+cyan_J9Z38  0.2431
+cyan_sink   0.1382
+JCZ38_JSE76 1.0000
+JCZ38_sink  0.0000
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   26.35 175.12    52.72   6.114    65.6
+JCZ38  22.20  73.75       NA      NA      NA
+J9Z38 108.23 359.53       NA      NA      NA
+JSE76 179.30 595.62       NA      NA      NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical SFORB path 1 fit with constant variance
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:43:36 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 507.042 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+             102.0643               -2.8987               -2.7077 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+              -3.4717               -3.4008               -5.0024 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+              -5.8613                0.6855                1.2366 
+       f_JCZ38_qlogis 
+              13.7395 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 4.466          0.0000                 0.000
+log_k_cyan_free             0.000          0.6158                 0.000
+log_k_cyan_free_bound       0.000          0.0000                 1.463
+log_k_cyan_bound_free       0.000          0.0000                 0.000
+log_k_JCZ38                 0.000          0.0000                 0.000
+log_k_J9Z38                 0.000          0.0000                 0.000
+log_k_JSE76                 0.000          0.0000                 0.000
+f_cyan_ilr_1                0.000          0.0000                 0.000
+f_cyan_ilr_2                0.000          0.0000                 0.000
+f_JCZ38_qlogis              0.000          0.0000                 0.000
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000       0.000       0.000       0.000
+log_k_cyan_free                       0.000       0.000       0.000       0.000
+log_k_cyan_free_bound                 0.000       0.000       0.000       0.000
+log_k_cyan_bound_free                 1.058       0.000       0.000       0.000
+log_k_JCZ38                           0.000       2.382       0.000       0.000
+log_k_J9Z38                           0.000       0.000       1.595       0.000
+log_k_JSE76                           0.000       0.000       0.000       1.245
+f_cyan_ilr_1                          0.000       0.000       0.000       0.000
+f_cyan_ilr_2                          0.000       0.000       0.000       0.000
+f_JCZ38_qlogis                        0.000       0.000       0.000       0.000
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis
+cyan_free_0                 0.0000         0.00           0.00
+log_k_cyan_free             0.0000         0.00           0.00
+log_k_cyan_free_bound       0.0000         0.00           0.00
+log_k_cyan_bound_free       0.0000         0.00           0.00
+log_k_JCZ38                 0.0000         0.00           0.00
+log_k_J9Z38                 0.0000         0.00           0.00
+log_k_JSE76                 0.0000         0.00           0.00
+f_cyan_ilr_1                0.6852         0.00           0.00
+f_cyan_ilr_2                0.0000         1.28           0.00
+f_JCZ38_qlogis              0.0000         0.00          16.13
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2401 2394  -1181
+
+Optimised parameters:
+                             est. lower upper
+cyan_free_0              102.8136    NA    NA
+log_k_cyan_free           -2.7935    NA    NA
+log_k_cyan_free_bound     -2.5440    NA    NA
+log_k_cyan_bound_free     -3.4303    NA    NA
+log_k_JCZ38               -3.5010    NA    NA
+log_k_J9Z38               -5.1226    NA    NA
+log_k_JSE76               -5.6314    NA    NA
+f_cyan_ilr_1               0.6609    NA    NA
+f_cyan_ilr_2               0.5085    NA    NA
+f_JCZ38_qlogis            44.0153    NA    NA
+a.1                        3.2318    NA    NA
+SD.log_k_cyan_free         0.3211    NA    NA
+SD.log_k_cyan_free_bound   0.8408    NA    NA
+SD.log_k_cyan_bound_free   0.5724    NA    NA
+SD.log_k_JCZ38             1.4925    NA    NA
+SD.log_k_J9Z38             0.5816    NA    NA
+SD.log_k_JSE76             0.6037    NA    NA
+SD.f_cyan_ilr_1            0.3115    NA    NA
+SD.f_cyan_ilr_2            0.3436    NA    NA
+SD.f_JCZ38_qlogis          4.8937    NA    NA
+
+Correlation is not available
+
+Random effects:
+                           est. lower upper
+SD.log_k_cyan_free       0.3211    NA    NA
+SD.log_k_cyan_free_bound 0.8408    NA    NA
+SD.log_k_cyan_bound_free 0.5724    NA    NA
+SD.log_k_JCZ38           1.4925    NA    NA
+SD.log_k_J9Z38           0.5816    NA    NA
+SD.log_k_JSE76           0.6037    NA    NA
+SD.f_cyan_ilr_1          0.3115    NA    NA
+SD.f_cyan_ilr_2          0.3436    NA    NA
+SD.f_JCZ38_qlogis        4.8937    NA    NA
+
+Variance model:
+     est. lower upper
+a.1 3.232    NA    NA
+
+Backtransformed parameters:
+                          est. lower upper
+cyan_free_0          1.028e+02    NA    NA
+k_cyan_free          6.120e-02    NA    NA
+k_cyan_free_bound    7.855e-02    NA    NA
+k_cyan_bound_free    3.238e-02    NA    NA
+k_JCZ38              3.017e-02    NA    NA
+k_J9Z38              5.961e-03    NA    NA
+k_JSE76              3.584e-03    NA    NA
+f_cyan_free_to_JCZ38 5.784e-01    NA    NA
+f_cyan_free_to_J9Z38 2.271e-01    NA    NA
+f_JCZ38_to_JSE76     1.000e+00    NA    NA
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+0.15973 0.01241 0.33124 
+
+Resulting formation fractions:
+                    ff
+cyan_free_JCZ38 0.5784
+cyan_free_J9Z38 0.2271
+cyan_free_sink  0.1945
+cyan_free       1.0000
+JCZ38_JSE76     1.0000
+JCZ38_sink      0.0000
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   24.51 153.18    46.11         4.34        55.87
+JCZ38  22.98  76.33       NA           NA           NA
+J9Z38 116.28 386.29       NA           NA           NA
+JSE76 193.42 642.53       NA           NA           NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical SFORB path 1 fit with two-component error
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:46:14 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 664.11 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+             101.3964               -2.9881               -2.7949 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+              -3.4376               -3.3626               -4.9792 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+              -5.8727                0.6814                6.7399 
+       f_JCZ38_qlogis 
+              13.7395 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 5.317          0.0000                 0.000
+log_k_cyan_free             0.000          0.7301                 0.000
+log_k_cyan_free_bound       0.000          0.0000                 1.384
+log_k_cyan_bound_free       0.000          0.0000                 0.000
+log_k_JCZ38                 0.000          0.0000                 0.000
+log_k_J9Z38                 0.000          0.0000                 0.000
+log_k_JSE76                 0.000          0.0000                 0.000
+f_cyan_ilr_1                0.000          0.0000                 0.000
+f_cyan_ilr_2                0.000          0.0000                 0.000
+f_JCZ38_qlogis              0.000          0.0000                 0.000
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000       0.000       0.000       0.000
+log_k_cyan_free                       0.000       0.000       0.000       0.000
+log_k_cyan_free_bound                 0.000       0.000       0.000       0.000
+log_k_cyan_bound_free                 1.109       0.000       0.000       0.000
+log_k_JCZ38                           0.000       2.272       0.000       0.000
+log_k_J9Z38                           0.000       0.000       1.633       0.000
+log_k_JSE76                           0.000       0.000       0.000       1.271
+f_cyan_ilr_1                          0.000       0.000       0.000       0.000
+f_cyan_ilr_2                          0.000       0.000       0.000       0.000
+f_JCZ38_qlogis                        0.000       0.000       0.000       0.000
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis
+cyan_free_0                 0.0000         0.00           0.00
+log_k_cyan_free             0.0000         0.00           0.00
+log_k_cyan_free_bound       0.0000         0.00           0.00
+log_k_cyan_bound_free       0.0000         0.00           0.00
+log_k_JCZ38                 0.0000         0.00           0.00
+log_k_J9Z38                 0.0000         0.00           0.00
+log_k_JSE76                 0.0000         0.00           0.00
+f_cyan_ilr_1                0.6838         0.00           0.00
+f_cyan_ilr_2                0.0000        11.69           0.00
+f_JCZ38_qlogis              0.0000         0.00          16.13
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2400 2392  -1180
+
+Optimised parameters:
+                              est. lower upper
+cyan_free_0              100.56004    NA    NA
+log_k_cyan_free           -3.12657    NA    NA
+log_k_cyan_free_bound     -3.16825    NA    NA
+log_k_cyan_bound_free     -3.66003    NA    NA
+log_k_JCZ38               -3.47278    NA    NA
+log_k_J9Z38               -5.06823    NA    NA
+log_k_JSE76               -5.54327    NA    NA
+f_cyan_ilr_1               0.66631    NA    NA
+f_cyan_ilr_2               0.82898    NA    NA
+f_JCZ38_qlogis            38.31115    NA    NA
+a.1                        2.98352    NA    NA
+b.1                        0.04388    NA    NA
+SD.log_k_cyan_free         0.49145    NA    NA
+SD.log_k_cyan_bound_free   0.27347    NA    NA
+SD.log_k_JCZ38             1.41193    NA    NA
+SD.log_k_J9Z38             0.66073    NA    NA
+SD.log_k_JSE76             0.55885    NA    NA
+SD.f_cyan_ilr_1            0.33020    NA    NA
+SD.f_cyan_ilr_2            0.51367    NA    NA
+SD.f_JCZ38_qlogis          5.52122    NA    NA
+
+Correlation is not available
+
+Random effects:
+                           est. lower upper
+SD.log_k_cyan_free       0.4914    NA    NA
+SD.log_k_cyan_bound_free 0.2735    NA    NA
+SD.log_k_JCZ38           1.4119    NA    NA
+SD.log_k_J9Z38           0.6607    NA    NA
+SD.log_k_JSE76           0.5589    NA    NA
+SD.f_cyan_ilr_1          0.3302    NA    NA
+SD.f_cyan_ilr_2          0.5137    NA    NA
+SD.f_JCZ38_qlogis        5.5212    NA    NA
+
+Variance model:
+       est. lower upper
+a.1 2.98352    NA    NA
+b.1 0.04388    NA    NA
+
+Backtransformed parameters:
+                          est. lower upper
+cyan_free_0          1.006e+02    NA    NA
+k_cyan_free          4.387e-02    NA    NA
+k_cyan_free_bound    4.208e-02    NA    NA
+k_cyan_bound_free    2.573e-02    NA    NA
+k_JCZ38              3.103e-02    NA    NA
+k_J9Z38              6.294e-03    NA    NA
+k_JSE76              3.914e-03    NA    NA
+f_cyan_free_to_JCZ38 6.188e-01    NA    NA
+f_cyan_free_to_J9Z38 2.412e-01    NA    NA
+f_JCZ38_to_JSE76     1.000e+00    NA    NA
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+0.10044 0.01124 0.36580 
+
+Resulting formation fractions:
+                    ff
+cyan_free_JCZ38 0.6188
+cyan_free_J9Z38 0.2412
+cyan_free_sink  0.1400
+cyan_free       1.0000
+JCZ38_JSE76     1.0000
+JCZ38_sink      0.0000
+
+Estimated disappearance times:
+        DT50  DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   26.05 164.4    49.48        6.901        61.67
+JCZ38  22.34  74.2       NA           NA           NA
+J9Z38 110.14 365.9       NA           NA           NA
+JSE76 177.11 588.3       NA           NA           NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical HS path 1 fit with constant variance
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:43:42 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - ifelse(time &lt;= tb, k1, k2) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ifelse(time &lt;= tb, k1, k2) * cyan -
+           k_JCZ38 * JCZ38
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ifelse(time &lt;= tb, k1, k2) * cyan -
+           k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 512.818 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      102.8845        -3.4495        -4.9355        -5.6040         0.6468 
+  f_cyan_ilr_2 f_JCZ38_qlogis         log_k1         log_k2         log_tb 
+        1.2396         9.7220        -2.9079        -4.1810         1.7813 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.406        0.00        0.00       0.000       0.0000
+log_k_JCZ38     0.000        2.33        0.00       0.000       0.0000
+log_k_J9Z38     0.000        0.00        1.59       0.000       0.0000
+log_k_JSE76     0.000        0.00        0.00       1.013       0.0000
+f_cyan_ilr_1    0.000        0.00        0.00       0.000       0.6367
+f_cyan_ilr_2    0.000        0.00        0.00       0.000       0.0000
+f_JCZ38_qlogis  0.000        0.00        0.00       0.000       0.0000
+log_k1          0.000        0.00        0.00       0.000       0.0000
+log_k2          0.000        0.00        0.00       0.000       0.0000
+log_tb          0.000        0.00        0.00       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis log_k1 log_k2 log_tb
+cyan_0                0.000           0.00 0.0000 0.0000 0.0000
+log_k_JCZ38           0.000           0.00 0.0000 0.0000 0.0000
+log_k_J9Z38           0.000           0.00 0.0000 0.0000 0.0000
+log_k_JSE76           0.000           0.00 0.0000 0.0000 0.0000
+f_cyan_ilr_1          0.000           0.00 0.0000 0.0000 0.0000
+f_cyan_ilr_2          2.038           0.00 0.0000 0.0000 0.0000
+f_JCZ38_qlogis        0.000          10.33 0.0000 0.0000 0.0000
+log_k1                0.000           0.00 0.7006 0.0000 0.0000
+log_k2                0.000           0.00 0.0000 0.8928 0.0000
+log_tb                0.000           0.00 0.0000 0.0000 0.6773
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2427 2419  -1194
+
+Optimised parameters:
+                      est.      lower      upper
+cyan_0            101.9660  1.005e+02  1.035e+02
+log_k_JCZ38        -3.4698 -4.716e+00 -2.224e+00
+log_k_J9Z38        -5.0947 -5.740e+00 -4.450e+00
+log_k_JSE76        -5.5977 -6.321e+00 -4.875e+00
+f_cyan_ilr_1        0.6595  3.734e-01  9.456e-01
+f_cyan_ilr_2        0.5905  1.664e-01  1.015e+00
+f_JCZ38_qlogis     25.8627 -4.224e+05  4.225e+05
+log_k1             -3.0884 -3.453e+00 -2.723e+00
+log_k2             -4.3877 -4.778e+00 -3.998e+00
+log_tb              2.3057  1.715e+00  2.896e+00
+a.1                 3.3228         NA         NA
+SD.log_k_JCZ38      1.4071         NA         NA
+SD.log_k_J9Z38      0.5774         NA         NA
+SD.log_k_JSE76      0.6214         NA         NA
+SD.f_cyan_ilr_1     0.3058         NA         NA
+SD.f_cyan_ilr_2     0.3470         NA         NA
+SD.f_JCZ38_qlogis   0.0644         NA         NA
+SD.log_k1           0.3994         NA         NA
+SD.log_k2           0.4373         NA         NA
+SD.log_tb           0.6419         NA         NA
+
+Correlation is not available
+
+Random effects:
+                    est. lower upper
+SD.log_k_JCZ38    1.4071    NA    NA
+SD.log_k_J9Z38    0.5774    NA    NA
+SD.log_k_JSE76    0.6214    NA    NA
+SD.f_cyan_ilr_1   0.3058    NA    NA
+SD.f_cyan_ilr_2   0.3470    NA    NA
+SD.f_JCZ38_qlogis 0.0644    NA    NA
+SD.log_k1         0.3994    NA    NA
+SD.log_k2         0.4373    NA    NA
+SD.log_tb         0.6419    NA    NA
+
+Variance model:
+     est. lower upper
+a.1 3.323    NA    NA
+
+Backtransformed parameters:
+                      est.     lower     upper
+cyan_0           1.020e+02 1.005e+02 1.035e+02
+k_JCZ38          3.112e-02 8.951e-03 1.082e-01
+k_J9Z38          6.129e-03 3.216e-03 1.168e-02
+k_JSE76          3.706e-03 1.798e-03 7.639e-03
+f_cyan_to_JCZ38  5.890e-01        NA        NA
+f_cyan_to_J9Z38  2.318e-01        NA        NA
+f_JCZ38_to_JSE76 1.000e+00 0.000e+00 1.000e+00
+k1               4.558e-02 3.164e-02 6.565e-02
+k2               1.243e-02 8.417e-03 1.835e-02
+tb               1.003e+01 5.557e+00 1.811e+01
+
+Resulting formation fractions:
+                   ff
+cyan_JCZ38  5.890e-01
+cyan_J9Z38  2.318e-01
+cyan_sink   1.793e-01
+JCZ38_JSE76 1.000e+00
+JCZ38_sink  5.861e-12
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   29.02 158.51    47.72   15.21   55.77
+JCZ38  22.27  73.98       NA      NA      NA
+J9Z38 113.09 375.69       NA      NA      NA
+JSE76 187.01 621.23       NA      NA      NA
+
+</code></pre>
+<p></p>
+</div>
+<div class="section level4">
+<h4 id="pathway-2">Pathway 2<a class="anchor" aria-label="anchor" href="#pathway-2"></a>
+</h4>
+<caption>
+Hierarchical FOMC path 2 fit with two-component error
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:54:36 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - (alpha/beta) * 1/((time/beta) + 1) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_JCZ38 * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 491.928 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      102.4477        -1.8631        -5.1087        -2.5114         0.6826 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis      log_alpha       log_beta 
+        4.7944        15.9616        13.1566        -0.1564         2.9781 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          7.701       0.000       0.000       0.000       0.0000
+log_k_JCZ38     0.000       1.448       0.000       0.000       0.0000
+log_k_J9Z38     0.000       0.000       1.724       0.000       0.0000
+log_k_JSE76     0.000       0.000       0.000       3.659       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000       0.000       0.6356
+f_cyan_ilr_2    0.000       0.000       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000       0.000       0.0000
+f_JSE76_qlogis  0.000       0.000       0.000       0.000       0.0000
+log_alpha       0.000       0.000       0.000       0.000       0.0000
+log_beta        0.000       0.000       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_alpha log_beta
+cyan_0                 0.00           0.00           0.00    0.0000   0.0000
+log_k_JCZ38            0.00           0.00           0.00    0.0000   0.0000
+log_k_J9Z38            0.00           0.00           0.00    0.0000   0.0000
+log_k_JSE76            0.00           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_1           0.00           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_2          10.32           0.00           0.00    0.0000   0.0000
+f_JCZ38_qlogis         0.00          12.23           0.00    0.0000   0.0000
+f_JSE76_qlogis         0.00           0.00          14.99    0.0000   0.0000
+log_alpha              0.00           0.00           0.00    0.3924   0.0000
+log_beta               0.00           0.00           0.00    0.0000   0.5639
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2249 2241  -1104
+
+Optimised parameters:
+                       est.      lower     upper
+cyan_0            101.55265  9.920e+01  103.9059
+log_k_JCZ38        -2.32302 -2.832e+00   -1.8142
+log_k_J9Z38        -5.13082 -5.942e+00   -4.3199
+log_k_JSE76        -3.01756 -4.262e+00   -1.7736
+f_cyan_ilr_1        0.70850  3.657e-01    1.0513
+f_cyan_ilr_2        0.95775  2.612e-01    1.6543
+f_JCZ38_qlogis      3.86105  9.248e-01    6.7973
+f_JSE76_qlogis      7.51583 -1.120e+02  127.0392
+log_alpha          -0.15308 -4.508e-01    0.1446
+log_beta            2.99165  2.711e+00    3.2720
+a.1                 2.04034  1.843e+00    2.2382
+b.1                 0.06924  5.749e-02    0.0810
+SD.log_k_JCZ38      0.50818  1.390e-01    0.8774
+SD.log_k_J9Z38      0.86597  2.652e-01    1.4667
+SD.log_k_JSE76      1.38092  4.864e-01    2.2754
+SD.f_cyan_ilr_1     0.38204  1.354e-01    0.6286
+SD.f_cyan_ilr_2     0.55129  7.198e-02    1.0306
+SD.f_JCZ38_qlogis   1.88457  1.711e-02    3.7520
+SD.f_JSE76_qlogis   2.64018 -2.450e+03 2454.9447
+SD.log_alpha        0.31860  1.047e-01    0.5325
+SD.log_beta         0.24195  1.273e-02    0.4712
+
+Correlation: 
+               cyan_0  l__JCZ3 l__J9Z3 l__JSE7 f_cy__1 f_cy__2 f_JCZ38 f_JSE76
+log_k_JCZ38    -0.0235                                                        
+log_k_J9Z38    -0.0442  0.0047                                                
+log_k_JSE76    -0.0023  0.0966  0.0006                                        
+f_cyan_ilr_1   -0.0032  0.0070 -0.0536 -0.0001                                
+f_cyan_ilr_2   -0.5189  0.0452  0.1152  0.0013 -0.0304                        
+f_JCZ38_qlogis  0.1088 -0.0848 -0.0240  0.0040 -0.0384 -0.2303                
+f_JSE76_qlogis -0.0545  0.1315  0.0195  0.0020  0.0252  0.1737 -0.5939        
+log_alpha      -0.0445  0.0056  0.0261  0.0019 -0.0055  0.0586 -0.0239 -0.0284
+log_beta       -0.2388  0.0163  0.0566  0.0040 -0.0078  0.2183 -0.0714 -0.0332
+               log_lph
+log_k_JCZ38           
+log_k_J9Z38           
+log_k_JSE76           
+f_cyan_ilr_1          
+f_cyan_ilr_2          
+f_JCZ38_qlogis        
+f_JSE76_qlogis        
+log_alpha             
+log_beta        0.2135
+
+Random effects:
+                    est.      lower     upper
+SD.log_k_JCZ38    0.5082  1.390e-01    0.8774
+SD.log_k_J9Z38    0.8660  2.652e-01    1.4667
+SD.log_k_JSE76    1.3809  4.864e-01    2.2754
+SD.f_cyan_ilr_1   0.3820  1.354e-01    0.6286
+SD.f_cyan_ilr_2   0.5513  7.198e-02    1.0306
+SD.f_JCZ38_qlogis 1.8846  1.711e-02    3.7520
+SD.f_JSE76_qlogis 2.6402 -2.450e+03 2454.9447
+SD.log_alpha      0.3186  1.047e-01    0.5325
+SD.log_beta       0.2420  1.273e-02    0.4712
+
+Variance model:
+       est.   lower upper
+a.1 2.04034 1.84252 2.238
+b.1 0.06924 0.05749 0.081
+
+Backtransformed parameters:
+                      est.     lower    upper
+cyan_0           1.016e+02 9.920e+01 103.9059
+k_JCZ38          9.798e-02 5.890e-02   0.1630
+k_J9Z38          5.912e-03 2.627e-03   0.0133
+k_JSE76          4.892e-02 1.410e-02   0.1697
+f_cyan_to_JCZ38  6.432e-01        NA       NA
+f_cyan_to_J9Z38  2.362e-01        NA       NA
+f_JCZ38_to_JSE76 9.794e-01 7.160e-01   0.9989
+f_JSE76_to_JCZ38 9.995e-01 2.268e-49   1.0000
+alpha            8.581e-01 6.371e-01   1.1556
+beta             1.992e+01 1.505e+01  26.3646
+
+Resulting formation fractions:
+                   ff
+cyan_JCZ38  0.6432301
+cyan_J9Z38  0.2361657
+cyan_sink   0.1206042
+JCZ38_JSE76 0.9793879
+JCZ38_sink  0.0206121
+JSE76_JCZ38 0.9994559
+JSE76_sink  0.0005441
+
+Estimated disappearance times:
+         DT50   DT90 DT50back
+cyan   24.759 271.61    81.76
+JCZ38   7.075  23.50       NA
+J9Z38 117.249 389.49       NA
+JSE76  14.169  47.07       NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical DFOP path 2 fit with constant variance
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:55:32 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38 +
+           f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 548.554 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      102.4380        -2.3107        -5.3123        -3.7120         0.6757 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis         log_k1         log_k2 
+        1.1439        13.1194        12.3492        -1.9317        -4.4557 
+      g_qlogis 
+       -0.5644 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          4.591      0.0000       0.000         0.0       0.0000
+log_k_JCZ38     0.000      0.7966       0.000         0.0       0.0000
+log_k_J9Z38     0.000      0.0000       1.561         0.0       0.0000
+log_k_JSE76     0.000      0.0000       0.000         0.8       0.0000
+f_cyan_ilr_1    0.000      0.0000       0.000         0.0       0.6349
+f_cyan_ilr_2    0.000      0.0000       0.000         0.0       0.0000
+f_JCZ38_qlogis  0.000      0.0000       0.000         0.0       0.0000
+f_JSE76_qlogis  0.000      0.0000       0.000         0.0       0.0000
+log_k1          0.000      0.0000       0.000         0.0       0.0000
+log_k2          0.000      0.0000       0.000         0.0       0.0000
+g_qlogis        0.000      0.0000       0.000         0.0       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_k1 log_k2
+cyan_0                0.000           0.00           0.00  0.000 0.0000
+log_k_JCZ38           0.000           0.00           0.00  0.000 0.0000
+log_k_J9Z38           0.000           0.00           0.00  0.000 0.0000
+log_k_JSE76           0.000           0.00           0.00  0.000 0.0000
+f_cyan_ilr_1          0.000           0.00           0.00  0.000 0.0000
+f_cyan_ilr_2          1.797           0.00           0.00  0.000 0.0000
+f_JCZ38_qlogis        0.000          13.86           0.00  0.000 0.0000
+f_JSE76_qlogis        0.000           0.00          13.91  0.000 0.0000
+log_k1                0.000           0.00           0.00  1.106 0.0000
+log_k2                0.000           0.00           0.00  0.000 0.6141
+g_qlogis              0.000           0.00           0.00  0.000 0.0000
+               g_qlogis
+cyan_0            0.000
+log_k_JCZ38       0.000
+log_k_J9Z38       0.000
+log_k_JSE76       0.000
+f_cyan_ilr_1      0.000
+f_cyan_ilr_2      0.000
+f_JCZ38_qlogis    0.000
+f_JSE76_qlogis    0.000
+log_k1            0.000
+log_k2            0.000
+g_qlogis          1.595
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2288 2280  -1122
+
+Optimised parameters:
+                      est.      lower      upper
+cyan_0            102.7204  1.014e+02  1.040e+02
+log_k_JCZ38        -2.8925 -4.044e+00 -1.741e+00
+log_k_J9Z38        -5.1430 -5.828e+00 -4.457e+00
+log_k_JSE76        -3.5577 -4.174e+00 -2.941e+00
+f_cyan_ilr_1        0.6929  3.788e-01  1.007e+00
+f_cyan_ilr_2        0.6066  5.342e-02  1.160e+00
+f_JCZ38_qlogis      9.8071 -2.819e+03  2.838e+03
+f_JSE76_qlogis      2.2229  5.684e-01  3.877e+00
+log_k1             -1.9339 -2.609e+00 -1.258e+00
+log_k2             -4.4709 -4.935e+00 -4.007e+00
+g_qlogis           -0.4987 -1.373e+00  3.757e-01
+a.1                 2.7368  2.545e+00  2.928e+00
+SD.log_k_JCZ38      1.2747  4.577e-01  2.092e+00
+SD.log_k_J9Z38      0.6758  1.418e-01  1.210e+00
+SD.log_k_JSE76      0.5869  1.169e-01  1.057e+00
+SD.f_cyan_ilr_1     0.3392  1.161e-01  5.622e-01
+SD.f_cyan_ilr_2     0.4200  8.501e-02  7.550e-01
+SD.f_JCZ38_qlogis   0.8511 -1.137e+06  1.137e+06
+SD.f_JSE76_qlogis   0.3767 -5.238e-01  1.277e+00
+SD.log_k1           0.7475  2.601e-01  1.235e+00
+SD.log_k2           0.5179  1.837e-01  8.521e-01
+SD.g_qlogis         0.9817  3.553e-01  1.608e+00
+
+Correlation: 
+               cyan_0  l__JCZ3 l__J9Z3 l__JSE7 f_cy__1 f_cy__2 f_JCZ38 f_JSE76
+log_k_JCZ38    -0.0351                                                        
+log_k_J9Z38    -0.0541  0.0043                                                
+log_k_JSE76    -0.0078  0.0900 -0.0014                                        
+f_cyan_ilr_1   -0.0249  0.0268 -0.0962  0.0000                                
+f_cyan_ilr_2   -0.3560  0.0848  0.1545 -0.0022  0.0463                        
+f_JCZ38_qlogis  0.2005 -0.1226 -0.0347  0.0514 -0.1840 -0.5906                
+f_JSE76_qlogis -0.1638  0.1307  0.0266  0.0001  0.1645  0.5181 -0.9297        
+log_k1          0.0881 -0.0071  0.0005 -0.0070 -0.0064 -0.0346  0.0316 -0.0341
+log_k2          0.0238 -0.0003  0.0082 -0.0022 -0.0017 -0.0017 -0.0002 -0.0076
+g_qlogis        0.0198 -0.0002 -0.0109  0.0034  0.0017 -0.0176  0.0044  0.0051
+               log_k1  log_k2 
+log_k_JCZ38                   
+log_k_J9Z38                   
+log_k_JSE76                   
+f_cyan_ilr_1                  
+f_cyan_ilr_2                  
+f_JCZ38_qlogis                
+f_JSE76_qlogis                
+log_k1                        
+log_k2          0.0276        
+g_qlogis       -0.0283 -0.0309
+
+Random effects:
+                    est.      lower     upper
+SD.log_k_JCZ38    1.2747  4.577e-01 2.092e+00
+SD.log_k_J9Z38    0.6758  1.418e-01 1.210e+00
+SD.log_k_JSE76    0.5869  1.169e-01 1.057e+00
+SD.f_cyan_ilr_1   0.3392  1.161e-01 5.622e-01
+SD.f_cyan_ilr_2   0.4200  8.501e-02 7.550e-01
+SD.f_JCZ38_qlogis 0.8511 -1.137e+06 1.137e+06
+SD.f_JSE76_qlogis 0.3767 -5.238e-01 1.277e+00
+SD.log_k1         0.7475  2.601e-01 1.235e+00
+SD.log_k2         0.5179  1.837e-01 8.521e-01
+SD.g_qlogis       0.9817  3.553e-01 1.608e+00
+
+Variance model:
+     est. lower upper
+a.1 2.737 2.545 2.928
+
+Backtransformed parameters:
+                      est.     lower     upper
+cyan_0           102.72037 1.014e+02 104.00464
+k_JCZ38            0.05544 1.752e-02   0.17539
+k_J9Z38            0.00584 2.942e-03   0.01159
+k_JSE76            0.02850 1.539e-02   0.05279
+f_cyan_to_JCZ38    0.59995        NA        NA
+f_cyan_to_J9Z38    0.22519        NA        NA
+f_JCZ38_to_JSE76   0.99994 0.000e+00   1.00000
+f_JSE76_to_JCZ38   0.90229 6.384e-01   0.97971
+k1                 0.14459 7.357e-02   0.28414
+k2                 0.01144 7.192e-03   0.01819
+g                  0.37784 2.021e-01   0.59284
+
+Resulting formation fractions:
+                   ff
+cyan_JCZ38  5.999e-01
+cyan_J9Z38  2.252e-01
+cyan_sink   1.749e-01
+JCZ38_JSE76 9.999e-01
+JCZ38_sink  5.506e-05
+JSE76_JCZ38 9.023e-01
+JSE76_sink  9.771e-02
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   21.93 159.83    48.11   4.794    60.6
+JCZ38  12.50  41.53       NA      NA      NA
+J9Z38 118.69 394.27       NA      NA      NA
+JSE76  24.32  80.78       NA      NA      NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical DFOP path 2 fit with two-component error
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:57:56 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38 +
+           f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 691.67 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      101.7393        -1.4493        -5.0118        -2.1269         0.6720 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis         log_k1         log_k2 
+        7.3362        13.4423        13.2659        -2.0061        -4.5527 
+      g_qlogis 
+       -0.5806 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.604        0.00       0.000       0.000       0.0000
+log_k_JCZ38     0.000        2.77       0.000       0.000       0.0000
+log_k_J9Z38     0.000        0.00       1.662       0.000       0.0000
+log_k_JSE76     0.000        0.00       0.000       5.021       0.0000
+f_cyan_ilr_1    0.000        0.00       0.000       0.000       0.6519
+f_cyan_ilr_2    0.000        0.00       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000        0.00       0.000       0.000       0.0000
+f_JSE76_qlogis  0.000        0.00       0.000       0.000       0.0000
+log_k1          0.000        0.00       0.000       0.000       0.0000
+log_k2          0.000        0.00       0.000       0.000       0.0000
+g_qlogis        0.000        0.00       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_k1 log_k2
+cyan_0                 0.00           0.00           0.00 0.0000 0.0000
+log_k_JCZ38            0.00           0.00           0.00 0.0000 0.0000
+log_k_J9Z38            0.00           0.00           0.00 0.0000 0.0000
+log_k_JSE76            0.00           0.00           0.00 0.0000 0.0000
+f_cyan_ilr_1           0.00           0.00           0.00 0.0000 0.0000
+f_cyan_ilr_2          13.37           0.00           0.00 0.0000 0.0000
+f_JCZ38_qlogis         0.00          14.21           0.00 0.0000 0.0000
+f_JSE76_qlogis         0.00           0.00          14.58 0.0000 0.0000
+log_k1                 0.00           0.00           0.00 0.8453 0.0000
+log_k2                 0.00           0.00           0.00 0.0000 0.5969
+g_qlogis               0.00           0.00           0.00 0.0000 0.0000
+               g_qlogis
+cyan_0             0.00
+log_k_JCZ38        0.00
+log_k_J9Z38        0.00
+log_k_JSE76        0.00
+f_cyan_ilr_1       0.00
+f_cyan_ilr_2       0.00
+f_JCZ38_qlogis     0.00
+f_JSE76_qlogis     0.00
+log_k1             0.00
+log_k2             0.00
+g_qlogis           1.69
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2234 2226  -1095
+
+Optimised parameters:
+                       est.     lower     upper
+cyan_0            101.25496  99.14662 103.36331
+log_k_JCZ38        -2.55593  -3.32972  -1.78215
+log_k_J9Z38        -5.07103  -5.85423  -4.28783
+log_k_JSE76        -3.25468  -4.17577  -2.33360
+f_cyan_ilr_1        0.70139   0.35924   1.04355
+f_cyan_ilr_2        1.07712   0.17789   1.97636
+f_JCZ38_qlogis      3.57483   0.05990   7.08976
+f_JSE76_qlogis      4.54884  -7.25628  16.35395
+log_k1             -2.38201  -2.51639  -2.24763
+log_k2             -4.66741  -4.91865  -4.41617
+g_qlogis           -0.28446  -1.14192   0.57300
+a.1                 2.05925   1.86481   2.25369
+b.1                 0.06172   0.05062   0.07282
+SD.log_k_JCZ38      0.81137   0.25296   1.36977
+SD.log_k_J9Z38      0.83542   0.25395   1.41689
+SD.log_k_JSE76      0.97903   0.30100   1.65707
+SD.f_cyan_ilr_1     0.37878   0.13374   0.62382
+SD.f_cyan_ilr_2     0.67274   0.10102   1.24446
+SD.f_JCZ38_qlogis   1.35327  -0.42359   3.13012
+SD.f_JSE76_qlogis   1.43956 -19.14972  22.02884
+SD.log_k2           0.25329   0.07521   0.43138
+SD.g_qlogis         0.95167   0.35149   1.55184
+
+Correlation: 
+               cyan_0  l__JCZ3 l__J9Z3 l__JSE7 f_cy__1 f_cy__2 f_JCZ38 f_JSE76
+log_k_JCZ38    -0.0265                                                        
+log_k_J9Z38    -0.0392  0.0024                                                
+log_k_JSE76     0.0011  0.1220 -0.0016                                        
+f_cyan_ilr_1   -0.0161  0.0217 -0.0552  0.0034                                
+f_cyan_ilr_2   -0.4718  0.0829  0.1102  0.0042  0.0095                        
+f_JCZ38_qlogis  0.1609 -0.1318 -0.0277  0.0081 -0.1040 -0.4559                
+f_JSE76_qlogis -0.1289  0.1494  0.0219  0.0012  0.1004  0.4309 -0.8543        
+log_k1          0.2618 -0.0739 -0.0167 -0.0148 -0.0444 -0.2768  0.3518 -0.3818
+log_k2          0.0603 -0.0217  0.0174 -0.0058 -0.0197 -0.0533  0.0923 -0.1281
+g_qlogis        0.0362  0.0115 -0.0111  0.0040  0.0095 -0.0116 -0.0439  0.0651
+               log_k1  log_k2 
+log_k_JCZ38                   
+log_k_J9Z38                   
+log_k_JSE76                   
+f_cyan_ilr_1                  
+f_cyan_ilr_2                  
+f_JCZ38_qlogis                
+f_JSE76_qlogis                
+log_k1                        
+log_k2          0.3269        
+g_qlogis       -0.1656 -0.0928
+
+Random effects:
+                    est.     lower   upper
+SD.log_k_JCZ38    0.8114   0.25296  1.3698
+SD.log_k_J9Z38    0.8354   0.25395  1.4169
+SD.log_k_JSE76    0.9790   0.30100  1.6571
+SD.f_cyan_ilr_1   0.3788   0.13374  0.6238
+SD.f_cyan_ilr_2   0.6727   0.10102  1.2445
+SD.f_JCZ38_qlogis 1.3533  -0.42359  3.1301
+SD.f_JSE76_qlogis 1.4396 -19.14972 22.0288
+SD.log_k2         0.2533   0.07521  0.4314
+SD.g_qlogis       0.9517   0.35149  1.5518
+
+Variance model:
+       est.   lower   upper
+a.1 2.05925 1.86481 2.25369
+b.1 0.06172 0.05062 0.07282
+
+Backtransformed parameters:
+                      est.     lower     upper
+cyan_0           1.013e+02 9.915e+01 103.36331
+k_JCZ38          7.762e-02 3.580e-02   0.16828
+k_J9Z38          6.276e-03 2.868e-03   0.01373
+k_JSE76          3.859e-02 1.536e-02   0.09695
+f_cyan_to_JCZ38  6.520e-01        NA        NA
+f_cyan_to_J9Z38  2.418e-01        NA        NA
+f_JCZ38_to_JSE76 9.727e-01 5.150e-01   0.99917
+f_JSE76_to_JCZ38 9.895e-01 7.052e-04   1.00000
+k1               9.236e-02 8.075e-02   0.10565
+k2               9.397e-03 7.309e-03   0.01208
+g                4.294e-01 2.420e-01   0.63945
+
+Resulting formation fractions:
+                 ff
+cyan_JCZ38  0.65203
+cyan_J9Z38  0.24181
+cyan_sink   0.10616
+JCZ38_JSE76 0.97274
+JCZ38_sink  0.02726
+JSE76_JCZ38 0.98953
+JSE76_sink  0.01047
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   24.26 185.34    55.79   7.504   73.77
+JCZ38   8.93  29.66       NA      NA      NA
+J9Z38 110.45 366.89       NA      NA      NA
+JSE76  17.96  59.66       NA      NA      NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical SFORB path 2 fit with constant variance
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:55:26 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 542.162 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+             102.4395               -2.7673               -2.8942 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+              -3.6201               -2.3107               -5.3123 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+              -3.7120                0.6754                1.1448 
+       f_JCZ38_qlogis        f_JSE76_qlogis 
+              14.8408               15.4734 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 4.589          0.0000                  0.00
+log_k_cyan_free             0.000          0.4849                  0.00
+log_k_cyan_free_bound       0.000          0.0000                  1.62
+log_k_cyan_bound_free       0.000          0.0000                  0.00
+log_k_JCZ38                 0.000          0.0000                  0.00
+log_k_J9Z38                 0.000          0.0000                  0.00
+log_k_JSE76                 0.000          0.0000                  0.00
+f_cyan_ilr_1                0.000          0.0000                  0.00
+f_cyan_ilr_2                0.000          0.0000                  0.00
+f_JCZ38_qlogis              0.000          0.0000                  0.00
+f_JSE76_qlogis              0.000          0.0000                  0.00
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000      0.0000       0.000         0.0
+log_k_cyan_free                       0.000      0.0000       0.000         0.0
+log_k_cyan_free_bound                 0.000      0.0000       0.000         0.0
+log_k_cyan_bound_free                 1.197      0.0000       0.000         0.0
+log_k_JCZ38                           0.000      0.7966       0.000         0.0
+log_k_J9Z38                           0.000      0.0000       1.561         0.0
+log_k_JSE76                           0.000      0.0000       0.000         0.8
+f_cyan_ilr_1                          0.000      0.0000       0.000         0.0
+f_cyan_ilr_2                          0.000      0.0000       0.000         0.0
+f_JCZ38_qlogis                        0.000      0.0000       0.000         0.0
+f_JSE76_qlogis                        0.000      0.0000       0.000         0.0
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis
+cyan_free_0                 0.0000        0.000            0.0           0.00
+log_k_cyan_free             0.0000        0.000            0.0           0.00
+log_k_cyan_free_bound       0.0000        0.000            0.0           0.00
+log_k_cyan_bound_free       0.0000        0.000            0.0           0.00
+log_k_JCZ38                 0.0000        0.000            0.0           0.00
+log_k_J9Z38                 0.0000        0.000            0.0           0.00
+log_k_JSE76                 0.0000        0.000            0.0           0.00
+f_cyan_ilr_1                0.6349        0.000            0.0           0.00
+f_cyan_ilr_2                0.0000        1.797            0.0           0.00
+f_JCZ38_qlogis              0.0000        0.000           15.6           0.00
+f_JSE76_qlogis              0.0000        0.000            0.0          17.52
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2283 2275  -1120
+
+Optimised parameters:
+                             est.     lower    upper
+cyan_free_0              102.6517 101.40815 103.8952
+log_k_cyan_free           -2.8729  -3.18649  -2.5593
+log_k_cyan_free_bound     -2.7803  -3.60525  -1.9552
+log_k_cyan_bound_free     -3.5845  -4.16644  -3.0026
+log_k_JCZ38               -2.3411  -2.89698  -1.7852
+log_k_J9Z38               -5.2487  -6.01271  -4.4847
+log_k_JSE76               -3.0259  -4.28274  -1.7690
+f_cyan_ilr_1               0.7289   0.38214   1.0756
+f_cyan_ilr_2               0.6891   0.18277   1.1954
+f_JCZ38_qlogis             4.2162   0.47015   7.9622
+f_JSE76_qlogis             5.8911 -20.19088  31.9730
+a.1                        2.7159   2.52587   2.9060
+SD.log_k_cyan_free         0.3354   0.10979   0.5610
+SD.log_k_cyan_free_bound   0.9061   0.30969   1.5025
+SD.log_k_cyan_bound_free   0.6376   0.21229   1.0628
+SD.log_k_JCZ38             0.5499   0.14533   0.9545
+SD.log_k_J9Z38             0.7457   0.15106   1.3404
+SD.log_k_JSE76             1.3822   0.47329   2.2912
+SD.f_cyan_ilr_1            0.3820   0.13280   0.6313
+SD.f_cyan_ilr_2            0.4317   0.06803   0.7953
+SD.f_JCZ38_qlogis          1.8258  -0.25423   3.9059
+SD.f_JSE76_qlogis          2.2348 -83.33679  87.8065
+
+Correlation: 
+                      cyn_f_0 lg_k_c_ lg_k_cyn_f_ lg_k_cyn_b_ l__JCZ3 l__J9Z3
+log_k_cyan_free        0.1944                                                
+log_k_cyan_free_bound  0.0815  0.0814                                        
+log_k_cyan_bound_free  0.0106  0.0426  0.0585                                
+log_k_JCZ38           -0.0231 -0.0106 -0.0089     -0.0051                    
+log_k_J9Z38           -0.0457 -0.0108  0.0019      0.0129      0.0032        
+log_k_JSE76           -0.0054 -0.0024 -0.0017     -0.0005      0.1108  0.0009
+f_cyan_ilr_1           0.0051 -0.0005 -0.0035     -0.0056      0.0131 -0.0967
+f_cyan_ilr_2          -0.3182 -0.0771 -0.0309     -0.0038      0.0680  0.1643
+f_JCZ38_qlogis         0.0834  0.0369  0.0302      0.0172     -0.1145 -0.0204
+f_JSE76_qlogis        -0.0553 -0.0365 -0.0441     -0.0414      0.1579  0.0175
+                      l__JSE7 f_cy__1 f_cy__2 f_JCZ38
+log_k_cyan_free                                      
+log_k_cyan_free_bound                                
+log_k_cyan_bound_free                                
+log_k_JCZ38                                          
+log_k_J9Z38                                          
+log_k_JSE76                                          
+f_cyan_ilr_1          -0.0002                        
+f_cyan_ilr_2           0.0020 -0.0415                
+f_JCZ38_qlogis         0.0052 -0.0665 -0.3437        
+f_JSE76_qlogis         0.0066  0.0635  0.3491 -0.7487
+
+Random effects:
+                           est.     lower   upper
+SD.log_k_cyan_free       0.3354   0.10979  0.5610
+SD.log_k_cyan_free_bound 0.9061   0.30969  1.5025
+SD.log_k_cyan_bound_free 0.6376   0.21229  1.0628
+SD.log_k_JCZ38           0.5499   0.14533  0.9545
+SD.log_k_J9Z38           0.7457   0.15106  1.3404
+SD.log_k_JSE76           1.3822   0.47329  2.2912
+SD.f_cyan_ilr_1          0.3820   0.13280  0.6313
+SD.f_cyan_ilr_2          0.4317   0.06803  0.7953
+SD.f_JCZ38_qlogis        1.8258  -0.25423  3.9059
+SD.f_JSE76_qlogis        2.2348 -83.33679 87.8065
+
+Variance model:
+     est. lower upper
+a.1 2.716 2.526 2.906
+
+Backtransformed parameters:
+                          est.     lower     upper
+cyan_free_0          1.027e+02 1.014e+02 103.89517
+k_cyan_free          5.654e-02 4.132e-02   0.07736
+k_cyan_free_bound    6.202e-02 2.718e-02   0.14153
+k_cyan_bound_free    2.775e-02 1.551e-02   0.04966
+k_JCZ38              9.622e-02 5.519e-02   0.16777
+k_J9Z38              5.254e-03 2.447e-03   0.01128
+k_JSE76              4.852e-02 1.380e-02   0.17051
+f_cyan_free_to_JCZ38 6.197e-01 5.643e-01   0.84429
+f_cyan_free_to_J9Z38 2.211e-01 5.643e-01   0.84429
+f_JCZ38_to_JSE76     9.855e-01 6.154e-01   0.99965
+f_JSE76_to_JCZ38     9.972e-01 1.703e-09   1.00000
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+0.13466 0.01165 0.36490 
+
+Resulting formation fractions:
+                      ff
+cyan_free_JCZ38 0.619745
+cyan_free_J9Z38 0.221083
+cyan_free_sink  0.159172
+cyan_free       1.000000
+JCZ38_JSE76     0.985460
+JCZ38_sink      0.014540
+JSE76_JCZ38     0.997244
+JSE76_sink      0.002756
+
+Estimated disappearance times:
+         DT50   DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   23.293 158.67    47.77        5.147         59.5
+JCZ38   7.203  23.93       NA           NA           NA
+J9Z38 131.918 438.22       NA           NA           NA
+JSE76  14.287  47.46       NA           NA           NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical SFORB path 2 fit with two-component error
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 07:58:06 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 701.582 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+             101.7511               -2.8370               -3.0162 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+              -3.6600               -2.2988               -5.3129 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+              -3.6991                0.6722                4.8596 
+       f_JCZ38_qlogis        f_JSE76_qlogis 
+              13.4678               14.2149 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 5.629           0.000                 0.000
+log_k_cyan_free             0.000           0.446                 0.000
+log_k_cyan_free_bound       0.000           0.000                 1.449
+log_k_cyan_bound_free       0.000           0.000                 0.000
+log_k_JCZ38                 0.000           0.000                 0.000
+log_k_J9Z38                 0.000           0.000                 0.000
+log_k_JSE76                 0.000           0.000                 0.000
+f_cyan_ilr_1                0.000           0.000                 0.000
+f_cyan_ilr_2                0.000           0.000                 0.000
+f_JCZ38_qlogis              0.000           0.000                 0.000
+f_JSE76_qlogis              0.000           0.000                 0.000
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000      0.0000       0.000      0.0000
+log_k_cyan_free                       0.000      0.0000       0.000      0.0000
+log_k_cyan_free_bound                 0.000      0.0000       0.000      0.0000
+log_k_cyan_bound_free                 1.213      0.0000       0.000      0.0000
+log_k_JCZ38                           0.000      0.7801       0.000      0.0000
+log_k_J9Z38                           0.000      0.0000       1.575      0.0000
+log_k_JSE76                           0.000      0.0000       0.000      0.8078
+f_cyan_ilr_1                          0.000      0.0000       0.000      0.0000
+f_cyan_ilr_2                          0.000      0.0000       0.000      0.0000
+f_JCZ38_qlogis                        0.000      0.0000       0.000      0.0000
+f_JSE76_qlogis                        0.000      0.0000       0.000      0.0000
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis
+cyan_free_0                 0.0000        0.000           0.00           0.00
+log_k_cyan_free             0.0000        0.000           0.00           0.00
+log_k_cyan_free_bound       0.0000        0.000           0.00           0.00
+log_k_cyan_bound_free       0.0000        0.000           0.00           0.00
+log_k_JCZ38                 0.0000        0.000           0.00           0.00
+log_k_J9Z38                 0.0000        0.000           0.00           0.00
+log_k_JSE76                 0.0000        0.000           0.00           0.00
+f_cyan_ilr_1                0.6518        0.000           0.00           0.00
+f_cyan_ilr_2                0.0000        9.981           0.00           0.00
+f_JCZ38_qlogis              0.0000        0.000          14.26           0.00
+f_JSE76_qlogis              0.0000        0.000           0.00          16.17
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2240 2231  -1098
+
+Optimised parameters:
+                              est.      lower      upper
+cyan_free_0              100.73014  9.873e+01  1.027e+02
+log_k_cyan_free           -3.19634 -3.641e+00 -2.752e+00
+log_k_cyan_free_bound     -3.43533 -3.674e+00 -3.197e+00
+log_k_cyan_bound_free     -3.83282 -4.163e+00 -3.503e+00
+log_k_JCZ38               -2.51065 -3.225e+00 -1.796e+00
+log_k_J9Z38               -5.02539 -5.825e+00 -4.226e+00
+log_k_JSE76               -3.24777 -4.163e+00 -2.333e+00
+f_cyan_ilr_1               0.70640  3.562e-01  1.057e+00
+f_cyan_ilr_2               1.42704  3.170e-01  2.537e+00
+f_JCZ38_qlogis             2.84779  1.042e+00  4.654e+00
+f_JSE76_qlogis             8.63674 -6.407e+02  6.580e+02
+a.1                        2.07082  1.877e+00  2.265e+00
+b.1                        0.06227  5.098e-02  7.355e-02
+SD.log_k_cyan_free         0.49674  1.865e-01  8.069e-01
+SD.log_k_cyan_bound_free   0.28537  6.809e-02  5.027e-01
+SD.log_k_JCZ38             0.74846  2.305e-01  1.266e+00
+SD.log_k_J9Z38             0.86077  2.713e-01  1.450e+00
+SD.log_k_JSE76             0.97613  3.030e-01  1.649e+00
+SD.f_cyan_ilr_1            0.38994  1.382e-01  6.417e-01
+SD.f_cyan_ilr_2            0.82869  3.917e-02  1.618e+00
+SD.f_JCZ38_qlogis          1.05000 -2.808e-02  2.128e+00
+SD.f_JSE76_qlogis          0.44681 -3.985e+05  3.985e+05
+
+Correlation: 
+                      cyn_f_0 lg_k_c_ lg_k_cyn_f_ lg_k_cyn_b_ l__JCZ3 l__J9Z3
+log_k_cyan_free        0.0936                                                
+log_k_cyan_free_bound  0.1302  0.1627                                        
+log_k_cyan_bound_free  0.0029  0.0525  0.5181                                
+log_k_JCZ38           -0.0116 -0.0077 -0.0430     -0.0236                    
+log_k_J9Z38           -0.0192 -0.0077 -0.0048      0.0229     -0.0005        
+log_k_JSE76            0.0007 -0.0020 -0.0134     -0.0072      0.1225 -0.0016
+f_cyan_ilr_1          -0.0118 -0.0027 -0.0132     -0.0118      0.0127 -0.0505
+f_cyan_ilr_2          -0.4643 -0.0762 -0.1245      0.0137      0.0497  0.1003
+f_JCZ38_qlogis         0.0710  0.0371  0.1826      0.0925     -0.0869 -0.0130
+f_JSE76_qlogis        -0.0367 -0.0270 -0.2274     -0.1865      0.1244  0.0098
+                      l__JSE7 f_cy__1 f_cy__2 f_JCZ38
+log_k_cyan_free                                      
+log_k_cyan_free_bound                                
+log_k_cyan_bound_free                                
+log_k_JCZ38                                          
+log_k_J9Z38                                          
+log_k_JSE76                                          
+f_cyan_ilr_1           0.0036                        
+f_cyan_ilr_2           0.0050 -0.0201                
+f_JCZ38_qlogis         0.0142 -0.0529 -0.2698        
+f_JSE76_qlogis         0.0064  0.0345  0.2015 -0.7058
+
+Random effects:
+                           est.      lower     upper
+SD.log_k_cyan_free       0.4967  1.865e-01 8.069e-01
+SD.log_k_cyan_bound_free 0.2854  6.809e-02 5.027e-01
+SD.log_k_JCZ38           0.7485  2.305e-01 1.266e+00
+SD.log_k_J9Z38           0.8608  2.713e-01 1.450e+00
+SD.log_k_JSE76           0.9761  3.030e-01 1.649e+00
+SD.f_cyan_ilr_1          0.3899  1.382e-01 6.417e-01
+SD.f_cyan_ilr_2          0.8287  3.917e-02 1.618e+00
+SD.f_JCZ38_qlogis        1.0500 -2.808e-02 2.128e+00
+SD.f_JSE76_qlogis        0.4468 -3.985e+05 3.985e+05
+
+Variance model:
+       est.   lower   upper
+a.1 2.07082 1.87680 2.26483
+b.1 0.06227 0.05098 0.07355
+
+Backtransformed parameters:
+                          est.      lower     upper
+cyan_free_0          1.007e+02  9.873e+01 102.72898
+k_cyan_free          4.091e-02  2.623e-02   0.06382
+k_cyan_free_bound    3.221e-02  2.537e-02   0.04090
+k_cyan_bound_free    2.165e-02  1.557e-02   0.03011
+k_JCZ38              8.122e-02  3.975e-02   0.16594
+k_J9Z38              6.569e-03  2.954e-03   0.01461
+k_JSE76              3.886e-02  1.556e-02   0.09703
+f_cyan_free_to_JCZ38 6.785e-01  6.102e-01   0.97309
+f_cyan_free_to_J9Z38 2.498e-01  6.102e-01   0.97309
+f_JCZ38_to_JSE76     9.452e-01  7.392e-01   0.99056
+f_JSE76_to_JCZ38     9.998e-01 5.580e-279   1.00000
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+0.08426 0.01051 0.41220 
+
+Resulting formation fractions:
+                       ff
+cyan_free_JCZ38 0.6784541
+cyan_free_J9Z38 0.2498405
+cyan_free_sink  0.0717054
+cyan_free       1.0000000
+JCZ38_JSE76     0.9452043
+JCZ38_sink      0.0547957
+JSE76_JCZ38     0.9998226
+JSE76_sink      0.0001774
+
+Estimated disappearance times:
+         DT50   DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   25.237 168.51    50.73        8.226        65.95
+JCZ38   8.535  28.35       NA           NA           NA
+J9Z38 105.517 350.52       NA           NA           NA
+JSE76  17.837  59.25       NA           NA           NA
+
+</code></pre>
+<p></p>
+</div>
+<div class="section level4">
+<h4 id="pathway-2-refined-fits">Pathway 2, refined fits<a class="anchor" aria-label="anchor" href="#pathway-2-refined-fits"></a>
+</h4>
+<caption>
+Hierarchical FOMC path 2 fit with reduced random effects, two-component
+error
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:11:01 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - (alpha/beta) * 1/((time/beta) + 1) * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_JCZ38 * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * (alpha/beta) * 1/((time/beta) + 1) *
+           cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 773.693 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      102.4477        -1.8631        -5.1087        -2.5114         0.6826 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis      log_alpha       log_beta 
+        4.7944        15.9616        13.1566        -0.1564         2.9781 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          7.701       0.000       0.000       0.000       0.0000
+log_k_JCZ38     0.000       1.448       0.000       0.000       0.0000
+log_k_J9Z38     0.000       0.000       1.724       0.000       0.0000
+log_k_JSE76     0.000       0.000       0.000       3.659       0.0000
+f_cyan_ilr_1    0.000       0.000       0.000       0.000       0.6356
+f_cyan_ilr_2    0.000       0.000       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000       0.000       0.000       0.000       0.0000
+f_JSE76_qlogis  0.000       0.000       0.000       0.000       0.0000
+log_alpha       0.000       0.000       0.000       0.000       0.0000
+log_beta        0.000       0.000       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_alpha log_beta
+cyan_0                 0.00           0.00           0.00    0.0000   0.0000
+log_k_JCZ38            0.00           0.00           0.00    0.0000   0.0000
+log_k_J9Z38            0.00           0.00           0.00    0.0000   0.0000
+log_k_JSE76            0.00           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_1           0.00           0.00           0.00    0.0000   0.0000
+f_cyan_ilr_2          10.32           0.00           0.00    0.0000   0.0000
+f_JCZ38_qlogis         0.00          12.23           0.00    0.0000   0.0000
+f_JSE76_qlogis         0.00           0.00          14.99    0.0000   0.0000
+log_alpha              0.00           0.00           0.00    0.3924   0.0000
+log_beta               0.00           0.00           0.00    0.0000   0.5639
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2249 2242  -1106
+
+Optimised parameters:
+                     est.   lower  upper
+cyan_0          101.24524      NA     NA
+log_k_JCZ38      -2.85375      NA     NA
+log_k_J9Z38      -5.07729      NA     NA
+log_k_JSE76      -3.53511      NA     NA
+f_cyan_ilr_1      0.67478      NA     NA
+f_cyan_ilr_2      0.97152      NA     NA
+f_JCZ38_qlogis  213.48001      NA     NA
+f_JSE76_qlogis    2.02040      NA     NA
+log_alpha        -0.11041      NA     NA
+log_beta          3.06575      NA     NA
+a.1               2.05279 1.85495 2.2506
+b.1               0.07116 0.05912 0.0832
+SD.log_k_JCZ38    1.21713 0.44160 1.9927
+SD.log_k_J9Z38    0.88268 0.27541 1.4900
+SD.log_k_JSE76    0.59452 0.15005 1.0390
+SD.f_cyan_ilr_1   0.35370 0.12409 0.5833
+SD.f_cyan_ilr_2   0.78186 0.18547 1.3782
+SD.log_alpha      0.27781 0.08168 0.4739
+SD.log_beta       0.32608 0.06490 0.5873
+
+Correlation is not available
+
+Random effects:
+                  est.   lower  upper
+SD.log_k_JCZ38  1.2171 0.44160 1.9927
+SD.log_k_J9Z38  0.8827 0.27541 1.4900
+SD.log_k_JSE76  0.5945 0.15005 1.0390
+SD.f_cyan_ilr_1 0.3537 0.12409 0.5833
+SD.f_cyan_ilr_2 0.7819 0.18547 1.3782
+SD.log_alpha    0.2778 0.08168 0.4739
+SD.log_beta     0.3261 0.06490 0.5873
+
+Variance model:
+       est.   lower  upper
+a.1 2.05279 1.85495 2.2506
+b.1 0.07116 0.05912 0.0832
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.012e+02    NA    NA
+k_JCZ38          5.763e-02    NA    NA
+k_J9Z38          6.237e-03    NA    NA
+k_JSE76          2.916e-02    NA    NA
+f_cyan_to_JCZ38  6.354e-01    NA    NA
+f_cyan_to_J9Z38  2.447e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+f_JSE76_to_JCZ38 8.829e-01    NA    NA
+alpha            8.955e-01    NA    NA
+beta             2.145e+01    NA    NA
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.6354
+cyan_J9Z38  0.2447
+cyan_sink   0.1200
+JCZ38_JSE76 1.0000
+JCZ38_sink  0.0000
+JSE76_JCZ38 0.8829
+JSE76_sink  0.1171
+
+Estimated disappearance times:
+        DT50   DT90 DT50back
+cyan   25.07 259.21    78.03
+JCZ38  12.03  39.96       NA
+J9Z38 111.14 369.19       NA
+JSE76  23.77  78.98       NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical DFOP path 2 fit with reduced random effects, constant
+variance
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:12:27 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38 +
+           f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 859.906 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      102.4380        -2.3107        -5.3123        -3.7120         0.6757 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis         log_k1         log_k2 
+        1.1439        13.1194        12.3492        -1.9317        -4.4557 
+      g_qlogis 
+       -0.5644 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          4.591      0.0000       0.000         0.0       0.0000
+log_k_JCZ38     0.000      0.7966       0.000         0.0       0.0000
+log_k_J9Z38     0.000      0.0000       1.561         0.0       0.0000
+log_k_JSE76     0.000      0.0000       0.000         0.8       0.0000
+f_cyan_ilr_1    0.000      0.0000       0.000         0.0       0.6349
+f_cyan_ilr_2    0.000      0.0000       0.000         0.0       0.0000
+f_JCZ38_qlogis  0.000      0.0000       0.000         0.0       0.0000
+f_JSE76_qlogis  0.000      0.0000       0.000         0.0       0.0000
+log_k1          0.000      0.0000       0.000         0.0       0.0000
+log_k2          0.000      0.0000       0.000         0.0       0.0000
+g_qlogis        0.000      0.0000       0.000         0.0       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_k1 log_k2
+cyan_0                0.000           0.00           0.00  0.000 0.0000
+log_k_JCZ38           0.000           0.00           0.00  0.000 0.0000
+log_k_J9Z38           0.000           0.00           0.00  0.000 0.0000
+log_k_JSE76           0.000           0.00           0.00  0.000 0.0000
+f_cyan_ilr_1          0.000           0.00           0.00  0.000 0.0000
+f_cyan_ilr_2          1.797           0.00           0.00  0.000 0.0000
+f_JCZ38_qlogis        0.000          13.86           0.00  0.000 0.0000
+f_JSE76_qlogis        0.000           0.00          13.91  0.000 0.0000
+log_k1                0.000           0.00           0.00  1.106 0.0000
+log_k2                0.000           0.00           0.00  0.000 0.6141
+g_qlogis              0.000           0.00           0.00  0.000 0.0000
+               g_qlogis
+cyan_0            0.000
+log_k_JCZ38       0.000
+log_k_J9Z38       0.000
+log_k_JSE76       0.000
+f_cyan_ilr_1      0.000
+f_cyan_ilr_2      0.000
+f_JCZ38_qlogis    0.000
+f_JSE76_qlogis    0.000
+log_k1            0.000
+log_k2            0.000
+g_qlogis          1.595
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2282 2274  -1121
+
+Optimised parameters:
+                     est.   lower  upper
+cyan_0           102.6036      NA     NA
+log_k_JCZ38       -2.9348      NA     NA
+log_k_J9Z38       -5.1617      NA     NA
+log_k_JSE76       -3.6396      NA     NA
+f_cyan_ilr_1       0.6991      NA     NA
+f_cyan_ilr_2       0.6341      NA     NA
+f_JCZ38_qlogis  4232.3011      NA     NA
+f_JSE76_qlogis     1.9658      NA     NA
+log_k1            -1.9503      NA     NA
+log_k2            -4.4745      NA     NA
+g_qlogis          -0.4967      NA     NA
+a.1                2.7461 2.59274 2.8994
+SD.log_k_JCZ38     1.3178 0.47602 2.1596
+SD.log_k_J9Z38     0.7022 0.15061 1.2538
+SD.log_k_JSE76     0.6566 0.15613 1.1570
+SD.f_cyan_ilr_1    0.3409 0.11666 0.5652
+SD.f_cyan_ilr_2    0.4385 0.09482 0.7821
+SD.log_k1          0.7381 0.25599 1.2202
+SD.log_k2          0.5133 0.18152 0.8450
+SD.g_qlogis        0.9866 0.35681 1.6164
+
+Correlation is not available
+
+Random effects:
+                  est.   lower  upper
+SD.log_k_JCZ38  1.3178 0.47602 2.1596
+SD.log_k_J9Z38  0.7022 0.15061 1.2538
+SD.log_k_JSE76  0.6566 0.15613 1.1570
+SD.f_cyan_ilr_1 0.3409 0.11666 0.5652
+SD.f_cyan_ilr_2 0.4385 0.09482 0.7821
+SD.log_k1       0.7381 0.25599 1.2202
+SD.log_k2       0.5133 0.18152 0.8450
+SD.g_qlogis     0.9866 0.35681 1.6164
+
+Variance model:
+     est. lower upper
+a.1 2.746 2.593 2.899
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.026e+02    NA    NA
+k_JCZ38          5.314e-02    NA    NA
+k_J9Z38          5.732e-03    NA    NA
+k_JSE76          2.626e-02    NA    NA
+f_cyan_to_JCZ38  6.051e-01    NA    NA
+f_cyan_to_J9Z38  2.251e-01    NA    NA
+f_JCZ38_to_JSE76 1.000e+00    NA    NA
+f_JSE76_to_JCZ38 8.772e-01    NA    NA
+k1               1.422e-01    NA    NA
+k2               1.140e-02    NA    NA
+g                3.783e-01    NA    NA
+
+Resulting formation fractions:
+                ff
+cyan_JCZ38  0.6051
+cyan_J9Z38  0.2251
+cyan_sink   0.1698
+JCZ38_JSE76 1.0000
+JCZ38_sink  0.0000
+JSE76_JCZ38 0.8772
+JSE76_sink  0.1228
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   22.05 160.35    48.27   4.873   60.83
+JCZ38  13.04  43.33       NA      NA      NA
+J9Z38 120.93 401.73       NA      NA      NA
+JSE76  26.39  87.68       NA      NA      NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical DFOP path 2 fit with reduced random effects, two-component
+error
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:12:54 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * cyan
+d_JCZ38/dt = + f_cyan_to_JCZ38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_JCZ38 * JCZ38 +
+           f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_to_J9Z38 * ((k1 * g * exp(-k1 * time) + k2 * (1 -
+           g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
+           exp(-k2 * time))) * cyan - k_J9Z38 * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 886.298 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+        cyan_0    log_k_JCZ38    log_k_J9Z38    log_k_JSE76   f_cyan_ilr_1 
+      101.7393        -1.4493        -5.0118        -2.1269         0.6720 
+  f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis         log_k1         log_k2 
+        7.3362        13.4423        13.2659        -2.0061        -4.5527 
+      g_qlogis 
+       -0.5806 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+               cyan_0 log_k_JCZ38 log_k_J9Z38 log_k_JSE76 f_cyan_ilr_1
+cyan_0          5.604        0.00       0.000       0.000       0.0000
+log_k_JCZ38     0.000        2.77       0.000       0.000       0.0000
+log_k_J9Z38     0.000        0.00       1.662       0.000       0.0000
+log_k_JSE76     0.000        0.00       0.000       5.021       0.0000
+f_cyan_ilr_1    0.000        0.00       0.000       0.000       0.6519
+f_cyan_ilr_2    0.000        0.00       0.000       0.000       0.0000
+f_JCZ38_qlogis  0.000        0.00       0.000       0.000       0.0000
+f_JSE76_qlogis  0.000        0.00       0.000       0.000       0.0000
+log_k1          0.000        0.00       0.000       0.000       0.0000
+log_k2          0.000        0.00       0.000       0.000       0.0000
+g_qlogis        0.000        0.00       0.000       0.000       0.0000
+               f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis log_k1 log_k2
+cyan_0                 0.00           0.00           0.00 0.0000 0.0000
+log_k_JCZ38            0.00           0.00           0.00 0.0000 0.0000
+log_k_J9Z38            0.00           0.00           0.00 0.0000 0.0000
+log_k_JSE76            0.00           0.00           0.00 0.0000 0.0000
+f_cyan_ilr_1           0.00           0.00           0.00 0.0000 0.0000
+f_cyan_ilr_2          13.37           0.00           0.00 0.0000 0.0000
+f_JCZ38_qlogis         0.00          14.21           0.00 0.0000 0.0000
+f_JSE76_qlogis         0.00           0.00          14.58 0.0000 0.0000
+log_k1                 0.00           0.00           0.00 0.8453 0.0000
+log_k2                 0.00           0.00           0.00 0.0000 0.5969
+g_qlogis               0.00           0.00           0.00 0.0000 0.0000
+               g_qlogis
+cyan_0             0.00
+log_k_JCZ38        0.00
+log_k_J9Z38        0.00
+log_k_JSE76        0.00
+f_cyan_ilr_1       0.00
+f_cyan_ilr_2       0.00
+f_JCZ38_qlogis     0.00
+f_JSE76_qlogis     0.00
+log_k1             0.00
+log_k2             0.00
+g_qlogis           1.69
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2237 2229  -1099
+
+Optimised parameters:
+                     est.   lower   upper
+cyan_0          101.00243      NA      NA
+log_k_JCZ38      -2.80828      NA      NA
+log_k_J9Z38      -5.04449      NA      NA
+log_k_JSE76      -3.66981      NA      NA
+f_cyan_ilr_1      0.72564      NA      NA
+f_cyan_ilr_2      1.37978      NA      NA
+f_JCZ38_qlogis    1.98726      NA      NA
+f_JSE76_qlogis  414.80884      NA      NA
+log_k1           -2.38601      NA      NA
+log_k2           -4.63632      NA      NA
+g_qlogis         -0.33920      NA      NA
+a.1               2.10837 1.91261 2.30413
+b.1               0.06223 0.05085 0.07361
+SD.log_k_JCZ38    1.30902 0.48128 2.13675
+SD.log_k_J9Z38    0.83882 0.25790 1.41974
+SD.log_k_JSE76    0.58104 0.14201 1.02008
+SD.f_cyan_ilr_1   0.35421 0.12398 0.58443
+SD.f_cyan_ilr_2   0.79373 0.12007 1.46739
+SD.log_k2         0.27476 0.08557 0.46394
+SD.g_qlogis       0.96170 0.35463 1.56878
+
+Correlation is not available
+
+Random effects:
+                  est.   lower  upper
+SD.log_k_JCZ38  1.3090 0.48128 2.1367
+SD.log_k_J9Z38  0.8388 0.25790 1.4197
+SD.log_k_JSE76  0.5810 0.14201 1.0201
+SD.f_cyan_ilr_1 0.3542 0.12398 0.5844
+SD.f_cyan_ilr_2 0.7937 0.12007 1.4674
+SD.log_k2       0.2748 0.08557 0.4639
+SD.g_qlogis     0.9617 0.35463 1.5688
+
+Variance model:
+       est.   lower   upper
+a.1 2.10837 1.91261 2.30413
+b.1 0.06223 0.05085 0.07361
+
+Backtransformed parameters:
+                      est. lower upper
+cyan_0           1.010e+02    NA    NA
+k_JCZ38          6.031e-02    NA    NA
+k_J9Z38          6.445e-03    NA    NA
+k_JSE76          2.548e-02    NA    NA
+f_cyan_to_JCZ38  6.808e-01    NA    NA
+f_cyan_to_J9Z38  2.440e-01    NA    NA
+f_JCZ38_to_JSE76 8.795e-01    NA    NA
+f_JSE76_to_JCZ38 1.000e+00    NA    NA
+k1               9.200e-02    NA    NA
+k2               9.693e-03    NA    NA
+g                4.160e-01    NA    NA
+
+Resulting formation fractions:
+                 ff
+cyan_JCZ38  0.68081
+cyan_J9Z38  0.24398
+cyan_sink   0.07521
+JCZ38_JSE76 0.87945
+JCZ38_sink  0.12055
+JSE76_JCZ38 1.00000
+JSE76_sink  0.00000
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_k1 DT50_k2
+cyan   25.00 182.05     54.8   7.535   71.51
+JCZ38  11.49  38.18       NA      NA      NA
+J9Z38 107.55 357.28       NA      NA      NA
+JSE76  27.20  90.36       NA      NA      NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical SFORB path 2 fit with reduced random effects, constant
+variance
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:12:32 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 865.121 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+             102.4395               -2.7673               -2.8942 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+              -3.6201               -2.3107               -5.3123 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+              -3.7120                0.6754                1.1448 
+       f_JCZ38_qlogis        f_JSE76_qlogis 
+              14.8408               15.4734 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 4.589          0.0000                  0.00
+log_k_cyan_free             0.000          0.4849                  0.00
+log_k_cyan_free_bound       0.000          0.0000                  1.62
+log_k_cyan_bound_free       0.000          0.0000                  0.00
+log_k_JCZ38                 0.000          0.0000                  0.00
+log_k_J9Z38                 0.000          0.0000                  0.00
+log_k_JSE76                 0.000          0.0000                  0.00
+f_cyan_ilr_1                0.000          0.0000                  0.00
+f_cyan_ilr_2                0.000          0.0000                  0.00
+f_JCZ38_qlogis              0.000          0.0000                  0.00
+f_JSE76_qlogis              0.000          0.0000                  0.00
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000      0.0000       0.000         0.0
+log_k_cyan_free                       0.000      0.0000       0.000         0.0
+log_k_cyan_free_bound                 0.000      0.0000       0.000         0.0
+log_k_cyan_bound_free                 1.197      0.0000       0.000         0.0
+log_k_JCZ38                           0.000      0.7966       0.000         0.0
+log_k_J9Z38                           0.000      0.0000       1.561         0.0
+log_k_JSE76                           0.000      0.0000       0.000         0.8
+f_cyan_ilr_1                          0.000      0.0000       0.000         0.0
+f_cyan_ilr_2                          0.000      0.0000       0.000         0.0
+f_JCZ38_qlogis                        0.000      0.0000       0.000         0.0
+f_JSE76_qlogis                        0.000      0.0000       0.000         0.0
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis
+cyan_free_0                 0.0000        0.000            0.0           0.00
+log_k_cyan_free             0.0000        0.000            0.0           0.00
+log_k_cyan_free_bound       0.0000        0.000            0.0           0.00
+log_k_cyan_bound_free       0.0000        0.000            0.0           0.00
+log_k_JCZ38                 0.0000        0.000            0.0           0.00
+log_k_J9Z38                 0.0000        0.000            0.0           0.00
+log_k_JSE76                 0.0000        0.000            0.0           0.00
+f_cyan_ilr_1                0.6349        0.000            0.0           0.00
+f_cyan_ilr_2                0.0000        1.797            0.0           0.00
+f_JCZ38_qlogis              0.0000        0.000           15.6           0.00
+f_JSE76_qlogis              0.0000        0.000            0.0          17.52
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2280 2272  -1120
+
+Optimised parameters:
+                              est.   lower  upper
+cyan_free_0               102.6532      NA     NA
+log_k_cyan_free            -2.8547      NA     NA
+log_k_cyan_free_bound      -2.7004      NA     NA
+log_k_cyan_bound_free      -3.5078      NA     NA
+log_k_JCZ38                -2.9255      NA     NA
+log_k_J9Z38                -5.1089      NA     NA
+log_k_JSE76                -3.6263      NA     NA
+f_cyan_ilr_1                0.6873      NA     NA
+f_cyan_ilr_2                0.6498      NA     NA
+f_JCZ38_qlogis           3624.2149      NA     NA
+f_JSE76_qlogis              1.9991      NA     NA
+a.1                         2.7472 2.55559 2.9388
+SD.log_k_cyan_free          0.3227 0.10296 0.5423
+SD.log_k_cyan_free_bound    0.8757 0.29525 1.4562
+SD.log_k_cyan_bound_free    0.6128 0.20220 1.0233
+SD.log_k_JCZ38              1.3431 0.48474 2.2014
+SD.log_k_J9Z38              0.6881 0.14714 1.2291
+SD.log_k_JSE76              0.6461 0.15321 1.1390
+SD.f_cyan_ilr_1             0.3361 0.11376 0.5585
+SD.f_cyan_ilr_2             0.4286 0.08419 0.7730
+
+Correlation is not available
+
+Random effects:
+                           est.   lower  upper
+SD.log_k_cyan_free       0.3227 0.10296 0.5423
+SD.log_k_cyan_free_bound 0.8757 0.29525 1.4562
+SD.log_k_cyan_bound_free 0.6128 0.20220 1.0233
+SD.log_k_JCZ38           1.3431 0.48474 2.2014
+SD.log_k_J9Z38           0.6881 0.14714 1.2291
+SD.log_k_JSE76           0.6461 0.15321 1.1390
+SD.f_cyan_ilr_1          0.3361 0.11376 0.5585
+SD.f_cyan_ilr_2          0.4286 0.08419 0.7730
+
+Variance model:
+     est. lower upper
+a.1 2.747 2.556 2.939
+
+Backtransformed parameters:
+                          est. lower upper
+cyan_free_0          1.027e+02    NA    NA
+k_cyan_free          5.758e-02    NA    NA
+k_cyan_free_bound    6.718e-02    NA    NA
+k_cyan_bound_free    2.996e-02    NA    NA
+k_JCZ38              5.364e-02    NA    NA
+k_J9Z38              6.042e-03    NA    NA
+k_JSE76              2.662e-02    NA    NA
+f_cyan_free_to_JCZ38 6.039e-01    NA    NA
+f_cyan_free_to_J9Z38 2.285e-01    NA    NA
+f_JCZ38_to_JSE76     1.000e+00    NA    NA
+f_JSE76_to_JCZ38     8.807e-01    NA    NA
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+ 0.1426  0.0121  0.3484 
+
+Resulting formation fractions:
+                    ff
+cyan_free_JCZ38 0.6039
+cyan_free_J9Z38 0.2285
+cyan_free_sink  0.1676
+cyan_free       1.0000
+JCZ38_JSE76     1.0000
+JCZ38_sink      0.0000
+JSE76_JCZ38     0.8807
+JSE76_sink      0.1193
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   23.84 154.95    46.65         4.86        57.31
+JCZ38  12.92  42.93       NA           NA           NA
+J9Z38 114.71 381.07       NA           NA           NA
+JSE76  26.04  86.51       NA           NA           NA
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical SFORB path 2 fit with reduced random effects, two-component
+error
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:12:36 2025 
+Date of summary: Fri Feb 14 08:12:55 2025 
+
+Equations:
+d_cyan_free/dt = - k_cyan_free * cyan_free - k_cyan_free_bound *
+           cyan_free + k_cyan_bound_free * cyan_bound
+d_cyan_bound/dt = + k_cyan_free_bound * cyan_free - k_cyan_bound_free *
+           cyan_bound
+d_JCZ38/dt = + f_cyan_free_to_JCZ38 * k_cyan_free * cyan_free - k_JCZ38
+           * JCZ38 + f_JSE76_to_JCZ38 * k_JSE76 * JSE76
+d_J9Z38/dt = + f_cyan_free_to_J9Z38 * k_cyan_free * cyan_free - k_J9Z38
+           * J9Z38
+d_JSE76/dt = + f_JCZ38_to_JSE76 * k_JCZ38 * JCZ38 - k_JSE76 * JSE76
+
+Data:
+433 observations of 4 variable(s) grouped in 5 datasets
+
+Model predictions using solution type deSolve 
+
+Fitted in 868.952 s
+Using 300, 100 iterations and 10 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+          cyan_free_0       log_k_cyan_free log_k_cyan_free_bound 
+             101.7511               -2.8370               -3.0162 
+log_k_cyan_bound_free           log_k_JCZ38           log_k_J9Z38 
+              -3.6600               -2.2988               -5.3129 
+          log_k_JSE76          f_cyan_ilr_1          f_cyan_ilr_2 
+              -3.6991                0.6722                4.8596 
+       f_JCZ38_qlogis        f_JSE76_qlogis 
+              13.4678               14.2149 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+                      cyan_free_0 log_k_cyan_free log_k_cyan_free_bound
+cyan_free_0                 5.629           0.000                 0.000
+log_k_cyan_free             0.000           0.446                 0.000
+log_k_cyan_free_bound       0.000           0.000                 1.449
+log_k_cyan_bound_free       0.000           0.000                 0.000
+log_k_JCZ38                 0.000           0.000                 0.000
+log_k_J9Z38                 0.000           0.000                 0.000
+log_k_JSE76                 0.000           0.000                 0.000
+f_cyan_ilr_1                0.000           0.000                 0.000
+f_cyan_ilr_2                0.000           0.000                 0.000
+f_JCZ38_qlogis              0.000           0.000                 0.000
+f_JSE76_qlogis              0.000           0.000                 0.000
+                      log_k_cyan_bound_free log_k_JCZ38 log_k_J9Z38 log_k_JSE76
+cyan_free_0                           0.000      0.0000       0.000      0.0000
+log_k_cyan_free                       0.000      0.0000       0.000      0.0000
+log_k_cyan_free_bound                 0.000      0.0000       0.000      0.0000
+log_k_cyan_bound_free                 1.213      0.0000       0.000      0.0000
+log_k_JCZ38                           0.000      0.7801       0.000      0.0000
+log_k_J9Z38                           0.000      0.0000       1.575      0.0000
+log_k_JSE76                           0.000      0.0000       0.000      0.8078
+f_cyan_ilr_1                          0.000      0.0000       0.000      0.0000
+f_cyan_ilr_2                          0.000      0.0000       0.000      0.0000
+f_JCZ38_qlogis                        0.000      0.0000       0.000      0.0000
+f_JSE76_qlogis                        0.000      0.0000       0.000      0.0000
+                      f_cyan_ilr_1 f_cyan_ilr_2 f_JCZ38_qlogis f_JSE76_qlogis
+cyan_free_0                 0.0000        0.000           0.00           0.00
+log_k_cyan_free             0.0000        0.000           0.00           0.00
+log_k_cyan_free_bound       0.0000        0.000           0.00           0.00
+log_k_cyan_bound_free       0.0000        0.000           0.00           0.00
+log_k_JCZ38                 0.0000        0.000           0.00           0.00
+log_k_J9Z38                 0.0000        0.000           0.00           0.00
+log_k_JSE76                 0.0000        0.000           0.00           0.00
+f_cyan_ilr_1                0.6518        0.000           0.00           0.00
+f_cyan_ilr_2                0.0000        9.981           0.00           0.00
+f_JCZ38_qlogis              0.0000        0.000          14.26           0.00
+f_JSE76_qlogis              0.0000        0.000           0.00          16.17
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+   AIC  BIC logLik
+  2241 2233  -1101
+
+Optimised parameters:
+                              est.   lower   upper
+cyan_free_0              100.95469      NA      NA
+log_k_cyan_free           -3.18706      NA      NA
+log_k_cyan_free_bound     -3.38455      NA      NA
+log_k_cyan_bound_free     -3.75788      NA      NA
+log_k_JCZ38               -2.77024      NA      NA
+log_k_J9Z38               -5.03665      NA      NA
+log_k_JSE76               -3.60289      NA      NA
+f_cyan_ilr_1               0.72263      NA      NA
+f_cyan_ilr_2               1.45352      NA      NA
+f_JCZ38_qlogis             2.00778      NA      NA
+f_JSE76_qlogis           941.58570      NA      NA
+a.1                        2.11130 1.91479 2.30780
+b.1                        0.06299 0.05152 0.07445
+SD.log_k_cyan_free         0.50098 0.18805 0.81390
+SD.log_k_cyan_bound_free   0.31671 0.08467 0.54875
+SD.log_k_JCZ38             1.25865 0.45932 2.05798
+SD.log_k_J9Z38             0.86833 0.27222 1.46444
+SD.log_k_JSE76             0.59325 0.14711 1.03940
+SD.f_cyan_ilr_1            0.35705 0.12521 0.58890
+SD.f_cyan_ilr_2            0.88541 0.13797 1.63286
+
+Correlation is not available
+
+Random effects:
+                           est.   lower  upper
+SD.log_k_cyan_free       0.5010 0.18805 0.8139
+SD.log_k_cyan_bound_free 0.3167 0.08467 0.5487
+SD.log_k_JCZ38           1.2587 0.45932 2.0580
+SD.log_k_J9Z38           0.8683 0.27222 1.4644
+SD.log_k_JSE76           0.5933 0.14711 1.0394
+SD.f_cyan_ilr_1          0.3571 0.12521 0.5889
+SD.f_cyan_ilr_2          0.8854 0.13797 1.6329
+
+Variance model:
+       est.   lower   upper
+a.1 2.11130 1.91479 2.30780
+b.1 0.06299 0.05152 0.07445
+
+Backtransformed parameters:
+                          est. lower upper
+cyan_free_0          1.010e+02    NA    NA
+k_cyan_free          4.129e-02    NA    NA
+k_cyan_free_bound    3.389e-02    NA    NA
+k_cyan_bound_free    2.333e-02    NA    NA
+k_JCZ38              6.265e-02    NA    NA
+k_J9Z38              6.495e-03    NA    NA
+k_JSE76              2.724e-02    NA    NA
+f_cyan_free_to_JCZ38 6.844e-01    NA    NA
+f_cyan_free_to_J9Z38 2.463e-01    NA    NA
+f_JCZ38_to_JSE76     8.816e-01    NA    NA
+f_JSE76_to_JCZ38     1.000e+00    NA    NA
+
+Estimated Eigenvalues of SFORB model(s):
+cyan_b1 cyan_b2  cyan_g 
+0.08751 0.01101 0.39586 
+
+Resulting formation fractions:
+                     ff
+cyan_free_JCZ38 0.68444
+cyan_free_J9Z38 0.24633
+cyan_free_sink  0.06923
+cyan_free       1.00000
+JCZ38_JSE76     0.88161
+JCZ38_sink      0.11839
+JSE76_JCZ38     1.00000
+JSE76_sink      0.00000
+
+Estimated disappearance times:
+        DT50   DT90 DT50back DT50_cyan_b1 DT50_cyan_b2
+cyan   25.36 163.36    49.18        7.921        62.95
+JCZ38  11.06  36.75       NA           NA           NA
+J9Z38 106.71 354.49       NA           NA           NA
+JSE76  25.44  84.51       NA           NA           NA
+
+</code></pre>
+<p></p>
+</div>
+</div>
+<div class="section level3">
+<h3 id="session-info">Session info<a class="anchor" aria-label="anchor" href="#session-info"></a>
+</h3>
+<pre><code>R version 4.4.2 (2024-10-31)
+Platform: x86_64-pc-linux-gnu
+Running under: Debian GNU/Linux 12 (bookworm)
+
+Matrix products: default
+BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.11.0 
+LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.11.0
+
+locale:
+ [1] LC_CTYPE=de_DE.UTF-8       LC_NUMERIC=C              
+ [3] LC_TIME=de_DE.UTF-8        LC_COLLATE=de_DE.UTF-8    
+ [5] LC_MONETARY=de_DE.UTF-8    LC_MESSAGES=de_DE.UTF-8   
+ [7] LC_PAPER=de_DE.UTF-8       LC_NAME=C                 
+ [9] LC_ADDRESS=C               LC_TELEPHONE=C            
+[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C       
+
+time zone: Europe/Berlin
+tzcode source: system (glibc)
+
+attached base packages:
+[1] parallel  stats     graphics  grDevices utils     datasets  methods  
+[8] base     
+
+other attached packages:
+[1] rmarkdown_2.29  nvimcom_0.9-167 saemix_3.3      npde_3.5       
+[5] knitr_1.49      mkin_1.2.10    
+
+loaded via a namespace (and not attached):
+ [1] sass_0.4.9        utf8_1.2.4        generics_0.1.3    lattice_0.22-6   
+ [5] digest_0.6.37     magrittr_2.0.3    evaluate_1.0.1    grid_4.4.2       
+ [9] fastmap_1.2.0     cellranger_1.1.0  jsonlite_1.8.9    processx_3.8.4   
+[13] pkgbuild_1.4.5    deSolve_1.40      mclust_6.1.1      ps_1.8.1         
+[17] gridExtra_2.3     fansi_1.0.6       scales_1.3.0      codetools_0.2-20 
+[21] textshaping_0.4.1 jquerylib_0.1.4   cli_3.6.3         rlang_1.1.4      
+[25] munsell_0.5.1     cachem_1.1.0      yaml_2.3.10       inline_0.3.20    
+[29] tools_4.4.2       dplyr_1.1.4       colorspace_2.1-1  ggplot2_3.5.1    
+[33] vctrs_0.6.5       R6_2.5.1          zoo_1.8-12        lifecycle_1.0.4  
+[37] fs_1.6.5          htmlwidgets_1.6.4 MASS_7.3-61       ragg_1.3.3       
+[41] callr_3.7.6       pkgconfig_2.0.3   desc_1.4.3        pkgdown_2.1.1    
+[45] pillar_1.9.0      bslib_0.8.0       gtable_0.3.6      glue_1.8.0       
+[49] systemfonts_1.1.0 xfun_0.49         tibble_3.2.1      lmtest_0.9-40    
+[53] tidyselect_1.2.1  htmltools_0.5.8.1 nlme_3.1-166      compiler_4.4.2   
+[57] readxl_1.4.3     </code></pre>
+</div>
+<div class="section level3">
+<h3 id="hardware-info">Hardware info<a class="anchor" aria-label="anchor" href="#hardware-info"></a>
+</h3>
+<pre><code>CPU model: AMD Ryzen 9 7950X 16-Core Processor</code></pre>
+<pre><code>MemTotal:       64927788 kB</code></pre>
+</div>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
+</div>
+
+    </footer>
+</div>
+
+
+
+
+
+  </body>
+</html>
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+
+      <h1>Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change on 5 January
+2023, last compiled on 14 Februar 2025</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/prebuilt/2022_dmta_parent.rmd" class="external-link"><code>vignettes/prebuilt/2022_dmta_parent.rmd</code></a></small>
+      <div class="d-none name"><code>2022_dmta_parent.rmd</code></div>
+    </div>
+
+    
+    
+<div class="section level2">
+<h2 id="introduction">Introduction<a class="anchor" aria-label="anchor" href="#introduction"></a>
+</h2>
+<p>The purpose of this document is to demonstrate how nonlinear
+hierarchical models (NLHM) based on the parent degradation models SFO,
+FOMC, DFOP and HS can be fitted with the mkin package.</p>
+<p>It was assembled in the course of work package 1.1 of Project Number
+173340 (Application of nonlinear hierarchical models to the kinetic
+evaluation of chemical degradation data) of the German Environment
+Agency carried out in 2022 and 2023.</p>
+<p>The mkin package is used in version 1.2.10. It contains the test data
+and the functions used in the evaluations. The <code>saemix</code>
+package is used as a backend for fitting the NLHM, but is also loaded to
+make the convergence plot function available.</p>
+<p>This document is processed with the <code>knitr</code> package, which
+also provides the <code>kable</code> function that is used to improve
+the display of tabular data in R markdown documents. For parallel
+processing, the <code>parallel</code> package is used.</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://pkgdown.jrwb.de/mkin/">mkin</a></span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://yihui.org/knitr/" class="external-link">knitr</a></span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">saemix</span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">parallel</span><span class="op">)</span></span>
+<span><span class="va">n_cores</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/detectCores.html" class="external-link">detectCores</a></span><span class="op">(</span><span class="op">)</span></span>
+<span><span class="kw">if</span> <span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/Sys.info.html" class="external-link">Sys.info</a></span><span class="op">(</span><span class="op">)</span><span class="op">[</span><span class="st">"sysname"</span><span class="op">]</span> <span class="op">==</span> <span class="st">"Windows"</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="va">cl</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">makePSOCKcluster</a></span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
+<span><span class="op">}</span> <span class="kw">else</span> <span class="op">{</span></span>
+<span>  <span class="va">cl</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">makeForkCluster</a></span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
+<span><span class="op">}</span></span></code></pre></div>
+</div>
+<div class="section level2">
+<h2 id="data">Data<a class="anchor" aria-label="anchor" href="#data"></a>
+</h2>
+<p>The test data are available in the mkin package as an object of class
+<code>mkindsg</code> (mkin dataset group) under the identifier
+<code>dimethenamid_2018</code>. The following preprocessing steps are
+still necessary:</p>
+<ul>
+<li>The data available for the enantiomer dimethenamid-P (DMTAP) are
+renamed to have the same substance name as the data for the racemic
+mixture dimethenamid (DMTA). The reason for this is that no difference
+between their degradation behaviour was identified in the EU risk
+assessment.</li>
+<li>The data for transformation products and unnecessary columns are
+discarded</li>
+<li>The observation times of each dataset are multiplied with the
+corresponding normalisation factor also available in the dataset, in
+order to make it possible to describe all datasets with a single set of
+parameters that are independent of temperature</li>
+<li>Finally, datasets observed in the same soil (<code>Elliot 1</code>
+and <code>Elliot 2</code>) are combined, resulting in dimethenamid
+(DMTA) data from six soils.</li>
+</ul>
+<p>The following commented R code performs this preprocessing.</p>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="co"># Apply a function to each of the seven datasets in the mkindsg object to create a list</span></span>
+<span><span class="va">dmta_ds</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">lapply</a></span><span class="op">(</span><span class="fl">1</span><span class="op">:</span><span class="fl">7</span>, <span class="kw">function</span><span class="op">(</span><span class="va">i</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="va">ds_i</span> <span class="op">&lt;-</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span><span class="op">[[</span><span class="va">i</span><span class="op">]</span><span class="op">]</span><span class="op">$</span><span class="va">data</span>                     <span class="co"># Get a dataset</span></span>
+<span>  <span class="va">ds_i</span><span class="op">[</span><span class="va">ds_i</span><span class="op">$</span><span class="va">name</span> <span class="op">==</span> <span class="st">"DMTAP"</span>, <span class="st">"name"</span><span class="op">]</span> <span class="op">&lt;-</span>  <span class="st">"DMTA"</span>              <span class="co"># Rename DMTAP to DMTA</span></span>
+<span>  <span class="va">ds_i</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/subset.html" class="external-link">subset</a></span><span class="op">(</span><span class="va">ds_i</span>, <span class="va">name</span> <span class="op">==</span> <span class="st">"DMTA"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"name"</span>, <span class="st">"time"</span>, <span class="st">"value"</span><span class="op">)</span><span class="op">)</span> <span class="co"># Select data</span></span>
+<span>  <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">&lt;-</span> <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">*</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">f_time_norm</span><span class="op">[</span><span class="va">i</span><span class="op">]</span>  <span class="co"># Normalise time</span></span>
+<span>  <span class="va">ds_i</span>                                                       <span class="co"># Return the dataset</span></span>
+<span><span class="op">}</span><span class="op">)</span></span>
+<span></span>
+<span><span class="co"># Use dataset titles as names for the list elements</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">)</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">sapply</a></span><span class="op">(</span><span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span>, <span class="kw">function</span><span class="op">(</span><span class="va">ds</span><span class="op">)</span> <span class="va">ds</span><span class="op">$</span><span class="va">title</span><span class="op">)</span></span>
+<span></span>
+<span><span class="co"># Combine data for Elliot soil to obtain a named list with six elements</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span>, <span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span> <span class="co">#</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="cn">NULL</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="cn">NULL</span></span></code></pre></div>
+<p>The following tables show the 6 datasets.</p>
+<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw">for</span> <span class="op">(</span><span class="va">ds_name</span> <span class="kw">in</span> <span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">)</span><span class="op">)</span> <span class="op">{</span></span>
+<span>    <span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="fu"><a href="../../reference/mkin_long_to_wide.html">mkin_long_to_wide</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">[[</span><span class="va">ds_name</span><span class="op">]</span><span class="op">]</span><span class="op">)</span>,</span>
+<span>      caption <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/paste.html" class="external-link">paste</a></span><span class="op">(</span><span class="st">"Dataset"</span>, <span class="va">ds_name</span><span class="op">)</span>,</span>
+<span>      label <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/paste.html" class="external-link">paste0</a></span><span class="op">(</span><span class="st">"tab:"</span>, <span class="va">ds_name</span><span class="op">)</span>, booktabs <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span><span class="op">)</span></span>
+<span>    <span class="fu"><a href="https://rdrr.io/r/base/cat.html" class="external-link">cat</a></span><span class="op">(</span><span class="st">"\n\\clearpage\n"</span><span class="op">)</span></span>
+<span><span class="op">}</span></span></code></pre></div>
+<table class="table">
+<caption>Dataset Calke</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0</td>
+<td align="right">95.8</td>
+</tr>
+<tr class="even">
+<td align="right">0</td>
+<td align="right">98.7</td>
+</tr>
+<tr class="odd">
+<td align="right">14</td>
+<td align="right">60.5</td>
+</tr>
+<tr class="even">
+<td align="right">30</td>
+<td align="right">39.1</td>
+</tr>
+<tr class="odd">
+<td align="right">59</td>
+<td align="right">15.2</td>
+</tr>
+<tr class="even">
+<td align="right">120</td>
+<td align="right">4.8</td>
+</tr>
+<tr class="odd">
+<td align="right">120</td>
+<td align="right">4.6</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Borstel</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">100.5</td>
+</tr>
+<tr class="even">
+<td align="right">0.000000</td>
+<td align="right">99.6</td>
+</tr>
+<tr class="odd">
+<td align="right">1.941295</td>
+<td align="right">91.9</td>
+</tr>
+<tr class="even">
+<td align="right">1.941295</td>
+<td align="right">91.3</td>
+</tr>
+<tr class="odd">
+<td align="right">6.794534</td>
+<td align="right">81.8</td>
+</tr>
+<tr class="even">
+<td align="right">6.794534</td>
+<td align="right">82.1</td>
+</tr>
+<tr class="odd">
+<td align="right">13.589067</td>
+<td align="right">69.1</td>
+</tr>
+<tr class="even">
+<td align="right">13.589067</td>
+<td align="right">68.0</td>
+</tr>
+<tr class="odd">
+<td align="right">27.178135</td>
+<td align="right">51.4</td>
+</tr>
+<tr class="even">
+<td align="right">27.178135</td>
+<td align="right">51.4</td>
+</tr>
+<tr class="odd">
+<td align="right">56.297565</td>
+<td align="right">27.6</td>
+</tr>
+<tr class="even">
+<td align="right">56.297565</td>
+<td align="right">26.8</td>
+</tr>
+<tr class="odd">
+<td align="right">86.387643</td>
+<td align="right">15.7</td>
+</tr>
+<tr class="even">
+<td align="right">86.387643</td>
+<td align="right">15.3</td>
+</tr>
+<tr class="odd">
+<td align="right">115.507073</td>
+<td align="right">7.9</td>
+</tr>
+<tr class="even">
+<td align="right">115.507073</td>
+<td align="right">8.1</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Flaach</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.0000000</td>
+<td align="right">96.5</td>
+</tr>
+<tr class="even">
+<td align="right">0.0000000</td>
+<td align="right">96.8</td>
+</tr>
+<tr class="odd">
+<td align="right">0.0000000</td>
+<td align="right">97.0</td>
+</tr>
+<tr class="even">
+<td align="right">0.6233856</td>
+<td align="right">82.9</td>
+</tr>
+<tr class="odd">
+<td align="right">0.6233856</td>
+<td align="right">86.7</td>
+</tr>
+<tr class="even">
+<td align="right">0.6233856</td>
+<td align="right">87.4</td>
+</tr>
+<tr class="odd">
+<td align="right">1.8701567</td>
+<td align="right">72.8</td>
+</tr>
+<tr class="even">
+<td align="right">1.8701567</td>
+<td align="right">69.9</td>
+</tr>
+<tr class="odd">
+<td align="right">1.8701567</td>
+<td align="right">71.9</td>
+</tr>
+<tr class="even">
+<td align="right">4.3636989</td>
+<td align="right">51.4</td>
+</tr>
+<tr class="odd">
+<td align="right">4.3636989</td>
+<td align="right">52.9</td>
+</tr>
+<tr class="even">
+<td align="right">4.3636989</td>
+<td align="right">48.6</td>
+</tr>
+<tr class="odd">
+<td align="right">8.7273979</td>
+<td align="right">28.5</td>
+</tr>
+<tr class="even">
+<td align="right">8.7273979</td>
+<td align="right">27.3</td>
+</tr>
+<tr class="odd">
+<td align="right">8.7273979</td>
+<td align="right">27.5</td>
+</tr>
+<tr class="even">
+<td align="right">13.0910968</td>
+<td align="right">14.8</td>
+</tr>
+<tr class="odd">
+<td align="right">13.0910968</td>
+<td align="right">13.4</td>
+</tr>
+<tr class="even">
+<td align="right">13.0910968</td>
+<td align="right">14.4</td>
+</tr>
+<tr class="odd">
+<td align="right">17.4547957</td>
+<td align="right">7.7</td>
+</tr>
+<tr class="even">
+<td align="right">17.4547957</td>
+<td align="right">7.3</td>
+</tr>
+<tr class="odd">
+<td align="right">17.4547957</td>
+<td align="right">8.1</td>
+</tr>
+<tr class="even">
+<td align="right">26.1821936</td>
+<td align="right">2.0</td>
+</tr>
+<tr class="odd">
+<td align="right">26.1821936</td>
+<td align="right">1.5</td>
+</tr>
+<tr class="even">
+<td align="right">26.1821936</td>
+<td align="right">1.9</td>
+</tr>
+<tr class="odd">
+<td align="right">34.9095915</td>
+<td align="right">1.3</td>
+</tr>
+<tr class="even">
+<td align="right">34.9095915</td>
+<td align="right">1.0</td>
+</tr>
+<tr class="odd">
+<td align="right">34.9095915</td>
+<td align="right">1.1</td>
+</tr>
+<tr class="even">
+<td align="right">43.6369893</td>
+<td align="right">0.9</td>
+</tr>
+<tr class="odd">
+<td align="right">43.6369893</td>
+<td align="right">0.7</td>
+</tr>
+<tr class="even">
+<td align="right">43.6369893</td>
+<td align="right">0.7</td>
+</tr>
+<tr class="odd">
+<td align="right">52.3643872</td>
+<td align="right">0.6</td>
+</tr>
+<tr class="even">
+<td align="right">52.3643872</td>
+<td align="right">0.4</td>
+</tr>
+<tr class="odd">
+<td align="right">52.3643872</td>
+<td align="right">0.5</td>
+</tr>
+<tr class="even">
+<td align="right">74.8062674</td>
+<td align="right">0.4</td>
+</tr>
+<tr class="odd">
+<td align="right">74.8062674</td>
+<td align="right">0.3</td>
+</tr>
+<tr class="even">
+<td align="right">74.8062674</td>
+<td align="right">0.3</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset BBA 2.2</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.0000000</td>
+<td align="right">98.09</td>
+</tr>
+<tr class="even">
+<td align="right">0.0000000</td>
+<td align="right">98.77</td>
+</tr>
+<tr class="odd">
+<td align="right">0.7678922</td>
+<td align="right">93.52</td>
+</tr>
+<tr class="even">
+<td align="right">0.7678922</td>
+<td align="right">92.03</td>
+</tr>
+<tr class="odd">
+<td align="right">2.3036765</td>
+<td align="right">88.39</td>
+</tr>
+<tr class="even">
+<td align="right">2.3036765</td>
+<td align="right">87.18</td>
+</tr>
+<tr class="odd">
+<td align="right">5.3752452</td>
+<td align="right">69.38</td>
+</tr>
+<tr class="even">
+<td align="right">5.3752452</td>
+<td align="right">71.06</td>
+</tr>
+<tr class="odd">
+<td align="right">10.7504904</td>
+<td align="right">45.21</td>
+</tr>
+<tr class="even">
+<td align="right">10.7504904</td>
+<td align="right">46.81</td>
+</tr>
+<tr class="odd">
+<td align="right">16.1257355</td>
+<td align="right">30.54</td>
+</tr>
+<tr class="even">
+<td align="right">16.1257355</td>
+<td align="right">30.07</td>
+</tr>
+<tr class="odd">
+<td align="right">21.5009807</td>
+<td align="right">21.60</td>
+</tr>
+<tr class="even">
+<td align="right">21.5009807</td>
+<td align="right">20.41</td>
+</tr>
+<tr class="odd">
+<td align="right">32.2514711</td>
+<td align="right">9.10</td>
+</tr>
+<tr class="even">
+<td align="right">32.2514711</td>
+<td align="right">9.70</td>
+</tr>
+<tr class="odd">
+<td align="right">43.0019614</td>
+<td align="right">6.58</td>
+</tr>
+<tr class="even">
+<td align="right">43.0019614</td>
+<td align="right">6.31</td>
+</tr>
+<tr class="odd">
+<td align="right">53.7524518</td>
+<td align="right">3.47</td>
+</tr>
+<tr class="even">
+<td align="right">53.7524518</td>
+<td align="right">3.52</td>
+</tr>
+<tr class="odd">
+<td align="right">64.5029421</td>
+<td align="right">3.40</td>
+</tr>
+<tr class="even">
+<td align="right">64.5029421</td>
+<td align="right">3.67</td>
+</tr>
+<tr class="odd">
+<td align="right">91.3791680</td>
+<td align="right">1.62</td>
+</tr>
+<tr class="even">
+<td align="right">91.3791680</td>
+<td align="right">1.62</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset BBA 2.3</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.0000000</td>
+<td align="right">99.33</td>
+</tr>
+<tr class="even">
+<td align="right">0.0000000</td>
+<td align="right">97.44</td>
+</tr>
+<tr class="odd">
+<td align="right">0.6733938</td>
+<td align="right">93.73</td>
+</tr>
+<tr class="even">
+<td align="right">0.6733938</td>
+<td align="right">93.77</td>
+</tr>
+<tr class="odd">
+<td align="right">2.0201814</td>
+<td align="right">87.84</td>
+</tr>
+<tr class="even">
+<td align="right">2.0201814</td>
+<td align="right">89.82</td>
+</tr>
+<tr class="odd">
+<td align="right">4.7137565</td>
+<td align="right">71.61</td>
+</tr>
+<tr class="even">
+<td align="right">4.7137565</td>
+<td align="right">71.42</td>
+</tr>
+<tr class="odd">
+<td align="right">9.4275131</td>
+<td align="right">45.60</td>
+</tr>
+<tr class="even">
+<td align="right">9.4275131</td>
+<td align="right">45.42</td>
+</tr>
+<tr class="odd">
+<td align="right">14.1412696</td>
+<td align="right">31.12</td>
+</tr>
+<tr class="even">
+<td align="right">14.1412696</td>
+<td align="right">31.68</td>
+</tr>
+<tr class="odd">
+<td align="right">18.8550262</td>
+<td align="right">23.20</td>
+</tr>
+<tr class="even">
+<td align="right">18.8550262</td>
+<td align="right">24.13</td>
+</tr>
+<tr class="odd">
+<td align="right">28.2825393</td>
+<td align="right">9.43</td>
+</tr>
+<tr class="even">
+<td align="right">28.2825393</td>
+<td align="right">9.82</td>
+</tr>
+<tr class="odd">
+<td align="right">37.7100523</td>
+<td align="right">7.08</td>
+</tr>
+<tr class="even">
+<td align="right">37.7100523</td>
+<td align="right">8.64</td>
+</tr>
+<tr class="odd">
+<td align="right">47.1375654</td>
+<td align="right">4.41</td>
+</tr>
+<tr class="even">
+<td align="right">47.1375654</td>
+<td align="right">4.78</td>
+</tr>
+<tr class="odd">
+<td align="right">56.5650785</td>
+<td align="right">4.92</td>
+</tr>
+<tr class="even">
+<td align="right">56.5650785</td>
+<td align="right">5.08</td>
+</tr>
+<tr class="odd">
+<td align="right">80.1338612</td>
+<td align="right">2.13</td>
+</tr>
+<tr class="even">
+<td align="right">80.1338612</td>
+<td align="right">2.23</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Elliot</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">97.5</td>
+</tr>
+<tr class="even">
+<td align="right">0.000000</td>
+<td align="right">100.7</td>
+</tr>
+<tr class="odd">
+<td align="right">1.228478</td>
+<td align="right">86.4</td>
+</tr>
+<tr class="even">
+<td align="right">1.228478</td>
+<td align="right">88.5</td>
+</tr>
+<tr class="odd">
+<td align="right">3.685435</td>
+<td align="right">69.8</td>
+</tr>
+<tr class="even">
+<td align="right">3.685435</td>
+<td align="right">77.1</td>
+</tr>
+<tr class="odd">
+<td align="right">8.599349</td>
+<td align="right">59.0</td>
+</tr>
+<tr class="even">
+<td align="right">8.599349</td>
+<td align="right">54.2</td>
+</tr>
+<tr class="odd">
+<td align="right">17.198697</td>
+<td align="right">31.3</td>
+</tr>
+<tr class="even">
+<td align="right">17.198697</td>
+<td align="right">33.5</td>
+</tr>
+<tr class="odd">
+<td align="right">25.798046</td>
+<td align="right">19.6</td>
+</tr>
+<tr class="even">
+<td align="right">25.798046</td>
+<td align="right">20.9</td>
+</tr>
+<tr class="odd">
+<td align="right">34.397395</td>
+<td align="right">13.3</td>
+</tr>
+<tr class="even">
+<td align="right">34.397395</td>
+<td align="right">15.8</td>
+</tr>
+<tr class="odd">
+<td align="right">51.596092</td>
+<td align="right">6.7</td>
+</tr>
+<tr class="even">
+<td align="right">51.596092</td>
+<td align="right">8.7</td>
+</tr>
+<tr class="odd">
+<td align="right">68.794789</td>
+<td align="right">8.8</td>
+</tr>
+<tr class="even">
+<td align="right">68.794789</td>
+<td align="right">8.7</td>
+</tr>
+<tr class="odd">
+<td align="right">103.192184</td>
+<td align="right">6.0</td>
+</tr>
+<tr class="even">
+<td align="right">103.192184</td>
+<td align="right">4.4</td>
+</tr>
+<tr class="odd">
+<td align="right">146.188928</td>
+<td align="right">3.3</td>
+</tr>
+<tr class="even">
+<td align="right">146.188928</td>
+<td align="right">2.8</td>
+</tr>
+<tr class="odd">
+<td align="right">223.583066</td>
+<td align="right">1.4</td>
+</tr>
+<tr class="even">
+<td align="right">223.583066</td>
+<td align="right">1.8</td>
+</tr>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">93.4</td>
+</tr>
+<tr class="even">
+<td align="right">0.000000</td>
+<td align="right">103.2</td>
+</tr>
+<tr class="odd">
+<td align="right">1.228478</td>
+<td align="right">89.2</td>
+</tr>
+<tr class="even">
+<td align="right">1.228478</td>
+<td align="right">86.6</td>
+</tr>
+<tr class="odd">
+<td align="right">3.685435</td>
+<td align="right">78.2</td>
+</tr>
+<tr class="even">
+<td align="right">3.685435</td>
+<td align="right">78.1</td>
+</tr>
+<tr class="odd">
+<td align="right">8.599349</td>
+<td align="right">55.6</td>
+</tr>
+<tr class="even">
+<td align="right">8.599349</td>
+<td align="right">53.0</td>
+</tr>
+<tr class="odd">
+<td align="right">17.198697</td>
+<td align="right">33.7</td>
+</tr>
+<tr class="even">
+<td align="right">17.198697</td>
+<td align="right">33.2</td>
+</tr>
+<tr class="odd">
+<td align="right">25.798046</td>
+<td align="right">20.9</td>
+</tr>
+<tr class="even">
+<td align="right">25.798046</td>
+<td align="right">19.9</td>
+</tr>
+<tr class="odd">
+<td align="right">34.397395</td>
+<td align="right">18.2</td>
+</tr>
+<tr class="even">
+<td align="right">34.397395</td>
+<td align="right">12.7</td>
+</tr>
+<tr class="odd">
+<td align="right">51.596092</td>
+<td align="right">7.8</td>
+</tr>
+<tr class="even">
+<td align="right">51.596092</td>
+<td align="right">9.0</td>
+</tr>
+<tr class="odd">
+<td align="right">68.794789</td>
+<td align="right">11.4</td>
+</tr>
+<tr class="even">
+<td align="right">68.794789</td>
+<td align="right">9.0</td>
+</tr>
+<tr class="odd">
+<td align="right">103.192184</td>
+<td align="right">3.9</td>
+</tr>
+<tr class="even">
+<td align="right">103.192184</td>
+<td align="right">4.4</td>
+</tr>
+<tr class="odd">
+<td align="right">146.188928</td>
+<td align="right">2.6</td>
+</tr>
+<tr class="even">
+<td align="right">146.188928</td>
+<td align="right">3.4</td>
+</tr>
+<tr class="odd">
+<td align="right">223.583066</td>
+<td align="right">2.0</td>
+</tr>
+<tr class="even">
+<td align="right">223.583066</td>
+<td align="right">1.7</td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="section level2">
+<h2 id="separate-evaluations">Separate evaluations<a class="anchor" aria-label="anchor" href="#separate-evaluations"></a>
+</h2>
+<p>In order to obtain suitable starting parameters for the NLHM fits,
+separate fits of the four models to the data for each soil are generated
+using the <code>mmkin</code> function from the <code>mkin</code>
+package. In a first step, constant variance is assumed. Convergence is
+checked with the <code>status</code> function.</p>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">deg_mods</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"FOMC"</span>, <span class="st">"DFOP"</span>, <span class="st">"HS"</span><span class="op">)</span></span>
+<span><span class="va">f_sep_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span></span>
+<span>  <span class="va">deg_mods</span>,</span>
+<span>  <span class="va">dmta_ds</span>,</span>
+<span>  error_model <span class="op">=</span> <span class="st">"const"</span>,</span>
+<span>  quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span></span>
+<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">Calke</th>
+<th align="left">Borstel</th>
+<th align="left">Flaach</th>
+<th align="left">BBA 2.2</th>
+<th align="left">BBA 2.3</th>
+<th align="left">Elliot</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">HS</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>In the table above, OK indicates convergence, and C indicates failure
+to converge. All separate fits with constant variance converged, with
+the sole exception of the HS fit to the BBA 2.2 data. To prepare for
+fitting NLHM using the two-component error model, the separate fits are
+updated assuming two-component error.</p>
+<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_sep_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_sep_const</span>, error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_tc</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">Calke</th>
+<th align="left">Borstel</th>
+<th align="left">Flaach</th>
+<th align="left">BBA 2.2</th>
+<th align="left">BBA 2.3</th>
+<th align="left">Elliot</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">HS</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>Using the two-component error model, the one fit that did not
+converge with constant variance did converge, but other non-SFO fits
+failed to converge.</p>
+</div>
+<div class="section level2">
+<h2 id="hierarchichal-model-fits">Hierarchichal model fits<a class="anchor" aria-label="anchor" href="#hierarchichal-model-fits"></a>
+</h2>
+<p>The following code fits eight versions of hierarchical models to the
+data, using SFO, FOMC, DFOP and HS for the parent compound, and using
+either constant variance or two-component error for the error model. The
+default parameter distribution model in mkin allows for variation of all
+degradation parameters across the assumed population of soils. In other
+words, each degradation parameter is associated with a random effect as
+a first step. The <code>mhmkin</code> function makes it possible to fit
+all eight versions in parallel (given a sufficient number of computing
+cores being available) to save execution time.</p>
+<p>Convergence plots and summaries for these fits are shown in the
+appendix.</p>
+<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_saem</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mhmkin.html">mhmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">f_sep_const</span>, <span class="va">f_sep_tc</span><span class="op">)</span>, transformations <span class="op">=</span> <span class="st">"saemix"</span><span class="op">)</span></span></code></pre></div>
+<p>The output of the <code>status</code> function shows that all fits
+terminated successfully.</p>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">HS</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>The AIC and BIC values show that the biphasic models DFOP and HS give
+the best fits.</p>
+<div class="sourceCode" id="cb8"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO const</td>
+<td align="right">5</td>
+<td align="right">796.3</td>
+<td align="right">795.3</td>
+<td align="right">-393.2</td>
+</tr>
+<tr class="even">
+<td align="left">SFO tc</td>
+<td align="right">6</td>
+<td align="right">798.3</td>
+<td align="right">797.1</td>
+<td align="right">-393.2</td>
+</tr>
+<tr class="odd">
+<td align="left">FOMC const</td>
+<td align="right">7</td>
+<td align="right">734.2</td>
+<td align="right">732.7</td>
+<td align="right">-360.1</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC tc</td>
+<td align="right">8</td>
+<td align="right">720.7</td>
+<td align="right">719.1</td>
+<td align="right">-352.4</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP const</td>
+<td align="right">9</td>
+<td align="right">711.8</td>
+<td align="right">710.0</td>
+<td align="right">-346.9</td>
+</tr>
+<tr class="even">
+<td align="left">HS const</td>
+<td align="right">9</td>
+<td align="right">714.0</td>
+<td align="right">712.1</td>
+<td align="right">-348.0</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP tc</td>
+<td align="right">10</td>
+<td align="right">665.7</td>
+<td align="right">663.6</td>
+<td align="right">-322.9</td>
+</tr>
+<tr class="even">
+<td align="left">HS tc</td>
+<td align="right">10</td>
+<td align="right">667.1</td>
+<td align="right">665.0</td>
+<td align="right">-323.6</td>
+</tr>
+</tbody>
+</table>
+<p>The DFOP model is preferred here, as it has a better mechanistic
+basis for batch experiments with constant incubation conditions. Also,
+it shows the lowest AIC and BIC values in the first set of fits when
+combined with the two-component error model. Therefore, the DFOP model
+was selected for further refinements of the fits with the aim to make
+the model fully identifiable.</p>
+<div class="section level3">
+<h3 id="parameter-identifiability-based-on-the-fisher-information-matrix">Parameter identifiability based on the Fisher Information
+Matrix<a class="anchor" aria-label="anchor" href="#parameter-identifiability-based-on-the-fisher-information-matrix"></a>
+</h3>
+<p>Using the <code>illparms</code> function, ill-defined statistical
+model parameters such as standard deviations of the degradation
+parameters in the population and error model parameters can be
+found.</p>
+<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left"></td>
+<td align="left">b.1</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left"></td>
+<td align="left">sd(DMTA_0)</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">sd(k2)</td>
+<td align="left">sd(k2)</td>
+</tr>
+<tr class="even">
+<td align="left">HS</td>
+<td align="left"></td>
+<td align="left">sd(tb)</td>
+</tr>
+</tbody>
+</table>
+<p>According to the <code>illparms</code> function, the fitted standard
+deviation of the second kinetic rate constant <code>k2</code> is
+ill-defined in both DFOP fits. This suggests that different values would
+be obtained for this standard deviation when using different starting
+values.</p>
+<p>The thus identified overparameterisation is addressed by removing the
+random effect for <code>k2</code> from the parameter model.</p>
+<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_saem_dfop_tc_no_ranef_k2</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span>,</span>
+<span>  no_random_effect <span class="op">=</span> <span class="st">"k2"</span><span class="op">)</span></span></code></pre></div>
+<p>For the resulting fit, it is checked whether there are still
+ill-defined parameters,</p>
+<div class="sourceCode" id="cb11"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2</span><span class="op">)</span></span></code></pre></div>
+<p>which is not the case. Below, the refined model is compared with the
+previous best model. The model without random effect for <code>k2</code>
+is a reduced version of the previous model. Therefore, the models are
+nested and can be compared using the likelihood ratio test. This is
+achieved with the argument <code>test = TRUE</code> to the
+<code>anova</code> function.</p>
+<div class="sourceCode" id="cb12"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span>, <span class="va">f_saem_dfop_tc_no_ranef_k2</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span> <span class="op">|&gt;</span></span>
+<span>  <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>format.args <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">4</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<colgroup>
+<col width="38%">
+<col width="7%">
+<col width="8%">
+<col width="8%">
+<col width="9%">
+<col width="8%">
+<col width="4%">
+<col width="15%">
+</colgroup>
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+<th align="right">Chisq</th>
+<th align="right">Df</th>
+<th align="right">Pr(&gt;Chisq)</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">f_saem_dfop_tc_no_ranef_k2</td>
+<td align="right">9</td>
+<td align="right">663.7</td>
+<td align="right">661.8</td>
+<td align="right">-322.9</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="left">f_saem[[“DFOP”, “tc”]]</td>
+<td align="right">10</td>
+<td align="right">665.7</td>
+<td align="right">663.6</td>
+<td align="right">-322.9</td>
+<td align="right">0</td>
+<td align="right">1</td>
+<td align="right">1</td>
+</tr>
+</tbody>
+</table>
+<p>The AIC and BIC criteria are lower after removal of the ill-defined
+random effect for <code>k2</code>. The p value of the likelihood ratio
+test is much greater than 0.05, indicating that the model with the
+higher likelihood (here the model with random effects for all
+degradation parameters <code>f_saem[["DFOP", "tc"]]</code>) does not fit
+significantly better than the model with the lower likelihood (the
+reduced model <code>f_saem_dfop_tc_no_ranef_k2</code>).</p>
+<p>Therefore, AIC, BIC and likelihood ratio test suggest the use of the
+reduced model.</p>
+<p>The convergence of the fit is checked visually.</p>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/convergence-saem-dfop-tc-no-ranef-k2-1.png" alt="Convergence plot for the NLHM DFOP fit with two-component error and without a random effect on 'k2'" width="864"><p class="caption">
+Convergence plot for the NLHM DFOP fit with two-component error and
+without a random effect on ‘k2’
+</p>
+</div>
+<p>All parameters appear to have converged to a satisfactory degree. The
+final fit is plotted using the plot method from the mkin package.</p>
+<div class="sourceCode" id="cb13"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/plot-saem-dfop-tc-no-ranef-k2-1.png" alt="Plot of the final NLHM DFOP fit" width="864"><p class="caption">
+Plot of the final NLHM DFOP fit
+</p>
+</div>
+<p>Finally, a summary report of the fit is produced.</p>
+<div class="sourceCode" id="cb14"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2</span><span class="op">)</span></span></code></pre></div>
+<pre><code>saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:13:19 2025 
+Date of summary: Fri Feb 14 08:13:19 2025 
+
+Equations:
+d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 4.154 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+  DMTA_0       k1       k2        g 
+98.71186  0.08675  0.01374  0.93491 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 g
+DMTA_0  98.71  0  0 0
+k1       0.00  1  0 0
+k2       0.00  0  1 0
+g        0.00  0  0 1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  663.7 661.8 -322.9
+
+Optimised parameters:
+               est.     lower     upper
+DMTA_0    98.256267 96.286112 100.22642
+k1         0.064037  0.033281   0.09479
+k2         0.008469  0.006002   0.01094
+g          0.954167  0.914460   0.99387
+a.1        1.061795  0.878608   1.24498
+b.1        0.029550  0.022593   0.03651
+SD.DMTA_0  2.068581  0.427178   3.70998
+SD.k1      0.598285  0.258235   0.93833
+SD.g       1.016689  0.360061   1.67332
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0213                
+k2  0.0541  0.0344        
+g  -0.0521 -0.0286 -0.2744
+
+Random effects:
+            est.  lower  upper
+SD.DMTA_0 2.0686 0.4272 3.7100
+SD.k1     0.5983 0.2582 0.9383
+SD.g      1.0167 0.3601 1.6733
+
+Variance model:
+       est.   lower   upper
+a.1 1.06180 0.87861 1.24498
+b.1 0.02955 0.02259 0.03651
+
+Estimated disappearance times:
+      DT50  DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.45 41.32    12.44   10.82   81.85</code></pre>
+</div>
+<div class="section level3">
+<h3 id="alternative-check-of-parameter-identifiability">Alternative check of parameter identifiability<a class="anchor" aria-label="anchor" href="#alternative-check-of-parameter-identifiability"></a>
+</h3>
+<p>The parameter check used in the <code>illparms</code> function is
+based on a quadratic approximation of the likelihood surface near its
+optimum, which is calculated using the Fisher Information Matrix (FIM).
+An alternative way to check parameter identifiability <span class="citation">(Duchesne et al. 2021)</span> based on a multistart
+approach has recently been implemented in mkin.</p>
+<p>The graph below shows boxplots of the parameters obtained in 50 runs
+of the saem algorithm with different parameter combinations, sampled
+from the range of the parameters obtained for the individual datasets
+fitted separately using nonlinear regression.</p>
+<div class="sourceCode" id="cb16"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_saem_dfop_tc_multi</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/multistart.html">multistart</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span>, n <span class="op">=</span> <span class="fl">50</span>, cores <span class="op">=</span> <span class="fl">15</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb17"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/graphics/par.html" class="external-link">par</a></span><span class="op">(</span>mar <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">6.1</span>, <span class="fl">4.1</span>, <span class="fl">2.1</span>, <span class="fl">2.1</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/parplot.html">parplot</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_multi</span>, lpos <span class="op">=</span> <span class="st">"bottomright"</span>, ylim <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.3</span>, <span class="fl">10</span><span class="op">)</span>, las <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/multistart-full-par-1.png" alt="Scaled parameters from the multistart runs, full model" width="960"><p class="caption">
+Scaled parameters from the multistart runs, full model
+</p>
+</div>
+<p>The graph clearly confirms the lack of identifiability of the
+variance of <code>k2</code> in the full model. The overparameterisation
+of the model also indicates a lack of identifiability of the variance of
+parameter <code>g</code>.</p>
+<p>The parameter boxplots of the multistart runs with the reduced model
+shown below indicate that all runs give similar results, regardless of
+the starting parameters.</p>
+<div class="sourceCode" id="cb18"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_saem_dfop_tc_no_ranef_k2_multi</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/multistart.html">multistart</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2</span>,</span>
+<span>  n <span class="op">=</span> <span class="fl">50</span>, cores <span class="op">=</span> <span class="fl">15</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb19"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/graphics/par.html" class="external-link">par</a></span><span class="op">(</span>mar <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">6.1</span>, <span class="fl">4.1</span>, <span class="fl">2.1</span>, <span class="fl">2.1</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/parplot.html">parplot</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2_multi</span>, ylim <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.5</span>, <span class="fl">2</span><span class="op">)</span>, las <span class="op">=</span> <span class="fl">2</span>,</span>
+<span>  lpos <span class="op">=</span> <span class="st">"bottomright"</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/multistart-reduced-par-1.png" alt="Scaled parameters from the multistart runs, reduced model" width="960"><p class="caption">
+Scaled parameters from the multistart runs, reduced model
+</p>
+</div>
+<p>When only the parameters of the top 25% of the fits are shown (based
+on a feature introduced in mkin 1.2.2 currently under development), the
+scatter is even less as shown below.</p>
+<div class="sourceCode" id="cb20"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/graphics/par.html" class="external-link">par</a></span><span class="op">(</span>mar <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">6.1</span>, <span class="fl">4.1</span>, <span class="fl">2.1</span>, <span class="fl">2.1</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/parplot.html">parplot</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2_multi</span>, ylim <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.5</span>, <span class="fl">2</span><span class="op">)</span>, las <span class="op">=</span> <span class="fl">2</span>, llquant <span class="op">=</span> <span class="fl">0.25</span>,</span>
+<span>  lpos <span class="op">=</span> <span class="st">"bottomright"</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/multistart-reduced-par-llquant-1.png" alt="Scaled parameters from the multistart runs, reduced model, fits with the top 25\% likelihood values" width="960"><p class="caption">
+Scaled parameters from the multistart runs, reduced model, fits with the
+top 25% likelihood values
+</p>
+</div>
+</div>
+</div>
+<div class="section level2">
+<h2 id="conclusions">Conclusions<a class="anchor" aria-label="anchor" href="#conclusions"></a>
+</h2>
+<p>Fitting the four parent degradation models SFO, FOMC, DFOP and HS as
+part of hierarchical model fits with two different error models and
+normal distributions of the transformed degradation parameters works
+without technical problems. The biphasic models DFOP and HS gave the
+best fit to the data, but the default parameter distribution model was
+not fully identifiable. Removing the random effect for the second
+kinetic rate constant of the DFOP model resulted in a reduced model that
+was fully identifiable and showed the lowest values for the model
+selection criteria AIC and BIC. The reliability of the identification of
+all model parameters was confirmed using multiple starting values.</p>
+</div>
+<div class="section level2">
+<h2 id="acknowledgements">Acknowledgements<a class="anchor" aria-label="anchor" href="#acknowledgements"></a>
+</h2>
+<p>The helpful comments by Janina Wöltjen of the German Environment
+Agency are gratefully acknowledged.</p>
+</div>
+<div class="section level2">
+<h2 id="references">References<a class="anchor" aria-label="anchor" href="#references"></a>
+</h2>
+<div id="refs" class="references csl-bib-body hanging-indent">
+<div id="ref-duchesne_2021" class="csl-entry">
+Duchesne, Ronan, Anissa Guillemin, Olivier Gandrillon, and Fabien
+Crauste. 2021. <span>“Practical Identifiability in the Frame of
+Nonlinear Mixed Effects Models: The Example of the in Vitro
+Erythropoiesis.”</span> <em>BMC Bioinformatics</em> 22 (478). <a href="https://doi.org/10.1186/s12859-021-04373-4" class="external-link">https://doi.org/10.1186/s12859-021-04373-4</a>.
+</div>
+</div>
+</div>
+<div class="section level2">
+<h2 id="appendix">Appendix<a class="anchor" aria-label="anchor" href="#appendix"></a>
+</h2>
+<div class="section level3">
+<h3 id="hierarchical-model-fit-listings">Hierarchical model fit listings<a class="anchor" aria-label="anchor" href="#hierarchical-model-fit-listings"></a>
+</h3>
+<caption>
+Hierarchical mkin fit of the SFO model with error model const
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:13:11 2025 
+Date of summary: Fri Feb 14 08:14:22 2025 
+
+Equations:
+d_DMTA/dt = - k_DMTA * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 0.869 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+ DMTA_0  k_DMTA 
+97.2953  0.0566 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k_DMTA
+DMTA_0   97.3      0
+k_DMTA    0.0      1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  796.3 795.3 -393.2
+
+Optimised parameters:
+              est.    lower   upper
+DMTA_0    97.28130 95.71113 98.8515
+k_DMTA     0.05665  0.02909  0.0842
+a.1        2.66442  2.35579  2.9731
+SD.DMTA_0  1.54776  0.15447  2.9411
+SD.k_DMTA  0.60690  0.26248  0.9513
+
+Correlation: 
+       DMTA_0
+k_DMTA 0.0168
+
+Random effects:
+            est.  lower  upper
+SD.DMTA_0 1.5478 0.1545 2.9411
+SD.k_DMTA 0.6069 0.2625 0.9513
+
+Variance model:
+     est. lower upper
+a.1 2.664 2.356 2.973
+
+Estimated disappearance times:
+      DT50  DT90
+DMTA 12.24 40.65
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical mkin fit of the SFO model with error model tc
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:13:13 2025 
+Date of summary: Fri Feb 14 08:14:22 2025 
+
+Equations:
+d_DMTA/dt = - k_DMTA * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 2.423 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+  DMTA_0   k_DMTA 
+96.99175  0.05603 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k_DMTA
+DMTA_0  96.99      0
+k_DMTA   0.00      1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  798.3 797.1 -393.2
+
+Optimised parameters:
+               est.    lower    upper
+DMTA_0    97.271822 95.70316 98.84049
+k_DMTA     0.056638  0.02911  0.08417
+a.1        2.660081  2.27492  3.04525
+b.1        0.001665 -0.14451  0.14784
+SD.DMTA_0  1.545520  0.14301  2.94803
+SD.k_DMTA  0.606422  0.26227  0.95057
+
+Correlation: 
+       DMTA_0
+k_DMTA 0.0169
+
+Random effects:
+            est.  lower  upper
+SD.DMTA_0 1.5455 0.1430 2.9480
+SD.k_DMTA 0.6064 0.2623 0.9506
+
+Variance model:
+        est.   lower  upper
+a.1 2.660081  2.2749 3.0452
+b.1 0.001665 -0.1445 0.1478
+
+Estimated disappearance times:
+      DT50  DT90
+DMTA 12.24 40.65
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical mkin fit of the FOMC model with error model const
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:13:11 2025 
+Date of summary: Fri Feb 14 08:14:22 2025 
+
+Equations:
+d_DMTA/dt = - (alpha/beta) * 1/((time/beta) + 1) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 1.228 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+ DMTA_0   alpha    beta 
+ 98.292   9.909 156.341 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 alpha beta
+DMTA_0  98.29     0    0
+alpha    0.00     1    0
+beta     0.00     0    1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  734.2 732.7 -360.1
+
+Optimised parameters:
+              est.   lower   upper
+DMTA_0     98.3435 96.9033  99.784
+alpha       7.2007  2.5889  11.812
+beta      112.8745 34.8816 190.867
+a.1         2.0459  1.8054   2.286
+SD.DMTA_0   1.4795  0.2717   2.687
+SD.alpha    0.6396  0.1509   1.128
+SD.beta     0.6874  0.1587   1.216
+
+Correlation: 
+      DMTA_0  alpha  
+alpha -0.1125        
+beta  -0.1227  0.3632
+
+Random effects:
+            est.  lower upper
+SD.DMTA_0 1.4795 0.2717 2.687
+SD.alpha  0.6396 0.1509 1.128
+SD.beta   0.6874 0.1587 1.216
+
+Variance model:
+     est. lower upper
+a.1 2.046 1.805 2.286
+
+Estimated disappearance times:
+      DT50  DT90 DT50back
+DMTA 11.41 42.53     12.8
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical mkin fit of the FOMC model with error model tc
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:13:13 2025 
+Date of summary: Fri Feb 14 08:14:22 2025 
+
+Equations:
+d_DMTA/dt = - (alpha/beta) * 1/((time/beta) + 1) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 2.87 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+DMTA_0  alpha   beta 
+98.772  4.663 92.597 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 alpha beta
+DMTA_0  98.77     0    0
+alpha    0.00     1    0
+beta     0.00     0    1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  720.7 719.1 -352.4
+
+Optimised parameters:
+              est.    lower     upper
+DMTA_0    99.10577 97.33296 100.87859
+alpha      5.46260  2.52199   8.40321
+beta      81.66080 30.46664 132.85497
+a.1        1.50219  1.25801   1.74636
+b.1        0.02893  0.02048   0.03739
+SD.DMTA_0  1.61887 -0.03843   3.27618
+SD.alpha   0.58145  0.17364   0.98925
+SD.beta    0.68205  0.21108   1.15302
+
+Correlation: 
+      DMTA_0  alpha  
+alpha -0.1321        
+beta  -0.1430  0.2467
+
+Random effects:
+            est.    lower  upper
+SD.DMTA_0 1.6189 -0.03843 3.2762
+SD.alpha  0.5814  0.17364 0.9892
+SD.beta   0.6821  0.21108 1.1530
+
+Variance model:
+       est.   lower   upper
+a.1 1.50219 1.25801 1.74636
+b.1 0.02893 0.02048 0.03739
+
+Estimated disappearance times:
+      DT50  DT90 DT50back
+DMTA 11.05 42.81    12.89
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical mkin fit of the DFOP model with error model const
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:13:12 2025 
+Date of summary: Fri Feb 14 08:14:22 2025 
+
+Equations:
+d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 1.843 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+  DMTA_0       k1       k2        g 
+98.64383  0.09211  0.02999  0.76814 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 g
+DMTA_0  98.64  0  0 0
+k1       0.00  1  0 0
+k2       0.00  0  1 0
+g        0.00  0  0 1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC BIC logLik
+  711.8 710 -346.9
+
+Optimised parameters:
+               est.     lower    upper
+DMTA_0    98.092481 96.573899 99.61106
+k1         0.062499  0.030336  0.09466
+k2         0.009065 -0.005133  0.02326
+g          0.948967  0.862080  1.03586
+a.1        1.821671  1.604774  2.03857
+SD.DMTA_0  1.677785  0.472066  2.88350
+SD.k1      0.634962  0.270788  0.99914
+SD.k2      1.033498 -0.205994  2.27299
+SD.g       1.710046  0.428642  2.99145
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0246                
+k2  0.0491  0.0953        
+g  -0.0552 -0.0889 -0.4795
+
+Random effects:
+           est.   lower  upper
+SD.DMTA_0 1.678  0.4721 2.8835
+SD.k1     0.635  0.2708 0.9991
+SD.k2     1.033 -0.2060 2.2730
+SD.g      1.710  0.4286 2.9914
+
+Variance model:
+     est. lower upper
+a.1 1.822 1.605 2.039
+
+Estimated disappearance times:
+      DT50 DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.79 42.8    12.88   11.09   76.46
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical mkin fit of the DFOP model with error model tc
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:13:14 2025 
+Date of summary: Fri Feb 14 08:14:22 2025 
+
+Equations:
+d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 3.469 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+  DMTA_0       k1       k2        g 
+98.71186  0.08675  0.01374  0.93491 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 g
+DMTA_0  98.71  0  0 0
+k1       0.00  1  0 0
+k2       0.00  0  1 0
+g        0.00  0  0 1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  665.7 663.6 -322.9
+
+Optimised parameters:
+               est.     lower     upper
+DMTA_0    98.347470 96.380815 100.31413
+k1         0.064524  0.034279   0.09477
+k2         0.008304  0.005843   0.01076
+g          0.952128  0.909578   0.99468
+a.1        1.068907  0.883665   1.25415
+b.1        0.029265  0.022318   0.03621
+SD.DMTA_0  2.065796  0.427951   3.70364
+SD.k1      0.583703  0.251796   0.91561
+SD.k2      0.004167 -7.831228   7.83956
+SD.g       1.064450  0.397479   1.73142
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0223                
+k2  0.0568  0.0394        
+g  -0.0464 -0.0269 -0.2713
+
+Random effects:
+              est.   lower  upper
+SD.DMTA_0 2.065796  0.4280 3.7036
+SD.k1     0.583703  0.2518 0.9156
+SD.k2     0.004167 -7.8312 7.8396
+SD.g      1.064450  0.3975 1.7314
+
+Variance model:
+       est.   lower   upper
+a.1 1.06891 0.88367 1.25415
+b.1 0.02927 0.02232 0.03621
+
+Estimated disappearance times:
+      DT50  DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.39 41.36    12.45   10.74   83.48
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical mkin fit of the HS model with error model const
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:13:12 2025 
+Date of summary: Fri Feb 14 08:14:22 2025 
+
+Equations:
+d_DMTA/dt = - ifelse(time &lt;= tb, k1, k2) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 1.698 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+  DMTA_0       k1       k2       tb 
+97.82176  0.06931  0.02997 11.13945 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 tb
+DMTA_0  97.82  0  0  0
+k1       0.00  1  0  0
+k2       0.00  0  1  0
+tb       0.00  0  0  1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+  AIC   BIC logLik
+  714 712.1   -348
+
+Optimised parameters:
+              est.    lower    upper
+DMTA_0    98.16102 96.47747 99.84456
+k1         0.07876  0.05261  0.10491
+k2         0.02227  0.01706  0.02747
+tb        13.99089 -7.40049 35.38228
+a.1        1.82305  1.60700  2.03910
+SD.DMTA_0  1.88413  0.56204  3.20622
+SD.k1      0.34292  0.10482  0.58102
+SD.k2      0.19851  0.01718  0.37985
+SD.tb      1.68168  0.58064  2.78272
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0142                
+k2  0.0001 -0.0025        
+tb  0.0165 -0.1256 -0.0301
+
+Random effects:
+            est.   lower  upper
+SD.DMTA_0 1.8841 0.56204 3.2062
+SD.k1     0.3429 0.10482 0.5810
+SD.k2     0.1985 0.01718 0.3798
+SD.tb     1.6817 0.58064 2.7827
+
+Variance model:
+     est. lower upper
+a.1 1.823 1.607 2.039
+
+Estimated disappearance times:
+      DT50  DT90 DT50back DT50_k1 DT50_k2
+DMTA 8.801 67.91    20.44   8.801   31.13
+
+</code></pre>
+<p></p>
+<caption>
+Hierarchical mkin fit of the HS model with error model tc
+</caption>
+<pre><code>
+saemix version used for fitting:      3.3 
+mkin version used for pre-fitting:  1.2.10 
+R version used for fitting:         4.4.2 
+Date of fit:     Fri Feb 14 08:13:13 2025 
+Date of summary: Fri Feb 14 08:14:22 2025 
+
+Equations:
+d_DMTA/dt = - ifelse(time &lt;= tb, k1, k2) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 3.307 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+  DMTA_0       k1       k2       tb 
+98.45190  0.07525  0.02576 19.19375 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 tb
+DMTA_0  98.45  0  0  0
+k1       0.00  1  0  0
+k2       0.00  0  1  0
+tb       0.00  0  0  1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC BIC logLik
+  667.1 665 -323.6
+
+Optimised parameters:
+              est.    lower    upper
+DMTA_0    97.76571 95.81350 99.71791
+k1         0.05855  0.03080  0.08630
+k2         0.02337  0.01664  0.03010
+tb        31.09638 29.38289 32.80987
+a.1        1.08835  0.90059  1.27611
+b.1        0.02964  0.02261  0.03667
+SD.DMTA_0  2.04877  0.42553  3.67200
+SD.k1      0.59166  0.25621  0.92711
+SD.k2      0.30698  0.09561  0.51835
+SD.tb      0.01274 -0.10915  0.13464
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0160                
+k2 -0.0070 -0.0024        
+tb -0.0668 -0.0103 -0.2013
+
+Random effects:
+             est.    lower  upper
+SD.DMTA_0 2.04877  0.42553 3.6720
+SD.k1     0.59166  0.25621 0.9271
+SD.k2     0.30698  0.09561 0.5183
+SD.tb     0.01274 -0.10915 0.1346
+
+Variance model:
+       est.   lower   upper
+a.1 1.08835 0.90059 1.27611
+b.1 0.02964 0.02261 0.03667
+
+Estimated disappearance times:
+      DT50  DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.84 51.71    15.57   11.84   29.66
+
+</code></pre>
+<p></p>
+</div>
+<div class="section level3">
+<h3 id="hierarchical-model-convergence-plots">Hierarchical model convergence plots<a class="anchor" aria-label="anchor" href="#hierarchical-model-convergence-plots"></a>
+</h3>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/convergence-saem-sfo-const-1.png" alt="Convergence plot for the NLHM SFO fit with constant variance" width="864"><p class="caption">
+Convergence plot for the NLHM SFO fit with constant variance
+</p>
+</div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/convergence-saem-sfo-tc-1.png" alt="Convergence plot for the NLHM SFO fit with two-component error" width="864"><p class="caption">
+Convergence plot for the NLHM SFO fit with two-component error
+</p>
+</div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/convergence-saem-fomc-const-1.png" alt="Convergence plot for the NLHM FOMC fit with constant variance" width="864"><p class="caption">
+Convergence plot for the NLHM FOMC fit with constant variance
+</p>
+</div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/convergence-saem-fomc-tc-1.png" alt="Convergence plot for the NLHM FOMC fit with two-component error" width="864"><p class="caption">
+Convergence plot for the NLHM FOMC fit with two-component error
+</p>
+</div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/convergence-saem-dfop-const-1.png" alt="Convergence plot for the NLHM DFOP fit with constant variance" width="864"><p class="caption">
+Convergence plot for the NLHM DFOP fit with constant variance
+</p>
+</div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/convergence-saem-dfop-tc-1.png" alt="Convergence plot for the NLHM DFOP fit with two-component error" width="864"><p class="caption">
+Convergence plot for the NLHM DFOP fit with two-component error
+</p>
+</div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/convergence-saem-hs-const-1.png" alt="Convergence plot for the NLHM HS fit with constant variance" width="864"><p class="caption">
+Convergence plot for the NLHM HS fit with constant variance
+</p>
+</div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_parent_files/figure-html/convergence-saem-hs-tc-1.png" alt="Convergence plot for the NLHM HS fit with two-component error" width="864"><p class="caption">
+Convergence plot for the NLHM HS fit with two-component error
+</p>
+</div>
+</div>
+<div class="section level3">
+<h3 id="session-info">Session info<a class="anchor" aria-label="anchor" href="#session-info"></a>
+</h3>
+<pre><code>R version 4.4.2 (2024-10-31)
+Platform: x86_64-pc-linux-gnu
+Running under: Debian GNU/Linux 12 (bookworm)
+
+Matrix products: default
+BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.11.0 
+LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.11.0
+
+locale:
+ [1] LC_CTYPE=de_DE.UTF-8       LC_NUMERIC=C              
+ [3] LC_TIME=de_DE.UTF-8        LC_COLLATE=de_DE.UTF-8    
+ [5] LC_MONETARY=de_DE.UTF-8    LC_MESSAGES=de_DE.UTF-8   
+ [7] LC_PAPER=de_DE.UTF-8       LC_NAME=C                 
+ [9] LC_ADDRESS=C               LC_TELEPHONE=C            
+[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C       
+
+time zone: Europe/Berlin
+tzcode source: system (glibc)
+
+attached base packages:
+[1] parallel  stats     graphics  grDevices utils     datasets  methods  
+[8] base     
+
+other attached packages:
+[1] rmarkdown_2.29  nvimcom_0.9-167 saemix_3.3      npde_3.5       
+[5] knitr_1.49      mkin_1.2.10    
+
+loaded via a namespace (and not attached):
+ [1] gtable_0.3.6      jsonlite_1.8.9    dplyr_1.1.4       compiler_4.4.2   
+ [5] tidyselect_1.2.1  gridExtra_2.3     jquerylib_0.1.4   systemfonts_1.1.0
+ [9] scales_1.3.0      textshaping_0.4.1 yaml_2.3.10       fastmap_1.2.0    
+[13] lattice_0.22-6    ggplot2_3.5.1     R6_2.5.1          generics_0.1.3   
+[17] lmtest_0.9-40     MASS_7.3-61       htmlwidgets_1.6.4 tibble_3.2.1     
+[21] desc_1.4.3        munsell_0.5.1     bslib_0.8.0       pillar_1.9.0     
+[25] rlang_1.1.4       utf8_1.2.4        cachem_1.1.0      xfun_0.49        
+[29] fs_1.6.5          sass_0.4.9        cli_3.6.3         pkgdown_2.1.1    
+[33] magrittr_2.0.3    digest_0.6.37     grid_4.4.2        mclust_6.1.1     
+[37] lifecycle_1.0.4   nlme_3.1-166      vctrs_0.6.5       evaluate_1.0.1   
+[41] glue_1.8.0        codetools_0.2-20  ragg_1.3.3        zoo_1.8-12       
+[45] fansi_1.0.6       colorspace_2.1-1  pkgconfig_2.0.3   tools_4.4.2      
+[49] htmltools_0.5.8.1</code></pre>
+</div>
+<div class="section level3">
+<h3 id="hardware-info">Hardware info<a class="anchor" aria-label="anchor" href="#hardware-info"></a>
+</h3>
+<pre><code>CPU model: AMD Ryzen 9 7950X 16-Core Processor</code></pre>
+<pre><code>MemTotal:       64927788 kB</code></pre>
+</div>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
+</div>
+
+    </footer>
+</div>
+
+
+
+
+
+  </body>
+</html>
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+    <li><a class="dropdown-item" href="../../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a></li>
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+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
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+    <li><a class="dropdown-item" href="../../articles/prebuilt/2023_mesotrione_parent.html">Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_cyan_pathway.html">Testing hierarchical pathway kinetics with residue data on cyantraniliprole</a></li>
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+      <h1>Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change on 20 April 2023,
+last compiled on 14 Februar 2025</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/prebuilt/2022_dmta_pathway.rmd" class="external-link"><code>vignettes/prebuilt/2022_dmta_pathway.rmd</code></a></small>
+      <div class="d-none name"><code>2022_dmta_pathway.rmd</code></div>
+    </div>
+
+    
+    
+<div class="section level2">
+<h2 id="introduction">Introduction<a class="anchor" aria-label="anchor" href="#introduction"></a>
+</h2>
+<p>The purpose of this document is to test demonstrate how nonlinear
+hierarchical models (NLHM) based on the parent degradation models SFO,
+FOMC, DFOP and HS, with parallel formation of two or more metabolites
+can be fitted with the mkin package.</p>
+<p>It was assembled in the course of work package 1.2 of Project Number
+173340 (Application of nonlinear hierarchical models to the kinetic
+evaluation of chemical degradation data) of the German Environment
+Agency carried out in 2022 and 2023.</p>
+<p>The mkin package is used in version 1.2.10, which is currently under
+development. It contains the test data, and the functions used in the
+evaluations. The <code>saemix</code> package is used as a backend for
+fitting the NLHM, but is also loaded to make the convergence plot
+function available.</p>
+<p>This document is processed with the <code>knitr</code> package, which
+also provides the <code>kable</code> function that is used to improve
+the display of tabular data in R markdown documents. For parallel
+processing, the <code>parallel</code> package is used.</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://pkgdown.jrwb.de/mkin/">mkin</a></span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://yihui.org/knitr/" class="external-link">knitr</a></span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">saemix</span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">parallel</span><span class="op">)</span></span>
+<span><span class="va">n_cores</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/detectCores.html" class="external-link">detectCores</a></span><span class="op">(</span><span class="op">)</span></span>
+<span></span>
+<span><span class="co"># We need to start a new cluster after defining a compiled model that is</span></span>
+<span><span class="co"># saved as a DLL to the user directory, therefore we define a function</span></span>
+<span><span class="co"># This is used again after defining the pathway model</span></span>
+<span><span class="va">start_cluster</span> <span class="op">&lt;-</span> <span class="kw">function</span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="kw">if</span> <span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/Sys.info.html" class="external-link">Sys.info</a></span><span class="op">(</span><span class="op">)</span><span class="op">[</span><span class="st">"sysname"</span><span class="op">]</span> <span class="op">==</span> <span class="st">"Windows"</span><span class="op">)</span> <span class="op">{</span></span>
+<span>    <span class="va">ret</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">makePSOCKcluster</a></span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
+<span>  <span class="op">}</span> <span class="kw">else</span> <span class="op">{</span></span>
+<span>    <span class="va">ret</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">makeForkCluster</a></span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
+<span>  <span class="op">}</span></span>
+<span>  <span class="kw"><a href="https://rdrr.io/r/base/function.html" class="external-link">return</a></span><span class="op">(</span><span class="va">ret</span><span class="op">)</span></span>
+<span><span class="op">}</span></span></code></pre></div>
+</div>
+<div class="section level2">
+<h2 id="data">Data<a class="anchor" aria-label="anchor" href="#data"></a>
+</h2>
+<p>The test data are available in the mkin package as an object of class
+<code>mkindsg</code> (mkin dataset group) under the identifier
+<code>dimethenamid_2018</code>. The following preprocessing steps are
+done in this document.</p>
+<ul>
+<li>The data available for the enantiomer dimethenamid-P (DMTAP) are
+renamed to have the same substance name as the data for the racemic
+mixture dimethenamid (DMTA). The reason for this is that no difference
+between their degradation behaviour was identified in the EU risk
+assessment.</li>
+<li>Unnecessary columns are discarded</li>
+<li>The observation times of each dataset are multiplied with the
+corresponding normalisation factor also available in the dataset, in
+order to make it possible to describe all datasets with a single set of
+parameters that are independent of temperature</li>
+<li>Finally, datasets observed in the same soil (<code>Elliot 1</code>
+and <code>Elliot 2</code>) are combined, resulting in dimethenamid
+(DMTA) data from six soils.</li>
+</ul>
+<p>The following commented R code performs this preprocessing.</p>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="co"># Apply a function to each of the seven datasets in the mkindsg object to create a list</span></span>
+<span><span class="va">dmta_ds</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">lapply</a></span><span class="op">(</span><span class="fl">1</span><span class="op">:</span><span class="fl">7</span>, <span class="kw">function</span><span class="op">(</span><span class="va">i</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="va">ds_i</span> <span class="op">&lt;-</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span><span class="op">[[</span><span class="va">i</span><span class="op">]</span><span class="op">]</span><span class="op">$</span><span class="va">data</span>                     <span class="co"># Get a dataset</span></span>
+<span>  <span class="va">ds_i</span><span class="op">[</span><span class="va">ds_i</span><span class="op">$</span><span class="va">name</span> <span class="op">==</span> <span class="st">"DMTAP"</span>, <span class="st">"name"</span><span class="op">]</span> <span class="op">&lt;-</span>  <span class="st">"DMTA"</span>              <span class="co"># Rename DMTAP to DMTA</span></span>
+<span>  <span class="va">ds_i</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/subset.html" class="external-link">subset</a></span><span class="op">(</span><span class="va">ds_i</span>, select <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"name"</span>, <span class="st">"time"</span>, <span class="st">"value"</span><span class="op">)</span><span class="op">)</span> <span class="co"># Select data</span></span>
+<span>  <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">&lt;-</span> <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">*</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">f_time_norm</span><span class="op">[</span><span class="va">i</span><span class="op">]</span>  <span class="co"># Normalise time</span></span>
+<span>  <span class="va">ds_i</span>                                                       <span class="co"># Return the dataset</span></span>
+<span><span class="op">}</span><span class="op">)</span></span>
+<span></span>
+<span><span class="co"># Use dataset titles as names for the list elements</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">)</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">sapply</a></span><span class="op">(</span><span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span>, <span class="kw">function</span><span class="op">(</span><span class="va">ds</span><span class="op">)</span> <span class="va">ds</span><span class="op">$</span><span class="va">title</span><span class="op">)</span></span>
+<span></span>
+<span><span class="co"># Combine data for Elliot soil to obtain a named list with six elements</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span>, <span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span> <span class="co">#</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="cn">NULL</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="cn">NULL</span></span></code></pre></div>
+<p>The following tables show the 6 datasets.</p>
+<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw">for</span> <span class="op">(</span><span class="va">ds_name</span> <span class="kw">in</span> <span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">)</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span></span>
+<span>    <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="fu"><a href="../../reference/mkin_long_to_wide.html">mkin_long_to_wide</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">[[</span><span class="va">ds_name</span><span class="op">]</span><span class="op">]</span><span class="op">)</span>,</span>
+<span>      caption <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/paste.html" class="external-link">paste</a></span><span class="op">(</span><span class="st">"Dataset"</span>, <span class="va">ds_name</span><span class="op">)</span>,</span>
+<span>      booktabs <span class="op">=</span> <span class="cn">TRUE</span>, row.names <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span><span class="op">)</span></span>
+<span>    <span class="fu"><a href="https://rdrr.io/r/base/cat.html" class="external-link">cat</a></span><span class="op">(</span><span class="st">"\n\\clearpage\n"</span><span class="op">)</span></span>
+<span><span class="op">}</span></span></code></pre></div>
+<table class="table">
+<caption>Dataset Calke</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+<th align="right">M23</th>
+<th align="right">M27</th>
+<th align="right">M31</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0</td>
+<td align="right">95.8</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">0</td>
+<td align="right">98.7</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">14</td>
+<td align="right">60.5</td>
+<td align="right">4.1</td>
+<td align="right">1.5</td>
+<td align="right">2.0</td>
+</tr>
+<tr class="even">
+<td align="right">30</td>
+<td align="right">39.1</td>
+<td align="right">5.3</td>
+<td align="right">2.4</td>
+<td align="right">2.1</td>
+</tr>
+<tr class="odd">
+<td align="right">59</td>
+<td align="right">15.2</td>
+<td align="right">6.0</td>
+<td align="right">3.2</td>
+<td align="right">2.2</td>
+</tr>
+<tr class="even">
+<td align="right">120</td>
+<td align="right">4.8</td>
+<td align="right">4.3</td>
+<td align="right">3.8</td>
+<td align="right">1.8</td>
+</tr>
+<tr class="odd">
+<td align="right">120</td>
+<td align="right">4.6</td>
+<td align="right">4.1</td>
+<td align="right">3.7</td>
+<td align="right">2.1</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Borstel</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+<th align="right">M23</th>
+<th align="right">M27</th>
+<th align="right">M31</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">100.5</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">0.000000</td>
+<td align="right">99.6</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">1.941295</td>
+<td align="right">91.9</td>
+<td align="right">0.4</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">1.941295</td>
+<td align="right">91.3</td>
+<td align="right">0.5</td>
+<td align="right">0.3</td>
+<td align="right">0.1</td>
+</tr>
+<tr class="odd">
+<td align="right">6.794534</td>
+<td align="right">81.8</td>
+<td align="right">1.2</td>
+<td align="right">0.8</td>
+<td align="right">1.0</td>
+</tr>
+<tr class="even">
+<td align="right">6.794534</td>
+<td align="right">82.1</td>
+<td align="right">1.3</td>
+<td align="right">0.9</td>
+<td align="right">0.9</td>
+</tr>
+<tr class="odd">
+<td align="right">13.589067</td>
+<td align="right">69.1</td>
+<td align="right">2.8</td>
+<td align="right">1.4</td>
+<td align="right">2.0</td>
+</tr>
+<tr class="even">
+<td align="right">13.589067</td>
+<td align="right">68.0</td>
+<td align="right">2.0</td>
+<td align="right">1.4</td>
+<td align="right">2.5</td>
+</tr>
+<tr class="odd">
+<td align="right">27.178135</td>
+<td align="right">51.4</td>
+<td align="right">2.9</td>
+<td align="right">2.7</td>
+<td align="right">4.3</td>
+</tr>
+<tr class="even">
+<td align="right">27.178135</td>
+<td align="right">51.4</td>
+<td align="right">4.9</td>
+<td align="right">2.6</td>
+<td align="right">3.2</td>
+</tr>
+<tr class="odd">
+<td align="right">56.297565</td>
+<td align="right">27.6</td>
+<td align="right">12.2</td>
+<td align="right">4.4</td>
+<td align="right">4.3</td>
+</tr>
+<tr class="even">
+<td align="right">56.297565</td>
+<td align="right">26.8</td>
+<td align="right">12.2</td>
+<td align="right">4.7</td>
+<td align="right">4.8</td>
+</tr>
+<tr class="odd">
+<td align="right">86.387643</td>
+<td align="right">15.7</td>
+<td align="right">12.2</td>
+<td align="right">5.4</td>
+<td align="right">5.0</td>
+</tr>
+<tr class="even">
+<td align="right">86.387643</td>
+<td align="right">15.3</td>
+<td align="right">12.0</td>
+<td align="right">5.2</td>
+<td align="right">5.1</td>
+</tr>
+<tr class="odd">
+<td align="right">115.507073</td>
+<td align="right">7.9</td>
+<td align="right">10.4</td>
+<td align="right">5.4</td>
+<td align="right">4.3</td>
+</tr>
+<tr class="even">
+<td align="right">115.507073</td>
+<td align="right">8.1</td>
+<td align="right">11.6</td>
+<td align="right">5.4</td>
+<td align="right">4.4</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Flaach</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+<th align="right">M23</th>
+<th align="right">M27</th>
+<th align="right">M31</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.0000000</td>
+<td align="right">96.5</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">0.0000000</td>
+<td align="right">96.8</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">0.0000000</td>
+<td align="right">97.0</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">0.6233856</td>
+<td align="right">82.9</td>
+<td align="right">0.7</td>
+<td align="right">1.1</td>
+<td align="right">0.3</td>
+</tr>
+<tr class="odd">
+<td align="right">0.6233856</td>
+<td align="right">86.7</td>
+<td align="right">0.7</td>
+<td align="right">1.1</td>
+<td align="right">0.3</td>
+</tr>
+<tr class="even">
+<td align="right">0.6233856</td>
+<td align="right">87.4</td>
+<td align="right">0.2</td>
+<td align="right">0.3</td>
+<td align="right">0.1</td>
+</tr>
+<tr class="odd">
+<td align="right">1.8701567</td>
+<td align="right">72.8</td>
+<td align="right">2.2</td>
+<td align="right">2.6</td>
+<td align="right">0.7</td>
+</tr>
+<tr class="even">
+<td align="right">1.8701567</td>
+<td align="right">69.9</td>
+<td align="right">1.8</td>
+<td align="right">2.4</td>
+<td align="right">0.6</td>
+</tr>
+<tr class="odd">
+<td align="right">1.8701567</td>
+<td align="right">71.9</td>
+<td align="right">1.6</td>
+<td align="right">2.3</td>
+<td align="right">0.7</td>
+</tr>
+<tr class="even">
+<td align="right">4.3636989</td>
+<td align="right">51.4</td>
+<td align="right">4.1</td>
+<td align="right">5.0</td>
+<td align="right">1.3</td>
+</tr>
+<tr class="odd">
+<td align="right">4.3636989</td>
+<td align="right">52.9</td>
+<td align="right">4.2</td>
+<td align="right">5.9</td>
+<td align="right">1.2</td>
+</tr>
+<tr class="even">
+<td align="right">4.3636989</td>
+<td align="right">48.6</td>
+<td align="right">4.2</td>
+<td align="right">4.8</td>
+<td align="right">1.4</td>
+</tr>
+<tr class="odd">
+<td align="right">8.7273979</td>
+<td align="right">28.5</td>
+<td align="right">7.5</td>
+<td align="right">8.5</td>
+<td align="right">2.4</td>
+</tr>
+<tr class="even">
+<td align="right">8.7273979</td>
+<td align="right">27.3</td>
+<td align="right">7.1</td>
+<td align="right">8.5</td>
+<td align="right">2.1</td>
+</tr>
+<tr class="odd">
+<td align="right">8.7273979</td>
+<td align="right">27.5</td>
+<td align="right">7.5</td>
+<td align="right">8.3</td>
+<td align="right">2.3</td>
+</tr>
+<tr class="even">
+<td align="right">13.0910968</td>
+<td align="right">14.8</td>
+<td align="right">8.4</td>
+<td align="right">9.3</td>
+<td align="right">3.3</td>
+</tr>
+<tr class="odd">
+<td align="right">13.0910968</td>
+<td align="right">13.4</td>
+<td align="right">6.8</td>
+<td align="right">8.7</td>
+<td align="right">2.4</td>
+</tr>
+<tr class="even">
+<td align="right">13.0910968</td>
+<td align="right">14.4</td>
+<td align="right">8.0</td>
+<td align="right">9.1</td>
+<td align="right">2.6</td>
+</tr>
+<tr class="odd">
+<td align="right">17.4547957</td>
+<td align="right">7.7</td>
+<td align="right">7.2</td>
+<td align="right">8.6</td>
+<td align="right">4.0</td>
+</tr>
+<tr class="even">
+<td align="right">17.4547957</td>
+<td align="right">7.3</td>
+<td align="right">7.2</td>
+<td align="right">8.5</td>
+<td align="right">3.6</td>
+</tr>
+<tr class="odd">
+<td align="right">17.4547957</td>
+<td align="right">8.1</td>
+<td align="right">6.9</td>
+<td align="right">8.9</td>
+<td align="right">3.3</td>
+</tr>
+<tr class="even">
+<td align="right">26.1821936</td>
+<td align="right">2.0</td>
+<td align="right">4.9</td>
+<td align="right">8.1</td>
+<td align="right">2.1</td>
+</tr>
+<tr class="odd">
+<td align="right">26.1821936</td>
+<td align="right">1.5</td>
+<td align="right">4.3</td>
+<td align="right">7.7</td>
+<td align="right">1.7</td>
+</tr>
+<tr class="even">
+<td align="right">26.1821936</td>
+<td align="right">1.9</td>
+<td align="right">4.5</td>
+<td align="right">7.4</td>
+<td align="right">1.8</td>
+</tr>
+<tr class="odd">
+<td align="right">34.9095915</td>
+<td align="right">1.3</td>
+<td align="right">3.8</td>
+<td align="right">5.9</td>
+<td align="right">1.6</td>
+</tr>
+<tr class="even">
+<td align="right">34.9095915</td>
+<td align="right">1.0</td>
+<td align="right">3.1</td>
+<td align="right">6.0</td>
+<td align="right">1.6</td>
+</tr>
+<tr class="odd">
+<td align="right">34.9095915</td>
+<td align="right">1.1</td>
+<td align="right">3.1</td>
+<td align="right">5.9</td>
+<td align="right">1.4</td>
+</tr>
+<tr class="even">
+<td align="right">43.6369893</td>
+<td align="right">0.9</td>
+<td align="right">2.7</td>
+<td align="right">5.6</td>
+<td align="right">1.8</td>
+</tr>
+<tr class="odd">
+<td align="right">43.6369893</td>
+<td align="right">0.7</td>
+<td align="right">2.3</td>
+<td align="right">5.2</td>
+<td align="right">1.5</td>
+</tr>
+<tr class="even">
+<td align="right">43.6369893</td>
+<td align="right">0.7</td>
+<td align="right">2.1</td>
+<td align="right">5.6</td>
+<td align="right">1.3</td>
+</tr>
+<tr class="odd">
+<td align="right">52.3643872</td>
+<td align="right">0.6</td>
+<td align="right">1.6</td>
+<td align="right">4.3</td>
+<td align="right">1.2</td>
+</tr>
+<tr class="even">
+<td align="right">52.3643872</td>
+<td align="right">0.4</td>
+<td align="right">1.1</td>
+<td align="right">3.7</td>
+<td align="right">0.9</td>
+</tr>
+<tr class="odd">
+<td align="right">52.3643872</td>
+<td align="right">0.5</td>
+<td align="right">1.3</td>
+<td align="right">3.9</td>
+<td align="right">1.1</td>
+</tr>
+<tr class="even">
+<td align="right">74.8062674</td>
+<td align="right">0.4</td>
+<td align="right">0.4</td>
+<td align="right">2.5</td>
+<td align="right">0.5</td>
+</tr>
+<tr class="odd">
+<td align="right">74.8062674</td>
+<td align="right">0.3</td>
+<td align="right">0.4</td>
+<td align="right">2.4</td>
+<td align="right">0.5</td>
+</tr>
+<tr class="even">
+<td align="right">74.8062674</td>
+<td align="right">0.3</td>
+<td align="right">0.3</td>
+<td align="right">2.2</td>
+<td align="right">0.3</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset BBA 2.2</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+<th align="right">M23</th>
+<th align="right">M27</th>
+<th align="right">M31</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.0000000</td>
+<td align="right">98.09</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">0.0000000</td>
+<td align="right">98.77</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">0.7678922</td>
+<td align="right">93.52</td>
+<td align="right">0.36</td>
+<td align="right">0.42</td>
+<td align="right">0.36</td>
+</tr>
+<tr class="even">
+<td align="right">0.7678922</td>
+<td align="right">92.03</td>
+<td align="right">0.40</td>
+<td align="right">0.47</td>
+<td align="right">0.33</td>
+</tr>
+<tr class="odd">
+<td align="right">2.3036765</td>
+<td align="right">88.39</td>
+<td align="right">1.03</td>
+<td align="right">0.71</td>
+<td align="right">0.55</td>
+</tr>
+<tr class="even">
+<td align="right">2.3036765</td>
+<td align="right">87.18</td>
+<td align="right">1.07</td>
+<td align="right">0.82</td>
+<td align="right">0.64</td>
+</tr>
+<tr class="odd">
+<td align="right">5.3752452</td>
+<td align="right">69.38</td>
+<td align="right">3.60</td>
+<td align="right">2.19</td>
+<td align="right">1.94</td>
+</tr>
+<tr class="even">
+<td align="right">5.3752452</td>
+<td align="right">71.06</td>
+<td align="right">3.66</td>
+<td align="right">2.28</td>
+<td align="right">1.62</td>
+</tr>
+<tr class="odd">
+<td align="right">10.7504904</td>
+<td align="right">45.21</td>
+<td align="right">6.97</td>
+<td align="right">5.45</td>
+<td align="right">4.22</td>
+</tr>
+<tr class="even">
+<td align="right">10.7504904</td>
+<td align="right">46.81</td>
+<td align="right">7.22</td>
+<td align="right">5.19</td>
+<td align="right">4.37</td>
+</tr>
+<tr class="odd">
+<td align="right">16.1257355</td>
+<td align="right">30.54</td>
+<td align="right">8.65</td>
+<td align="right">8.81</td>
+<td align="right">6.31</td>
+</tr>
+<tr class="even">
+<td align="right">16.1257355</td>
+<td align="right">30.07</td>
+<td align="right">8.38</td>
+<td align="right">7.93</td>
+<td align="right">6.85</td>
+</tr>
+<tr class="odd">
+<td align="right">21.5009807</td>
+<td align="right">21.60</td>
+<td align="right">9.10</td>
+<td align="right">10.25</td>
+<td align="right">7.05</td>
+</tr>
+<tr class="even">
+<td align="right">21.5009807</td>
+<td align="right">20.41</td>
+<td align="right">8.63</td>
+<td align="right">10.77</td>
+<td align="right">6.84</td>
+</tr>
+<tr class="odd">
+<td align="right">32.2514711</td>
+<td align="right">9.10</td>
+<td align="right">7.63</td>
+<td align="right">10.89</td>
+<td align="right">6.53</td>
+</tr>
+<tr class="even">
+<td align="right">32.2514711</td>
+<td align="right">9.70</td>
+<td align="right">8.01</td>
+<td align="right">10.85</td>
+<td align="right">7.11</td>
+</tr>
+<tr class="odd">
+<td align="right">43.0019614</td>
+<td align="right">6.58</td>
+<td align="right">6.40</td>
+<td align="right">10.41</td>
+<td align="right">6.06</td>
+</tr>
+<tr class="even">
+<td align="right">43.0019614</td>
+<td align="right">6.31</td>
+<td align="right">6.35</td>
+<td align="right">10.35</td>
+<td align="right">6.05</td>
+</tr>
+<tr class="odd">
+<td align="right">53.7524518</td>
+<td align="right">3.47</td>
+<td align="right">5.35</td>
+<td align="right">9.92</td>
+<td align="right">5.50</td>
+</tr>
+<tr class="even">
+<td align="right">53.7524518</td>
+<td align="right">3.52</td>
+<td align="right">5.06</td>
+<td align="right">9.42</td>
+<td align="right">5.07</td>
+</tr>
+<tr class="odd">
+<td align="right">64.5029421</td>
+<td align="right">3.40</td>
+<td align="right">5.14</td>
+<td align="right">9.15</td>
+<td align="right">4.94</td>
+</tr>
+<tr class="even">
+<td align="right">64.5029421</td>
+<td align="right">3.67</td>
+<td align="right">5.91</td>
+<td align="right">9.25</td>
+<td align="right">4.39</td>
+</tr>
+<tr class="odd">
+<td align="right">91.3791680</td>
+<td align="right">1.62</td>
+<td align="right">3.35</td>
+<td align="right">7.14</td>
+<td align="right">3.64</td>
+</tr>
+<tr class="even">
+<td align="right">91.3791680</td>
+<td align="right">1.62</td>
+<td align="right">2.87</td>
+<td align="right">7.13</td>
+<td align="right">3.55</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset BBA 2.3</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+<th align="right">M23</th>
+<th align="right">M27</th>
+<th align="right">M31</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.0000000</td>
+<td align="right">99.33</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">0.0000000</td>
+<td align="right">97.44</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">0.6733938</td>
+<td align="right">93.73</td>
+<td align="right">0.18</td>
+<td align="right">0.50</td>
+<td align="right">0.47</td>
+</tr>
+<tr class="even">
+<td align="right">0.6733938</td>
+<td align="right">93.77</td>
+<td align="right">0.18</td>
+<td align="right">0.83</td>
+<td align="right">0.34</td>
+</tr>
+<tr class="odd">
+<td align="right">2.0201814</td>
+<td align="right">87.84</td>
+<td align="right">0.52</td>
+<td align="right">1.25</td>
+<td align="right">1.00</td>
+</tr>
+<tr class="even">
+<td align="right">2.0201814</td>
+<td align="right">89.82</td>
+<td align="right">0.43</td>
+<td align="right">1.09</td>
+<td align="right">0.89</td>
+</tr>
+<tr class="odd">
+<td align="right">4.7137565</td>
+<td align="right">71.61</td>
+<td align="right">1.19</td>
+<td align="right">3.28</td>
+<td align="right">3.58</td>
+</tr>
+<tr class="even">
+<td align="right">4.7137565</td>
+<td align="right">71.42</td>
+<td align="right">1.11</td>
+<td align="right">3.24</td>
+<td align="right">3.41</td>
+</tr>
+<tr class="odd">
+<td align="right">9.4275131</td>
+<td align="right">45.60</td>
+<td align="right">2.26</td>
+<td align="right">7.17</td>
+<td align="right">8.74</td>
+</tr>
+<tr class="even">
+<td align="right">9.4275131</td>
+<td align="right">45.42</td>
+<td align="right">1.99</td>
+<td align="right">7.91</td>
+<td align="right">8.28</td>
+</tr>
+<tr class="odd">
+<td align="right">14.1412696</td>
+<td align="right">31.12</td>
+<td align="right">2.81</td>
+<td align="right">10.15</td>
+<td align="right">9.67</td>
+</tr>
+<tr class="even">
+<td align="right">14.1412696</td>
+<td align="right">31.68</td>
+<td align="right">2.83</td>
+<td align="right">9.55</td>
+<td align="right">8.95</td>
+</tr>
+<tr class="odd">
+<td align="right">18.8550262</td>
+<td align="right">23.20</td>
+<td align="right">3.39</td>
+<td align="right">12.09</td>
+<td align="right">10.34</td>
+</tr>
+<tr class="even">
+<td align="right">18.8550262</td>
+<td align="right">24.13</td>
+<td align="right">3.56</td>
+<td align="right">11.89</td>
+<td align="right">10.00</td>
+</tr>
+<tr class="odd">
+<td align="right">28.2825393</td>
+<td align="right">9.43</td>
+<td align="right">3.49</td>
+<td align="right">13.32</td>
+<td align="right">7.89</td>
+</tr>
+<tr class="even">
+<td align="right">28.2825393</td>
+<td align="right">9.82</td>
+<td align="right">3.28</td>
+<td align="right">12.05</td>
+<td align="right">8.13</td>
+</tr>
+<tr class="odd">
+<td align="right">37.7100523</td>
+<td align="right">7.08</td>
+<td align="right">2.80</td>
+<td align="right">10.04</td>
+<td align="right">5.06</td>
+</tr>
+<tr class="even">
+<td align="right">37.7100523</td>
+<td align="right">8.64</td>
+<td align="right">2.97</td>
+<td align="right">10.78</td>
+<td align="right">5.54</td>
+</tr>
+<tr class="odd">
+<td align="right">47.1375654</td>
+<td align="right">4.41</td>
+<td align="right">2.42</td>
+<td align="right">9.32</td>
+<td align="right">3.79</td>
+</tr>
+<tr class="even">
+<td align="right">47.1375654</td>
+<td align="right">4.78</td>
+<td align="right">2.51</td>
+<td align="right">9.62</td>
+<td align="right">4.11</td>
+</tr>
+<tr class="odd">
+<td align="right">56.5650785</td>
+<td align="right">4.92</td>
+<td align="right">2.22</td>
+<td align="right">8.00</td>
+<td align="right">3.11</td>
+</tr>
+<tr class="even">
+<td align="right">56.5650785</td>
+<td align="right">5.08</td>
+<td align="right">1.95</td>
+<td align="right">8.45</td>
+<td align="right">2.98</td>
+</tr>
+<tr class="odd">
+<td align="right">80.1338612</td>
+<td align="right">2.13</td>
+<td align="right">1.28</td>
+<td align="right">5.71</td>
+<td align="right">1.78</td>
+</tr>
+<tr class="even">
+<td align="right">80.1338612</td>
+<td align="right">2.23</td>
+<td align="right">0.99</td>
+<td align="right">3.33</td>
+<td align="right">1.55</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Elliot</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">DMTA</th>
+<th align="right">M23</th>
+<th align="right">M27</th>
+<th align="right">M31</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">97.5</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">0.000000</td>
+<td align="right">100.7</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">1.228478</td>
+<td align="right">86.4</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">1.228478</td>
+<td align="right">88.5</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">1.5</td>
+</tr>
+<tr class="odd">
+<td align="right">3.685435</td>
+<td align="right">69.8</td>
+<td align="right">2.8</td>
+<td align="right">2.3</td>
+<td align="right">5.0</td>
+</tr>
+<tr class="even">
+<td align="right">3.685435</td>
+<td align="right">77.1</td>
+<td align="right">1.7</td>
+<td align="right">2.1</td>
+<td align="right">2.4</td>
+</tr>
+<tr class="odd">
+<td align="right">8.599349</td>
+<td align="right">59.0</td>
+<td align="right">4.3</td>
+<td align="right">4.0</td>
+<td align="right">4.3</td>
+</tr>
+<tr class="even">
+<td align="right">8.599349</td>
+<td align="right">54.2</td>
+<td align="right">5.8</td>
+<td align="right">3.4</td>
+<td align="right">5.0</td>
+</tr>
+<tr class="odd">
+<td align="right">17.198697</td>
+<td align="right">31.3</td>
+<td align="right">8.2</td>
+<td align="right">6.6</td>
+<td align="right">8.0</td>
+</tr>
+<tr class="even">
+<td align="right">17.198697</td>
+<td align="right">33.5</td>
+<td align="right">5.2</td>
+<td align="right">6.9</td>
+<td align="right">7.7</td>
+</tr>
+<tr class="odd">
+<td align="right">25.798046</td>
+<td align="right">19.6</td>
+<td align="right">5.1</td>
+<td align="right">8.2</td>
+<td align="right">7.8</td>
+</tr>
+<tr class="even">
+<td align="right">25.798046</td>
+<td align="right">20.9</td>
+<td align="right">6.1</td>
+<td align="right">8.8</td>
+<td align="right">6.5</td>
+</tr>
+<tr class="odd">
+<td align="right">34.397395</td>
+<td align="right">13.3</td>
+<td align="right">6.0</td>
+<td align="right">9.7</td>
+<td align="right">8.0</td>
+</tr>
+<tr class="even">
+<td align="right">34.397395</td>
+<td align="right">15.8</td>
+<td align="right">6.0</td>
+<td align="right">8.8</td>
+<td align="right">7.4</td>
+</tr>
+<tr class="odd">
+<td align="right">51.596092</td>
+<td align="right">6.7</td>
+<td align="right">5.0</td>
+<td align="right">8.3</td>
+<td align="right">6.9</td>
+</tr>
+<tr class="even">
+<td align="right">51.596092</td>
+<td align="right">8.7</td>
+<td align="right">4.2</td>
+<td align="right">9.2</td>
+<td align="right">9.0</td>
+</tr>
+<tr class="odd">
+<td align="right">68.794789</td>
+<td align="right">8.8</td>
+<td align="right">3.9</td>
+<td align="right">9.3</td>
+<td align="right">5.5</td>
+</tr>
+<tr class="even">
+<td align="right">68.794789</td>
+<td align="right">8.7</td>
+<td align="right">2.9</td>
+<td align="right">8.5</td>
+<td align="right">6.1</td>
+</tr>
+<tr class="odd">
+<td align="right">103.192184</td>
+<td align="right">6.0</td>
+<td align="right">1.9</td>
+<td align="right">8.6</td>
+<td align="right">6.1</td>
+</tr>
+<tr class="even">
+<td align="right">103.192184</td>
+<td align="right">4.4</td>
+<td align="right">1.5</td>
+<td align="right">6.0</td>
+<td align="right">4.0</td>
+</tr>
+<tr class="odd">
+<td align="right">146.188928</td>
+<td align="right">3.3</td>
+<td align="right">2.0</td>
+<td align="right">5.6</td>
+<td align="right">3.1</td>
+</tr>
+<tr class="even">
+<td align="right">146.188928</td>
+<td align="right">2.8</td>
+<td align="right">2.3</td>
+<td align="right">4.5</td>
+<td align="right">2.9</td>
+</tr>
+<tr class="odd">
+<td align="right">223.583066</td>
+<td align="right">1.4</td>
+<td align="right">1.2</td>
+<td align="right">4.1</td>
+<td align="right">1.8</td>
+</tr>
+<tr class="even">
+<td align="right">223.583066</td>
+<td align="right">1.8</td>
+<td align="right">1.9</td>
+<td align="right">3.9</td>
+<td align="right">2.6</td>
+</tr>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">93.4</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="even">
+<td align="right">0.000000</td>
+<td align="right">103.2</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">1.228478</td>
+<td align="right">89.2</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">1.3</td>
+</tr>
+<tr class="even">
+<td align="right">1.228478</td>
+<td align="right">86.6</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+<td align="right">NA</td>
+</tr>
+<tr class="odd">
+<td align="right">3.685435</td>
+<td align="right">78.2</td>
+<td align="right">2.6</td>
+<td align="right">1.0</td>
+<td align="right">3.1</td>
+</tr>
+<tr class="even">
+<td align="right">3.685435</td>
+<td align="right">78.1</td>
+<td align="right">2.4</td>
+<td align="right">2.6</td>
+<td align="right">2.3</td>
+</tr>
+<tr class="odd">
+<td align="right">8.599349</td>
+<td align="right">55.6</td>
+<td align="right">5.5</td>
+<td align="right">4.5</td>
+<td align="right">3.4</td>
+</tr>
+<tr class="even">
+<td align="right">8.599349</td>
+<td align="right">53.0</td>
+<td align="right">5.6</td>
+<td align="right">4.6</td>
+<td align="right">4.3</td>
+</tr>
+<tr class="odd">
+<td align="right">17.198697</td>
+<td align="right">33.7</td>
+<td align="right">7.3</td>
+<td align="right">7.6</td>
+<td align="right">7.8</td>
+</tr>
+<tr class="even">
+<td align="right">17.198697</td>
+<td align="right">33.2</td>
+<td align="right">6.5</td>
+<td align="right">6.7</td>
+<td align="right">8.7</td>
+</tr>
+<tr class="odd">
+<td align="right">25.798046</td>
+<td align="right">20.9</td>
+<td align="right">5.8</td>
+<td align="right">8.7</td>
+<td align="right">7.7</td>
+</tr>
+<tr class="even">
+<td align="right">25.798046</td>
+<td align="right">19.9</td>
+<td align="right">7.7</td>
+<td align="right">7.6</td>
+<td align="right">6.5</td>
+</tr>
+<tr class="odd">
+<td align="right">34.397395</td>
+<td align="right">18.2</td>
+<td align="right">7.8</td>
+<td align="right">8.0</td>
+<td align="right">6.3</td>
+</tr>
+<tr class="even">
+<td align="right">34.397395</td>
+<td align="right">12.7</td>
+<td align="right">7.3</td>
+<td align="right">8.6</td>
+<td align="right">8.7</td>
+</tr>
+<tr class="odd">
+<td align="right">51.596092</td>
+<td align="right">7.8</td>
+<td align="right">7.0</td>
+<td align="right">7.4</td>
+<td align="right">5.7</td>
+</tr>
+<tr class="even">
+<td align="right">51.596092</td>
+<td align="right">9.0</td>
+<td align="right">6.3</td>
+<td align="right">7.2</td>
+<td align="right">4.2</td>
+</tr>
+<tr class="odd">
+<td align="right">68.794789</td>
+<td align="right">11.4</td>
+<td align="right">4.3</td>
+<td align="right">10.3</td>
+<td align="right">3.2</td>
+</tr>
+<tr class="even">
+<td align="right">68.794789</td>
+<td align="right">9.0</td>
+<td align="right">3.8</td>
+<td align="right">9.4</td>
+<td align="right">4.2</td>
+</tr>
+<tr class="odd">
+<td align="right">103.192184</td>
+<td align="right">3.9</td>
+<td align="right">2.6</td>
+<td align="right">6.5</td>
+<td align="right">3.8</td>
+</tr>
+<tr class="even">
+<td align="right">103.192184</td>
+<td align="right">4.4</td>
+<td align="right">2.8</td>
+<td align="right">6.9</td>
+<td align="right">4.0</td>
+</tr>
+<tr class="odd">
+<td align="right">146.188928</td>
+<td align="right">2.6</td>
+<td align="right">1.6</td>
+<td align="right">4.6</td>
+<td align="right">4.5</td>
+</tr>
+<tr class="even">
+<td align="right">146.188928</td>
+<td align="right">3.4</td>
+<td align="right">1.1</td>
+<td align="right">4.5</td>
+<td align="right">4.5</td>
+</tr>
+<tr class="odd">
+<td align="right">223.583066</td>
+<td align="right">2.0</td>
+<td align="right">1.4</td>
+<td align="right">4.3</td>
+<td align="right">3.8</td>
+</tr>
+<tr class="even">
+<td align="right">223.583066</td>
+<td align="right">1.7</td>
+<td align="right">1.3</td>
+<td align="right">4.2</td>
+<td align="right">2.3</td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="section level2">
+<h2 id="separate-evaluations">Separate evaluations<a class="anchor" aria-label="anchor" href="#separate-evaluations"></a>
+</h2>
+<p>As a first step to obtain suitable starting parameters for the NLHM
+fits, we do separate fits of several variants of the pathway model used
+previously <span class="citation">(Ranke et al. 2021)</span>, varying
+the kinetic model for the parent compound. Because the SFORB model often
+provides faster convergence than the DFOP model, and can sometimes be
+fitted where the DFOP model results in errors, it is included in the set
+of parent models tested here.</p>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw">if</span> <span class="op">(</span><span class="op">!</span><span class="fu"><a href="https://rdrr.io/r/base/files2.html" class="external-link">dir.exists</a></span><span class="op">(</span><span class="st">"dmta_dlls"</span><span class="op">)</span><span class="op">)</span> <span class="fu"><a href="https://rdrr.io/r/base/files2.html" class="external-link">dir.create</a></span><span class="op">(</span><span class="st">"dmta_dlls"</span><span class="op">)</span></span>
+<span><span class="va">m_sfo_path_1</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>  DMTA <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"M23"</span>, <span class="st">"M27"</span>, <span class="st">"M31"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>  M23 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>  M27 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>  M31 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M27"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>  name <span class="op">=</span> <span class="st">"m_sfo_path"</span>, dll_dir <span class="op">=</span> <span class="st">"dmta_dlls"</span>,</span>
+<span>  unload <span class="op">=</span> <span class="cn">TRUE</span>, overwrite <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>  quiet <span class="op">=</span> <span class="cn">TRUE</span></span>
+<span><span class="op">)</span></span>
+<span><span class="va">m_fomc_path_1</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>  DMTA <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"FOMC"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"M23"</span>, <span class="st">"M27"</span>, <span class="st">"M31"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>  M23 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>  M27 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>  M31 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M27"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>  name <span class="op">=</span> <span class="st">"m_fomc_path"</span>, dll_dir <span class="op">=</span> <span class="st">"dmta_dlls"</span>,</span>
+<span>  unload <span class="op">=</span> <span class="cn">TRUE</span>, overwrite <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>  quiet <span class="op">=</span> <span class="cn">TRUE</span></span>
+<span><span class="op">)</span></span>
+<span><span class="va">m_dfop_path_1</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>  DMTA <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"DFOP"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"M23"</span>, <span class="st">"M27"</span>, <span class="st">"M31"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>  M23 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>  M27 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>  M31 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M27"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>  name <span class="op">=</span> <span class="st">"m_dfop_path"</span>, dll_dir <span class="op">=</span> <span class="st">"dmta_dlls"</span>,</span>
+<span>  unload <span class="op">=</span> <span class="cn">TRUE</span>, overwrite <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>  quiet <span class="op">=</span> <span class="cn">TRUE</span></span>
+<span><span class="op">)</span></span>
+<span><span class="va">m_sforb_path_1</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>  DMTA <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFORB"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"M23"</span>, <span class="st">"M27"</span>, <span class="st">"M31"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>  M23 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>  M27 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>  M31 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M27"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>  name <span class="op">=</span> <span class="st">"m_sforb_path"</span>, dll_dir <span class="op">=</span> <span class="st">"dmta_dlls"</span>,</span>
+<span>  unload <span class="op">=</span> <span class="cn">TRUE</span>, overwrite <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>  quiet <span class="op">=</span> <span class="cn">TRUE</span></span>
+<span><span class="op">)</span></span>
+<span><span class="va">m_hs_path_1</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>  DMTA <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"HS"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"M23"</span>, <span class="st">"M27"</span>, <span class="st">"M31"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>  M23 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>  M27 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>  M31 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M27"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>  name <span class="op">=</span> <span class="st">"m_hs_path"</span>, dll_dir <span class="op">=</span> <span class="st">"dmta_dlls"</span>,</span>
+<span>  unload <span class="op">=</span> <span class="cn">TRUE</span>, overwrite <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>  quiet <span class="op">=</span> <span class="cn">TRUE</span></span>
+<span><span class="op">)</span></span>
+<span><span class="va">cl</span> <span class="op">&lt;-</span> <span class="fu">start_cluster</span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">deg_mods_1</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span></span>
+<span>  sfo_path_1 <span class="op">=</span> <span class="va">m_sfo_path_1</span>,</span>
+<span>  fomc_path_1 <span class="op">=</span> <span class="va">m_fomc_path_1</span>,</span>
+<span>  dfop_path_1 <span class="op">=</span> <span class="va">m_dfop_path_1</span>,</span>
+<span>  sforb_path_1 <span class="op">=</span> <span class="va">m_sforb_path_1</span>,</span>
+<span>  hs_path_1 <span class="op">=</span> <span class="va">m_hs_path_1</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">sep_1_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span></span>
+<span>  <span class="va">deg_mods_1</span>,</span>
+<span>  <span class="va">dmta_ds</span>,</span>
+<span>  error_model <span class="op">=</span> <span class="st">"const"</span>,</span>
+<span>  quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span></span>
+<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">sep_1_const</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">Calke</th>
+<th align="left">Borstel</th>
+<th align="left">Flaach</th>
+<th align="left">BBA 2.2</th>
+<th align="left">BBA 2.3</th>
+<th align="left">Elliot</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">fomc_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">hs_path_1</td>
+<td align="left">C</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">C</td>
+<td align="left">C</td>
+</tr>
+</tbody>
+</table>
+<p>All separate pathway fits with SFO or FOMC for the parent and
+constant variance converged (status OK). Most fits with DFOP or SFORB
+for the parent converged as well. The fits with HS for the parent did
+not converge with default settings.</p>
+<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">sep_1_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">sep_1_const</span>, error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">sep_1_tc</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">Calke</th>
+<th align="left">Borstel</th>
+<th align="left">Flaach</th>
+<th align="left">BBA 2.2</th>
+<th align="left">BBA 2.3</th>
+<th align="left">Elliot</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">fomc_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">hs_path_1</td>
+<td align="left">C</td>
+<td align="left">C</td>
+<td align="left">C</td>
+<td align="left">C</td>
+<td align="left">C</td>
+<td align="left">C</td>
+</tr>
+</tbody>
+</table>
+<p>With the two-component error model, the set of fits with convergence
+problems is slightly different, with convergence problems appearing for
+different data sets when applying the DFOP and SFORB model and some
+additional convergence problems when using the FOMC model for the
+parent.</p>
+</div>
+<div class="section level2">
+<h2 id="hierarchichal-model-fits">Hierarchichal model fits<a class="anchor" aria-label="anchor" href="#hierarchichal-model-fits"></a>
+</h2>
+<p>The following code fits two sets of the corresponding hierarchical
+models to the data, one assuming constant variance, and one assuming
+two-component error.</p>
+<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">saem_1</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mhmkin.html">mhmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">sep_1_const</span>, <span class="va">sep_1_tc</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<p>The run time for these fits was around two hours on five year old
+hardware. After a recent hardware upgrade these fits complete in less
+than twenty minutes.</p>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">saem_1</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">fomc_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">hs_path_1</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>According to the <code>status</code> function, all fits terminated
+successfully.</p>
+<div class="sourceCode" id="cb8"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">saem_1</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_path_1 const</td>
+<td align="right">17</td>
+<td align="right">2291.8</td>
+<td align="right">2288.3</td>
+<td align="right">-1128.9</td>
+</tr>
+<tr class="even">
+<td align="left">sfo_path_1 tc</td>
+<td align="right">18</td>
+<td align="right">2276.4</td>
+<td align="right">2272.7</td>
+<td align="right">-1120.2</td>
+</tr>
+<tr class="odd">
+<td align="left">fomc_path_1 const</td>
+<td align="right">19</td>
+<td align="right">2095.9</td>
+<td align="right">2091.9</td>
+<td align="right">-1028.9</td>
+</tr>
+<tr class="even">
+<td align="left">fomc_path_1 tc</td>
+<td align="right">20</td>
+<td align="right">1939.0</td>
+<td align="right">1934.8</td>
+<td align="right">-949.5</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1 const</td>
+<td align="right">21</td>
+<td align="right">2039.7</td>
+<td align="right">2035.3</td>
+<td align="right">-998.8</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1 const</td>
+<td align="right">21</td>
+<td align="right">2017.7</td>
+<td align="right">2013.4</td>
+<td align="right">-987.9</td>
+</tr>
+<tr class="odd">
+<td align="left">hs_path_1 const</td>
+<td align="right">21</td>
+<td align="right">2023.7</td>
+<td align="right">2019.3</td>
+<td align="right">-990.9</td>
+</tr>
+<tr class="even">
+<td align="left">dfop_path_1 tc</td>
+<td align="right">22</td>
+<td align="right">1881.7</td>
+<td align="right">1877.1</td>
+<td align="right">-918.9</td>
+</tr>
+<tr class="odd">
+<td align="left">sforb_path_1 tc</td>
+<td align="right">22</td>
+<td align="right">1832.7</td>
+<td align="right">1828.1</td>
+<td align="right">-894.3</td>
+</tr>
+<tr class="even">
+<td align="left">hs_path_1 tc</td>
+<td align="right">22</td>
+<td align="right">1831.6</td>
+<td align="right">1827.0</td>
+<td align="right">-893.8</td>
+</tr>
+</tbody>
+</table>
+<p>When the goodness-of-fit of the models is compared, a warning is
+obtained, indicating that the likelihood of the pathway fit with SFORB
+for the parent compound and constant variance could not be calculated
+with importance sampling (method ‘is’). As this is the default method on
+which all AIC and BIC comparisons are based, this variant is not
+included in the model comparison table. Comparing the goodness-of-fit of
+the remaining models, HS model model with two-component error provides
+the best fit. However, for batch experiments performed with constant
+conditions such as the experiments evaluated here, there is no reason to
+assume a discontinuity, so the SFORB model is preferable from a
+mechanistic viewpoint. In addition, the information criteria AIC and BIC
+are very similar for HS and SFORB. Therefore, the SFORB model is
+selected here for further refinements.</p>
+<div class="section level3">
+<h3 id="parameter-identifiability-based-on-the-fisher-information-matrix">Parameter identifiability based on the Fisher Information
+Matrix<a class="anchor" aria-label="anchor" href="#parameter-identifiability-based-on-the-fisher-information-matrix"></a>
+</h3>
+<p>Using the <code>illparms</code> function, ill-defined statistical
+model parameters such as standard deviations of the degradation
+parameters in the population and error model parameters can be
+found.</p>
+<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">saem_1</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_path_1</td>
+<td align="left"></td>
+<td align="left">sd(DMTA_0)</td>
+</tr>
+<tr class="even">
+<td align="left">fomc_path_1</td>
+<td align="left"></td>
+<td align="left">sd(DMTA_0)</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_path_1</td>
+<td align="left"></td>
+<td align="left"></td>
+</tr>
+<tr class="even">
+<td align="left">sforb_path_1</td>
+<td align="left">sd(log_k_DMTA_bound_free)</td>
+<td align="left">sd(log_k_DMTA_bound_free)</td>
+</tr>
+<tr class="odd">
+<td align="left">hs_path_1</td>
+<td align="left"></td>
+<td align="left">sd(log_tb)</td>
+</tr>
+</tbody>
+</table>
+<p>When using constant variance, no ill-defined variance parameters are
+identified with the <code>illparms</code> function in any of the
+degradation models. When using the two-component error model, there is
+one ill-defined variance parameter in all variants except for the
+variant using DFOP for the parent compound.</p>
+<p>For the selected combination of the SFORB pathway model with
+two-component error, the random effect for the rate constant from
+reversibly bound DMTA to the free DMTA (<code>k_DMTA_bound_free</code>)
+is not well-defined. Therefore, the fit is updated without assuming a
+random effect for this parameter.</p>
+<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">saem_sforb_path_1_tc_reduced</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">saem_1</span><span class="op">[[</span><span class="st">"sforb_path_1"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span>,</span>
+<span>  no_random_effect <span class="op">=</span> <span class="st">"log_k_DMTA_bound_free"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">saem_sforb_path_1_tc_reduced</span><span class="op">)</span></span></code></pre></div>
+<p>As expected, no ill-defined parameters remain. The model comparison
+below shows that the reduced model is preferable.</p>
+<div class="sourceCode" id="cb11"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">saem_1</span><span class="op">[[</span><span class="st">"sforb_path_1"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span>, <span class="va">saem_sforb_path_1_tc_reduced</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">saem_sforb_path_1_tc_reduced</td>
+<td align="right">21</td>
+<td align="right">1830.4</td>
+<td align="right">1826.0</td>
+<td align="right">-894.2</td>
+</tr>
+<tr class="even">
+<td align="left">saem_1[[“sforb_path_1”, “tc”]]</td>
+<td align="right">22</td>
+<td align="right">1832.7</td>
+<td align="right">1828.1</td>
+<td align="right">-894.3</td>
+</tr>
+</tbody>
+</table>
+<p>The convergence plot of the refined fit is shown below.</p>
+<div class="sourceCode" id="cb12"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">saem_sforb_path_1_tc_reduced</span><span class="op">$</span><span class="va">so</span>, plot.type <span class="op">=</span> <span class="st">"convergence"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="2022_dmta_pathway_files/figure-html/saem-sforb-path-1-tc-reduced-convergence-1.png" width="700" style="display: block; margin: auto;"></p>
+<p>For some parameters, for example for <code>f_DMTA_ilr_1</code> and
+<code>f_DMTA_ilr_2</code>, i.e. for two of the parameters determining
+the formation fractions of the parallel formation of the three
+metabolites, some movement of the parameters is still visible in the
+second phase of the algorithm. However, the amplitude of this movement
+is in the range of the amplitude towards the end of the first phase.
+Therefore, it is likely that an increase in iterations would not improve
+the parameter estimates very much, and it is proposed that the fit is
+acceptable. No numeric convergence criterion is implemented in
+saemix.</p>
+</div>
+<div class="section level3">
+<h3 id="alternative-check-of-parameter-identifiability">Alternative check of parameter identifiability<a class="anchor" aria-label="anchor" href="#alternative-check-of-parameter-identifiability"></a>
+</h3>
+<p>As an alternative check of parameter identifiability <span class="citation">(Duchesne et al. 2021)</span>, multistart runs were
+performed on the basis of the refined fit shown above.</p>
+<div class="sourceCode" id="cb13"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">saem_sforb_path_1_tc_reduced_multi</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/multistart.html">multistart</a></span><span class="op">(</span><span class="va">saem_sforb_path_1_tc_reduced</span>,</span>
+<span>  n <span class="op">=</span> <span class="fl">32</span>, cores <span class="op">=</span> <span class="fl">10</span><span class="op">)</span></span></code></pre></div>
+<pre><code> 
+  (subscript) logical subscript too long</code></pre>
+<div class="sourceCode" id="cb15"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">saem_sforb_path_1_tc_reduced_multi</span><span class="op">)</span></span></code></pre></div>
+<pre><code>&lt;multistart&gt; object with 32 fits:
+ E OK 
+ 7 25 
+OK: Fit terminated successfully
+E: Error</code></pre>
+<p>Out of the 32 fits that were initiated, only 17 terminated without an
+error. The reason for this is that the wide variation of starting
+parameters in combination with the parameter variation that is used in
+the SAEM algorithm leads to parameter combinations for the degradation
+model that the numerical integration routine cannot cope with. Because
+of this variation of initial parameters, some of the model fits take up
+to two times more time than the original fit.</p>
+<div class="sourceCode" id="cb17"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/graphics/par.html" class="external-link">par</a></span><span class="op">(</span>mar <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">12.1</span>, <span class="fl">4.1</span>, <span class="fl">2.1</span>, <span class="fl">2.1</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/parplot.html">parplot</a></span><span class="op">(</span><span class="va">saem_sforb_path_1_tc_reduced_multi</span>, ylim <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.5</span>, <span class="fl">2</span><span class="op">)</span>, las <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_pathway_files/figure-html/unnamed-chunk-2-1.png" alt="Parameter boxplots for the multistart runs that succeeded" width="960"><p class="caption">
+Parameter boxplots for the multistart runs that succeeded
+</p>
+</div>
+<p>However, visual analysis of the boxplot of the parameters obtained in
+the successful fits confirms that the results are sufficiently
+independent of the starting parameters, and there are no remaining
+ill-defined parameters.</p>
+</div>
+</div>
+<div class="section level2">
+<h2 id="plots-of-selected-fits">Plots of selected fits<a class="anchor" aria-label="anchor" href="#plots-of-selected-fits"></a>
+</h2>
+<p>The SFORB pathway fits with full and reduced parameter distribution
+model are shown below.</p>
+<div class="sourceCode" id="cb18"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">saem_1</span><span class="op">[[</span><span class="st">"sforb_path_1"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_pathway_files/figure-html/unnamed-chunk-3-1.png" alt="SFORB pathway fit with two-component error" width="700"><p class="caption">
+SFORB pathway fit with two-component error
+</p>
+</div>
+<div class="sourceCode" id="cb19"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">saem_sforb_path_1_tc_reduced</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_pathway_files/figure-html/unnamed-chunk-4-1.png" alt="SFORB pathway fit with two-component error, reduced parameter model" width="700"><p class="caption">
+SFORB pathway fit with two-component error, reduced parameter model
+</p>
+</div>
+<p>Plots of the remaining fits and listings for all successful fits are
+shown in the Appendix.</p>
+<div class="sourceCode" id="cb20"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">stopCluster</a></span><span class="op">(</span><span class="va">cl</span><span class="op">)</span></span></code></pre></div>
+</div>
+<div class="section level2">
+<h2 id="conclusions">Conclusions<a class="anchor" aria-label="anchor" href="#conclusions"></a>
+</h2>
+<p>Pathway fits with SFO, FOMC, DFOP, SFORB and HS models for the parent
+compound could be successfully performed.</p>
+</div>
+<div class="section level2">
+<h2 id="acknowledgements">Acknowledgements<a class="anchor" aria-label="anchor" href="#acknowledgements"></a>
+</h2>
+<p>The helpful comments by Janina Wöltjen of the German Environment
+Agency on earlier versions of this document are gratefully
+acknowledged.</p>
+</div>
+<div class="section level2">
+<h2 id="references">References<a class="anchor" aria-label="anchor" href="#references"></a>
+</h2>
+<div id="refs" class="references csl-bib-body hanging-indent">
+<div id="ref-duchesne_2021" class="csl-entry">
+Duchesne, Ronan, Anissa Guillemin, Olivier Gandrillon, and Fabien
+Crauste. 2021. <span>“Practical Identifiability in the Frame of
+Nonlinear Mixed Effects Models: The Example of the in Vitro
+Erythropoiesis.”</span> <em>BMC Bioinformatics</em> 22 (478). <a href="https://doi.org/10.1186/s12859-021-04373-4" class="external-link">https://doi.org/10.1186/s12859-021-04373-4</a>.
+</div>
+<div id="ref-ranke2021" class="csl-entry">
+Ranke, Johannes, Janina Wöltjen, Jana Schmidt, and Emmanuelle Comets.
+2021. <span>“Taking Kinetic Evaluations of Degradation Data to the Next
+Level with Nonlinear Mixed-Effects Models.”</span> <em>Environments</em>
+8 (8). <a href="https://doi.org/10.3390/environments8080071" class="external-link">https://doi.org/10.3390/environments8080071</a>.
+</div>
+</div>
+</div>
+<div class="section level2">
+<h2 id="appendix">Appendix<a class="anchor" aria-label="anchor" href="#appendix"></a>
+</h2>
+<div class="section level3">
+<h3 id="plots-of-hierarchical-fits-not-selected-for-refinement">Plots of hierarchical fits not selected for refinement<a class="anchor" aria-label="anchor" href="#plots-of-hierarchical-fits-not-selected-for-refinement"></a>
+</h3>
+<div class="sourceCode" id="cb21"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">saem_1</span><span class="op">[[</span><span class="st">"sfo_path_1"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_pathway_files/figure-html/unnamed-chunk-6-1.png" alt="SFO pathway fit with two-component error" width="700"><p class="caption">
+SFO pathway fit with two-component error
+</p>
+</div>
+<div class="sourceCode" id="cb22"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">saem_1</span><span class="op">[[</span><span class="st">"fomc_path_1"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_pathway_files/figure-html/unnamed-chunk-7-1.png" alt="FOMC pathway fit with two-component error" width="700"><p class="caption">
+FOMC pathway fit with two-component error
+</p>
+</div>
+<div class="sourceCode" id="cb23"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">saem_1</span><span class="op">[[</span><span class="st">"sforb_path_1"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<div class="figure" style="text-align: center">
+<img src="2022_dmta_pathway_files/figure-html/unnamed-chunk-8-1.png" alt="HS pathway fit with two-component error" width="700"><p class="caption">
+HS pathway fit with two-component error
+</p>
+</div>
+</div>
+<div class="section level3">
+<h3 id="hierarchical-model-fit-listings">Hierarchical model fit listings<a class="anchor" aria-label="anchor" href="#hierarchical-model-fit-listings"></a>
+</h3>
+<div class="section level4">
+<h4 id="fits-with-random-effects-for-all-degradation-parameters">Fits with random effects for all degradation parameters<a class="anchor" aria-label="anchor" href="#fits-with-random-effects-for-all-degradation-parameters"></a>
+</h4>
+
+</div>
+<div class="section level4">
+<h4 id="improved-fit-of-the-sforb-pathway-model-with-two-component-error">Improved fit of the SFORB pathway model with two-component
+error<a class="anchor" aria-label="anchor" href="#improved-fit-of-the-sforb-pathway-model-with-two-component-error"></a>
+</h4>
+
+</div>
+</div>
+<div class="section level3">
+<h3 id="session-info">Session info<a class="anchor" aria-label="anchor" href="#session-info"></a>
+</h3>
+<pre><code>R version 4.4.2 (2024-10-31)
+Platform: x86_64-pc-linux-gnu
+Running under: Debian GNU/Linux 12 (bookworm)
+
+Matrix products: default
+BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.11.0 
+LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.11.0
+
+locale:
+ [1] LC_CTYPE=de_DE.UTF-8       LC_NUMERIC=C              
+ [3] LC_TIME=de_DE.UTF-8        LC_COLLATE=de_DE.UTF-8    
+ [5] LC_MONETARY=de_DE.UTF-8    LC_MESSAGES=de_DE.UTF-8   
+ [7] LC_PAPER=de_DE.UTF-8       LC_NAME=C                 
+ [9] LC_ADDRESS=C               LC_TELEPHONE=C            
+[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C       
+
+time zone: Europe/Berlin
+tzcode source: system (glibc)
+
+attached base packages:
+[1] parallel  stats     graphics  grDevices utils     datasets  methods  
+[8] base     
+
+other attached packages:
+[1] rmarkdown_2.29  nvimcom_0.9-167 saemix_3.3      npde_3.5       
+[5] knitr_1.49      mkin_1.2.10    
+
+loaded via a namespace (and not attached):
+ [1] sass_0.4.9        utf8_1.2.4        generics_0.1.3    lattice_0.22-6   
+ [5] digest_0.6.37     magrittr_2.0.3    evaluate_1.0.1    grid_4.4.2       
+ [9] fastmap_1.2.0     jsonlite_1.8.9    processx_3.8.4    pkgbuild_1.4.5   
+[13] deSolve_1.40      mclust_6.1.1      ps_1.8.1          gridExtra_2.3    
+[17] fansi_1.0.6       scales_1.3.0      codetools_0.2-20  textshaping_0.4.1
+[21] jquerylib_0.1.4   cli_3.6.3         rlang_1.1.4       munsell_0.5.1    
+[25] cachem_1.1.0      yaml_2.3.10       inline_0.3.20     tools_4.4.2      
+[29] dplyr_1.1.4       colorspace_2.1-1  ggplot2_3.5.1     vctrs_0.6.5      
+[33] R6_2.5.1          zoo_1.8-12        lifecycle_1.0.4   fs_1.6.5         
+[37] htmlwidgets_1.6.4 MASS_7.3-61       ragg_1.3.3        pkgconfig_2.0.3  
+[41] desc_1.4.3        callr_3.7.6       pkgdown_2.1.1     pillar_1.9.0     
+[45] bslib_0.8.0       gtable_0.3.6      glue_1.8.0        systemfonts_1.1.0
+[49] xfun_0.49         tibble_3.2.1      lmtest_0.9-40     tidyselect_1.2.1 
+[53] htmltools_0.5.8.1 nlme_3.1-166      compiler_4.4.2   </code></pre>
+</div>
+<div class="section level3">
+<h3 id="hardware-info">Hardware info<a class="anchor" aria-label="anchor" href="#hardware-info"></a>
+</h3>
+<pre><code>CPU model: AMD Ryzen 9 7950X 16-Core Processor</code></pre>
+<pre><code>MemTotal:       64927788 kB</code></pre>
+</div>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
+</div>
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+
+      <h1>Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change on 4 August 2023,
+last compiled on 14 Februar 2025</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/prebuilt/2023_mesotrione_parent.rmd" class="external-link"><code>vignettes/prebuilt/2023_mesotrione_parent.rmd</code></a></small>
+      <div class="d-none name"><code>2023_mesotrione_parent.rmd</code></div>
+    </div>
+
+    
+    
+<div class="section level2">
+<h2 id="introduction">Introduction<a class="anchor" aria-label="anchor" href="#introduction"></a>
+</h2>
+<p>The purpose of this document is to test demonstrate how nonlinear
+hierarchical models (NLHM) based on the parent degradation models SFO,
+FOMC, DFOP and HS can be fitted with the mkin package, also considering
+the influence of covariates like soil pH on different degradation
+parameters. Because in some other case studies, the SFORB
+parameterisation of biexponential decline has shown some advantages over
+the DFOP parameterisation, SFORB was included in the list of tested
+models as well.</p>
+<p>The mkin package is used in version 1.2.10, which is contains the
+functions that were used for the evaluations. The <code>saemix</code>
+package is used as a backend for fitting the NLHM, but is also loaded to
+make the convergence plot function available.</p>
+<p>This document is processed with the <code>knitr</code> package, which
+also provides the <code>kable</code> function that is used to improve
+the display of tabular data in R markdown documents. For parallel
+processing, the <code>parallel</code> package is used.</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://pkgdown.jrwb.de/mkin/">mkin</a></span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://yihui.org/knitr/" class="external-link">knitr</a></span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">saemix</span><span class="op">)</span></span>
+<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">parallel</span><span class="op">)</span></span>
+<span><span class="va">n_cores</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/detectCores.html" class="external-link">detectCores</a></span><span class="op">(</span><span class="op">)</span></span>
+<span><span class="kw">if</span> <span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/Sys.info.html" class="external-link">Sys.info</a></span><span class="op">(</span><span class="op">)</span><span class="op">[</span><span class="st">"sysname"</span><span class="op">]</span> <span class="op">==</span> <span class="st">"Windows"</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="va">cl</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">makePSOCKcluster</a></span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
+<span><span class="op">}</span> <span class="kw">else</span> <span class="op">{</span></span>
+<span>  <span class="va">cl</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">makeForkCluster</a></span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
+<span><span class="op">}</span></span></code></pre></div>
+<div class="section level3">
+<h3 id="test-data">Test data<a class="anchor" aria-label="anchor" href="#test-data"></a>
+</h3>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">data_file</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.file.html" class="external-link">system.file</a></span><span class="op">(</span></span>
+<span>  <span class="st">"testdata"</span>, <span class="st">"mesotrione_soil_efsa_2016.xlsx"</span>, package <span class="op">=</span> <span class="st">"mkin"</span><span class="op">)</span></span>
+<span><span class="va">meso_ds</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/read_spreadsheet.html">read_spreadsheet</a></span><span class="op">(</span><span class="va">data_file</span>, parent_only <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<p>The following tables show the covariate data and the 18 datasets that
+were read in from the spreadsheet file.</p>
+<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">pH</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/attr.html" class="external-link">attr</a></span><span class="op">(</span><span class="va">meso_ds</span>, <span class="st">"covariates"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="va">pH</span>, caption <span class="op">=</span> <span class="st">"Covariate data"</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<caption>Covariate data</caption>
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">pH</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">Richmond</td>
+<td align="right">6.2</td>
+</tr>
+<tr class="even">
+<td align="left">Richmond 2</td>
+<td align="right">6.2</td>
+</tr>
+<tr class="odd">
+<td align="left">ERTC</td>
+<td align="right">6.4</td>
+</tr>
+<tr class="even">
+<td align="left">Toulouse</td>
+<td align="right">7.7</td>
+</tr>
+<tr class="odd">
+<td align="left">Picket Piece</td>
+<td align="right">7.1</td>
+</tr>
+<tr class="even">
+<td align="left">721</td>
+<td align="right">5.6</td>
+</tr>
+<tr class="odd">
+<td align="left">722</td>
+<td align="right">5.7</td>
+</tr>
+<tr class="even">
+<td align="left">723</td>
+<td align="right">5.4</td>
+</tr>
+<tr class="odd">
+<td align="left">724</td>
+<td align="right">4.8</td>
+</tr>
+<tr class="even">
+<td align="left">725</td>
+<td align="right">5.8</td>
+</tr>
+<tr class="odd">
+<td align="left">727</td>
+<td align="right">5.1</td>
+</tr>
+<tr class="even">
+<td align="left">728</td>
+<td align="right">5.9</td>
+</tr>
+<tr class="odd">
+<td align="left">729</td>
+<td align="right">5.6</td>
+</tr>
+<tr class="even">
+<td align="left">730</td>
+<td align="right">5.3</td>
+</tr>
+<tr class="odd">
+<td align="left">731</td>
+<td align="right">6.1</td>
+</tr>
+<tr class="even">
+<td align="left">732</td>
+<td align="right">5.0</td>
+</tr>
+<tr class="odd">
+<td align="left">741</td>
+<td align="right">5.7</td>
+</tr>
+<tr class="even">
+<td align="left">742</td>
+<td align="right">7.2</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw">for</span> <span class="op">(</span><span class="va">ds_name</span> <span class="kw">in</span> <span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">meso_ds</span><span class="op">)</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span></span>
+<span>    <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="fu"><a href="../../reference/mkin_long_to_wide.html">mkin_long_to_wide</a></span><span class="op">(</span><span class="va">meso_ds</span><span class="op">[[</span><span class="va">ds_name</span><span class="op">]</span><span class="op">]</span><span class="op">)</span>,</span>
+<span>      caption <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/paste.html" class="external-link">paste</a></span><span class="op">(</span><span class="st">"Dataset"</span>, <span class="va">ds_name</span><span class="op">)</span>,</span>
+<span>      booktabs <span class="op">=</span> <span class="cn">TRUE</span>, row.names <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="op">}</span></span></code></pre></div>
+<table class="table">
+<caption>Dataset Richmond</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">91.00</td>
+</tr>
+<tr class="even">
+<td align="right">1.179050</td>
+<td align="right">86.70</td>
+</tr>
+<tr class="odd">
+<td align="right">3.537149</td>
+<td align="right">73.60</td>
+</tr>
+<tr class="even">
+<td align="right">7.074299</td>
+<td align="right">61.50</td>
+</tr>
+<tr class="odd">
+<td align="right">10.611448</td>
+<td align="right">55.70</td>
+</tr>
+<tr class="even">
+<td align="right">15.327647</td>
+<td align="right">47.70</td>
+</tr>
+<tr class="odd">
+<td align="right">17.685747</td>
+<td align="right">39.50</td>
+</tr>
+<tr class="even">
+<td align="right">24.760046</td>
+<td align="right">29.80</td>
+</tr>
+<tr class="odd">
+<td align="right">35.371494</td>
+<td align="right">19.60</td>
+</tr>
+<tr class="even">
+<td align="right">68.384889</td>
+<td align="right">5.67</td>
+</tr>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">97.90</td>
+</tr>
+<tr class="even">
+<td align="right">1.179050</td>
+<td align="right">96.40</td>
+</tr>
+<tr class="odd">
+<td align="right">3.537149</td>
+<td align="right">89.10</td>
+</tr>
+<tr class="even">
+<td align="right">7.074299</td>
+<td align="right">74.40</td>
+</tr>
+<tr class="odd">
+<td align="right">10.611448</td>
+<td align="right">57.40</td>
+</tr>
+<tr class="even">
+<td align="right">15.327647</td>
+<td align="right">46.30</td>
+</tr>
+<tr class="odd">
+<td align="right">18.864797</td>
+<td align="right">35.50</td>
+</tr>
+<tr class="even">
+<td align="right">27.118146</td>
+<td align="right">27.20</td>
+</tr>
+<tr class="odd">
+<td align="right">35.371494</td>
+<td align="right">19.10</td>
+</tr>
+<tr class="even">
+<td align="right">74.280138</td>
+<td align="right">6.50</td>
+</tr>
+<tr class="odd">
+<td align="right">108.472582</td>
+<td align="right">3.40</td>
+</tr>
+<tr class="even">
+<td align="right">142.665027</td>
+<td align="right">2.20</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Richmond 2</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">96.0</td>
+</tr>
+<tr class="even">
+<td align="right">2.422004</td>
+<td align="right">82.4</td>
+</tr>
+<tr class="odd">
+<td align="right">5.651343</td>
+<td align="right">71.2</td>
+</tr>
+<tr class="even">
+<td align="right">8.073348</td>
+<td align="right">53.1</td>
+</tr>
+<tr class="odd">
+<td align="right">11.302687</td>
+<td align="right">48.5</td>
+</tr>
+<tr class="even">
+<td align="right">16.954030</td>
+<td align="right">33.4</td>
+</tr>
+<tr class="odd">
+<td align="right">22.605373</td>
+<td align="right">24.2</td>
+</tr>
+<tr class="even">
+<td align="right">45.210746</td>
+<td align="right">11.9</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset ERTC</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">99.9</td>
+</tr>
+<tr class="even">
+<td align="right">2.755193</td>
+<td align="right">80.0</td>
+</tr>
+<tr class="odd">
+<td align="right">6.428782</td>
+<td align="right">42.1</td>
+</tr>
+<tr class="even">
+<td align="right">9.183975</td>
+<td align="right">50.1</td>
+</tr>
+<tr class="odd">
+<td align="right">12.857565</td>
+<td align="right">28.4</td>
+</tr>
+<tr class="even">
+<td align="right">19.286347</td>
+<td align="right">39.8</td>
+</tr>
+<tr class="odd">
+<td align="right">25.715130</td>
+<td align="right">29.9</td>
+</tr>
+<tr class="even">
+<td align="right">51.430259</td>
+<td align="right">2.5</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Toulouse</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">96.8</td>
+</tr>
+<tr class="even">
+<td align="right">2.897983</td>
+<td align="right">63.3</td>
+</tr>
+<tr class="odd">
+<td align="right">6.761960</td>
+<td align="right">22.3</td>
+</tr>
+<tr class="even">
+<td align="right">9.659942</td>
+<td align="right">16.6</td>
+</tr>
+<tr class="odd">
+<td align="right">13.523919</td>
+<td align="right">16.1</td>
+</tr>
+<tr class="even">
+<td align="right">20.285879</td>
+<td align="right">17.2</td>
+</tr>
+<tr class="odd">
+<td align="right">27.047838</td>
+<td align="right">1.8</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset Picket Piece</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">102.0</td>
+</tr>
+<tr class="even">
+<td align="right">2.841195</td>
+<td align="right">73.7</td>
+</tr>
+<tr class="odd">
+<td align="right">6.629454</td>
+<td align="right">35.5</td>
+</tr>
+<tr class="even">
+<td align="right">9.470649</td>
+<td align="right">31.8</td>
+</tr>
+<tr class="odd">
+<td align="right">13.258909</td>
+<td align="right">18.0</td>
+</tr>
+<tr class="even">
+<td align="right">19.888364</td>
+<td align="right">3.7</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 721</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">86.4</td>
+</tr>
+<tr class="even">
+<td align="right">11.24366</td>
+<td align="right">61.4</td>
+</tr>
+<tr class="odd">
+<td align="right">22.48733</td>
+<td align="right">49.8</td>
+</tr>
+<tr class="even">
+<td align="right">33.73099</td>
+<td align="right">41.0</td>
+</tr>
+<tr class="odd">
+<td align="right">44.97466</td>
+<td align="right">35.1</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 722</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">90.3</td>
+</tr>
+<tr class="even">
+<td align="right">11.24366</td>
+<td align="right">52.1</td>
+</tr>
+<tr class="odd">
+<td align="right">22.48733</td>
+<td align="right">37.4</td>
+</tr>
+<tr class="even">
+<td align="right">33.73099</td>
+<td align="right">21.2</td>
+</tr>
+<tr class="odd">
+<td align="right">44.97466</td>
+<td align="right">14.3</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 723</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">89.3</td>
+</tr>
+<tr class="even">
+<td align="right">11.24366</td>
+<td align="right">70.8</td>
+</tr>
+<tr class="odd">
+<td align="right">22.48733</td>
+<td align="right">51.1</td>
+</tr>
+<tr class="even">
+<td align="right">33.73099</td>
+<td align="right">42.7</td>
+</tr>
+<tr class="odd">
+<td align="right">44.97466</td>
+<td align="right">26.7</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 724</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.000000</td>
+<td align="right">89.4</td>
+</tr>
+<tr class="even">
+<td align="right">9.008208</td>
+<td align="right">65.2</td>
+</tr>
+<tr class="odd">
+<td align="right">18.016415</td>
+<td align="right">55.8</td>
+</tr>
+<tr class="even">
+<td align="right">27.024623</td>
+<td align="right">46.0</td>
+</tr>
+<tr class="odd">
+<td align="right">36.032831</td>
+<td align="right">41.7</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 725</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">89.0</td>
+</tr>
+<tr class="even">
+<td align="right">10.99058</td>
+<td align="right">35.4</td>
+</tr>
+<tr class="odd">
+<td align="right">21.98116</td>
+<td align="right">18.6</td>
+</tr>
+<tr class="even">
+<td align="right">32.97174</td>
+<td align="right">11.6</td>
+</tr>
+<tr class="odd">
+<td align="right">43.96232</td>
+<td align="right">7.6</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 727</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">91.3</td>
+</tr>
+<tr class="even">
+<td align="right">10.96104</td>
+<td align="right">63.2</td>
+</tr>
+<tr class="odd">
+<td align="right">21.92209</td>
+<td align="right">51.1</td>
+</tr>
+<tr class="even">
+<td align="right">32.88313</td>
+<td align="right">42.0</td>
+</tr>
+<tr class="odd">
+<td align="right">43.84417</td>
+<td align="right">40.8</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 728</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">91.8</td>
+</tr>
+<tr class="even">
+<td align="right">11.24366</td>
+<td align="right">43.6</td>
+</tr>
+<tr class="odd">
+<td align="right">22.48733</td>
+<td align="right">22.0</td>
+</tr>
+<tr class="even">
+<td align="right">33.73099</td>
+<td align="right">15.9</td>
+</tr>
+<tr class="odd">
+<td align="right">44.97466</td>
+<td align="right">8.8</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 729</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">91.6</td>
+</tr>
+<tr class="even">
+<td align="right">11.24366</td>
+<td align="right">60.5</td>
+</tr>
+<tr class="odd">
+<td align="right">22.48733</td>
+<td align="right">43.5</td>
+</tr>
+<tr class="even">
+<td align="right">33.73099</td>
+<td align="right">28.4</td>
+</tr>
+<tr class="odd">
+<td align="right">44.97466</td>
+<td align="right">20.5</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 730</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">92.7</td>
+</tr>
+<tr class="even">
+<td align="right">11.07446</td>
+<td align="right">58.9</td>
+</tr>
+<tr class="odd">
+<td align="right">22.14893</td>
+<td align="right">44.0</td>
+</tr>
+<tr class="even">
+<td align="right">33.22339</td>
+<td align="right">46.0</td>
+</tr>
+<tr class="odd">
+<td align="right">44.29785</td>
+<td align="right">29.3</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 731</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">92.1</td>
+</tr>
+<tr class="even">
+<td align="right">11.24366</td>
+<td align="right">64.4</td>
+</tr>
+<tr class="odd">
+<td align="right">22.48733</td>
+<td align="right">45.3</td>
+</tr>
+<tr class="even">
+<td align="right">33.73099</td>
+<td align="right">33.6</td>
+</tr>
+<tr class="odd">
+<td align="right">44.97466</td>
+<td align="right">23.5</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 732</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">90.3</td>
+</tr>
+<tr class="even">
+<td align="right">11.24366</td>
+<td align="right">58.2</td>
+</tr>
+<tr class="odd">
+<td align="right">22.48733</td>
+<td align="right">40.1</td>
+</tr>
+<tr class="even">
+<td align="right">33.73099</td>
+<td align="right">33.1</td>
+</tr>
+<tr class="odd">
+<td align="right">44.97466</td>
+<td align="right">25.8</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 741</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">90.3</td>
+</tr>
+<tr class="even">
+<td align="right">10.84712</td>
+<td align="right">68.7</td>
+</tr>
+<tr class="odd">
+<td align="right">21.69424</td>
+<td align="right">58.0</td>
+</tr>
+<tr class="even">
+<td align="right">32.54136</td>
+<td align="right">52.2</td>
+</tr>
+<tr class="odd">
+<td align="right">43.38848</td>
+<td align="right">48.0</td>
+</tr>
+</tbody>
+</table>
+<table class="table">
+<caption>Dataset 742</caption>
+<thead><tr class="header">
+<th align="right">time</th>
+<th align="right">meso</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="right">0.00000</td>
+<td align="right">92.0</td>
+</tr>
+<tr class="even">
+<td align="right">11.24366</td>
+<td align="right">60.9</td>
+</tr>
+<tr class="odd">
+<td align="right">22.48733</td>
+<td align="right">36.2</td>
+</tr>
+<tr class="even">
+<td align="right">33.73099</td>
+<td align="right">18.3</td>
+</tr>
+<tr class="odd">
+<td align="right">44.97466</td>
+<td align="right">8.7</td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="section level2">
+<h2 id="separate-evaluations">Separate evaluations<a class="anchor" aria-label="anchor" href="#separate-evaluations"></a>
+</h2>
+<p>In order to obtain suitable starting parameters for the NLHM fits,
+separate fits of the five models to the data for each soil are generated
+using the <code>mmkin</code> function from the mkin package. In a first
+step, constant variance is assumed. Convergence is checked with the
+<code>status</code> function.</p>
+<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">deg_mods</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"FOMC"</span>, <span class="st">"DFOP"</span>, <span class="st">"SFORB"</span>, <span class="st">"HS"</span><span class="op">)</span></span>
+<span><span class="va">f_sep_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span></span>
+<span>  <span class="va">deg_mods</span>,</span>
+<span>  <span class="va">meso_ds</span>,</span>
+<span>  error_model <span class="op">=</span> <span class="st">"const"</span>,</span>
+<span>  cluster <span class="op">=</span> <span class="va">cl</span>,</span>
+<span>  quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span>, <span class="fl">1</span><span class="op">:</span><span class="fl">5</span><span class="op">]</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">Richmond</th>
+<th align="left">Richmond 2</th>
+<th align="left">ERTC</th>
+<th align="left">Toulouse</th>
+<th align="left">Picket Piece</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">HS</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span>, <span class="fl">6</span><span class="op">:</span><span class="fl">18</span><span class="op">]</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<colgroup>
+<col width="10%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+</colgroup>
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">721</th>
+<th align="left">722</th>
+<th align="left">723</th>
+<th align="left">724</th>
+<th align="left">725</th>
+<th align="left">727</th>
+<th align="left">728</th>
+<th align="left">729</th>
+<th align="left">730</th>
+<th align="left">731</th>
+<th align="left">732</th>
+<th align="left">741</th>
+<th align="left">742</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">HS</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>In the tables above, OK indicates convergence and C indicates failure
+to converge. Most separate fits with constant variance converged, with
+the exception of two FOMC fits, one SFORB fit and one HS fit.</p>
+<div class="sourceCode" id="cb8"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_sep_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_sep_const</span>, error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_tc</span><span class="op">[</span>, <span class="fl">1</span><span class="op">:</span><span class="fl">5</span><span class="op">]</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">Richmond</th>
+<th align="left">Richmond 2</th>
+<th align="left">ERTC</th>
+<th align="left">Toulouse</th>
+<th align="left">Picket Piece</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">HS</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_tc</span><span class="op">[</span>, <span class="fl">6</span><span class="op">:</span><span class="fl">18</span><span class="op">]</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<colgroup>
+<col width="10%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+<col width="6%">
+</colgroup>
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">721</th>
+<th align="left">722</th>
+<th align="left">723</th>
+<th align="left">724</th>
+<th align="left">725</th>
+<th align="left">727</th>
+<th align="left">728</th>
+<th align="left">729</th>
+<th align="left">730</th>
+<th align="left">731</th>
+<th align="left">732</th>
+<th align="left">741</th>
+<th align="left">742</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">HS</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">C</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>With the two-component error model, the set of fits that did not
+converge is larger, with convergence problems appearing for a number of
+non-SFO fits.</p>
+</div>
+<div class="section level2">
+<h2 id="hierarchical-model-fits-without-covariate-effect">Hierarchical model fits without covariate effect<a class="anchor" aria-label="anchor" href="#hierarchical-model-fits-without-covariate-effect"></a>
+</h2>
+<p>The following code fits hierarchical kinetic models for the ten
+combinations of the five different degradation models with the two
+different error models in parallel.</p>
+<div class="sourceCode" id="cb11"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_saem_1</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mhmkin.html">mhmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">f_sep_const</span>, <span class="va">f_sep_tc</span><span class="op">)</span>, cluster <span class="op">=</span> <span class="va">cl</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">HS</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>All fits terminate without errors (status OK).</p>
+<div class="sourceCode" id="cb12"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO const</td>
+<td align="right">5</td>
+<td align="right">800.0</td>
+<td align="right">804.5</td>
+<td align="right">-395.0</td>
+</tr>
+<tr class="even">
+<td align="left">SFO tc</td>
+<td align="right">6</td>
+<td align="right">801.9</td>
+<td align="right">807.2</td>
+<td align="right">-394.9</td>
+</tr>
+<tr class="odd">
+<td align="left">FOMC const</td>
+<td align="right">7</td>
+<td align="right">787.4</td>
+<td align="right">793.6</td>
+<td align="right">-386.7</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC tc</td>
+<td align="right">8</td>
+<td align="right">788.9</td>
+<td align="right">796.1</td>
+<td align="right">-386.5</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP const</td>
+<td align="right">9</td>
+<td align="right">787.6</td>
+<td align="right">795.6</td>
+<td align="right">-384.8</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB const</td>
+<td align="right">9</td>
+<td align="right">787.4</td>
+<td align="right">795.4</td>
+<td align="right">-384.7</td>
+</tr>
+<tr class="odd">
+<td align="left">HS const</td>
+<td align="right">9</td>
+<td align="right">781.9</td>
+<td align="right">789.9</td>
+<td align="right">-382.0</td>
+</tr>
+<tr class="even">
+<td align="left">DFOP tc</td>
+<td align="right">10</td>
+<td align="right">787.4</td>
+<td align="right">796.3</td>
+<td align="right">-383.7</td>
+</tr>
+<tr class="odd">
+<td align="left">SFORB tc</td>
+<td align="right">10</td>
+<td align="right">795.8</td>
+<td align="right">804.7</td>
+<td align="right">-387.9</td>
+</tr>
+<tr class="even">
+<td align="left">HS tc</td>
+<td align="right">10</td>
+<td align="right">783.7</td>
+<td align="right">792.7</td>
+<td align="right">-381.9</td>
+</tr>
+</tbody>
+</table>
+<p>The model comparisons show that the fits with constant variance are
+consistently preferable to the corresponding fits with two-component
+error for these data. This is confirmed by the fact that the parameter
+<code>b.1</code> (the relative standard deviation in the fits obtained
+with the saemix package), is ill-defined in all fits.</p>
+<div class="sourceCode" id="cb13"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<colgroup>
+<col width="6%">
+<col width="44%">
+<col width="49%">
+</colgroup>
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">sd(meso_0)</td>
+<td align="left">sd(meso_0), b.1</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">sd(meso_0), sd(log_beta)</td>
+<td align="left">sd(meso_0), sd(log_beta), b.1</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">sd(meso_0), sd(log_k1)</td>
+<td align="left">sd(meso_0), sd(g_qlogis), b.1</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB</td>
+<td align="left">sd(meso_free_0), sd(log_k_meso_free_bound)</td>
+<td align="left">sd(meso_free_0), sd(log_k_meso_free_bound), b.1</td>
+</tr>
+<tr class="odd">
+<td align="left">HS</td>
+<td align="left">sd(meso_0)</td>
+<td align="left">sd(meso_0), b.1</td>
+</tr>
+</tbody>
+</table>
+<p>For obtaining fits with only well-defined random effects, we update
+the set of fits, excluding random effects that were ill-defined
+according to the <code>illparms</code> function.</p>
+<div class="sourceCode" id="cb14"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_saem_2</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_saem_1</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+<tr class="odd">
+<td align="left">HS</td>
+<td align="left">OK</td>
+<td align="left">OK</td>
+</tr>
+</tbody>
+</table>
+<p>The updated fits terminate without errors.</p>
+<div class="sourceCode" id="cb15"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="left">const</th>
+<th align="left">tc</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left"></td>
+<td align="left">b.1</td>
+</tr>
+<tr class="even">
+<td align="left">FOMC</td>
+<td align="left"></td>
+<td align="left">b.1</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left"></td>
+<td align="left">b.1</td>
+</tr>
+<tr class="even">
+<td align="left">SFORB</td>
+<td align="left"></td>
+<td align="left">b.1</td>
+</tr>
+<tr class="odd">
+<td align="left">HS</td>
+<td align="left"></td>
+<td align="left">b.1</td>
+</tr>
+</tbody>
+</table>
+<p>No ill-defined errors remain in the fits with constant variance.</p>
+</div>
+<div class="section level2">
+<h2 id="hierarchical-model-fits-with-covariate-effect">Hierarchical model fits with covariate effect<a class="anchor" aria-label="anchor" href="#hierarchical-model-fits-with-covariate-effect"></a>
+</h2>
+<p>In the following sections, hierarchical fits including a model for
+the influence of pH on selected degradation parameters are shown for all
+parent models. Constant variance is selected as the error model based on
+the fits without covariate effects. Random effects that were ill-defined
+in the fits without pH influence are excluded. A potential influence of
+the soil pH is only included for parameters with a well-defined random
+effect, because experience has shown that only for such parameters a
+significant pH effect could be found.</p>
+<div class="section level3">
+<h3 id="sfo">SFO<a class="anchor" aria-label="anchor" href="#sfo"></a>
+</h3>
+<div class="sourceCode" id="cb16"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">sfo_pH</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"SFO"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="st">"meso_0"</span>, covariates <span class="op">=</span> <span class="va">pH</span>,</span>
+<span>  covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k_meso</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb17"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">sfo_pH</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">est.</th>
+<th align="right">lower</th>
+<th align="right">upper</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">meso_0</td>
+<td align="right">91.35</td>
+<td align="right">89.27</td>
+<td align="right">93.43</td>
+</tr>
+<tr class="even">
+<td align="left">log_k_meso</td>
+<td align="right">-6.66</td>
+<td align="right">-7.97</td>
+<td align="right">-5.35</td>
+</tr>
+<tr class="odd">
+<td align="left">beta_pH(log_k_meso)</td>
+<td align="right">0.59</td>
+<td align="right">0.37</td>
+<td align="right">0.81</td>
+</tr>
+<tr class="even">
+<td align="left">a.1</td>
+<td align="right">5.48</td>
+<td align="right">4.71</td>
+<td align="right">6.24</td>
+</tr>
+<tr class="odd">
+<td align="left">SD.log_k_meso</td>
+<td align="right">0.35</td>
+<td align="right">0.23</td>
+<td align="right">0.47</td>
+</tr>
+</tbody>
+</table>
+<p>The parameter showing the pH influence in the above table is
+<code>beta_pH(log_k_meso)</code>. Its confidence interval does not
+include zero, indicating that the influence of soil pH on the log of the
+degradation rate constant is significantly greater than zero.</p>
+<div class="sourceCode" id="cb18"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"SFO"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span>, <span class="va">sfo_pH</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
+
+                           npar    AIC    BIC     Lik  Chisq Df Pr(&gt;Chisq)    
+f_saem_2[["SFO", "const"]]    4 797.56 801.12 -394.78                         
+sfo_pH                        5 783.09 787.54 -386.54 16.473  1  4.934e-05 ***
+---
+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
+<p>The comparison with the SFO fit without covariate effect confirms
+that considering the soil pH improves the model, both by comparison of
+AIC and BIC and by the likelihood ratio test.</p>
+<div class="sourceCode" id="cb20"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">sfo_pH</span><span class="op">)</span></span></code></pre></div>
+<p><img src="2023_mesotrione_parent_files/figure-html/unnamed-chunk-8-1.png" width="700" style="display: block; margin: auto;"></p>
+<p>Endpoints for a model with covariates are by default calculated for
+the median of the covariate values. This quantile can be adapted, or a
+specific covariate value can be given as shown below.</p>
+<div class="sourceCode" id="cb21"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">sfo_pH</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+      pH
+50% 5.75
+
+$distimes
+         DT50     DT90
+meso 18.52069 61.52441</code></pre>
+<div class="sourceCode" id="cb23"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">sfo_pH</span>, covariate_quantile <span class="op">=</span> <span class="fl">0.9</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+      pH
+90% 7.13
+
+$distimes
+         DT50     DT90
+meso 8.237019 27.36278</code></pre>
+<div class="sourceCode" id="cb25"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">sfo_pH</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7.0</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+     pH
+User  7
+
+$distimes
+        DT50    DT90
+meso 8.89035 29.5331</code></pre>
+</div>
+<div class="section level3">
+<h3 id="fomc">FOMC<a class="anchor" aria-label="anchor" href="#fomc"></a>
+</h3>
+<div class="sourceCode" id="cb27"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">fomc_pH</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"FOMC"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="st">"meso_0"</span>, covariates <span class="op">=</span> <span class="va">pH</span>,</span>
+<span>  covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_alpha</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb28"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">fomc_pH</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">est.</th>
+<th align="right">lower</th>
+<th align="right">upper</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">meso_0</td>
+<td align="right">92.84</td>
+<td align="right">90.75</td>
+<td align="right">94.93</td>
+</tr>
+<tr class="even">
+<td align="left">log_alpha</td>
+<td align="right">-2.21</td>
+<td align="right">-3.49</td>
+<td align="right">-0.92</td>
+</tr>
+<tr class="odd">
+<td align="left">beta_pH(log_alpha)</td>
+<td align="right">0.58</td>
+<td align="right">0.37</td>
+<td align="right">0.79</td>
+</tr>
+<tr class="even">
+<td align="left">log_beta</td>
+<td align="right">4.21</td>
+<td align="right">3.44</td>
+<td align="right">4.99</td>
+</tr>
+<tr class="odd">
+<td align="left">a.1</td>
+<td align="right">5.03</td>
+<td align="right">4.32</td>
+<td align="right">5.73</td>
+</tr>
+<tr class="even">
+<td align="left">SD.log_alpha</td>
+<td align="right">0.00</td>
+<td align="right">-23.77</td>
+<td align="right">23.78</td>
+</tr>
+<tr class="odd">
+<td align="left">SD.log_beta</td>
+<td align="right">0.37</td>
+<td align="right">0.01</td>
+<td align="right">0.74</td>
+</tr>
+</tbody>
+</table>
+<p>As in the case of SFO, the confidence interval of the slope parameter
+(here <code>beta_pH(log_alpha)</code>) quantifying the influence of soil
+pH does not include zero, and the model comparison clearly indicates
+that the model with covariate influence is preferable. However, the
+random effect for <code>alpha</code> is not well-defined any more after
+inclusion of the covariate effect (the confidence interval of
+<code>SD.log_alpha</code> includes zero).</p>
+<div class="sourceCode" id="cb29"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">fomc_pH</span><span class="op">)</span></span></code></pre></div>
+<pre><code>[1] "sd(log_alpha)"</code></pre>
+<p>Therefore, the model is updated without this random effect, and no
+ill-defined parameters remain.</p>
+<div class="sourceCode" id="cb31"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">fomc_pH_2</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">fomc_pH</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span>, <span class="st">"log_alpha"</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">fomc_pH_2</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb32"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"FOMC"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span>, <span class="va">fomc_pH</span>, <span class="va">fomc_pH_2</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
+
+                            npar    AIC    BIC     Lik  Chisq Df Pr(&gt;Chisq)    
+f_saem_2[["FOMC", "const"]]    5 783.25 787.71 -386.63                         
+fomc_pH_2                      6 767.49 772.83 -377.75 17.762  1  2.503e-05 ***
+fomc_pH                        7 770.07 776.30 -378.04  0.000  1          1    
+---
+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
+<p>Model comparison indicates that including pH dependence significantly
+improves the fit, and that the reduced model with covariate influence
+results in the most preferable FOMC fit.</p>
+<div class="sourceCode" id="cb34"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">fomc_pH_2</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">est.</th>
+<th align="right">lower</th>
+<th align="right">upper</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">meso_0</td>
+<td align="right">93.05</td>
+<td align="right">90.98</td>
+<td align="right">95.13</td>
+</tr>
+<tr class="even">
+<td align="left">log_alpha</td>
+<td align="right">-2.91</td>
+<td align="right">-4.18</td>
+<td align="right">-1.63</td>
+</tr>
+<tr class="odd">
+<td align="left">beta_pH(log_alpha)</td>
+<td align="right">0.66</td>
+<td align="right">0.44</td>
+<td align="right">0.87</td>
+</tr>
+<tr class="even">
+<td align="left">log_beta</td>
+<td align="right">3.95</td>
+<td align="right">3.29</td>
+<td align="right">4.62</td>
+</tr>
+<tr class="odd">
+<td align="left">a.1</td>
+<td align="right">4.98</td>
+<td align="right">4.28</td>
+<td align="right">5.68</td>
+</tr>
+<tr class="even">
+<td align="left">SD.log_beta</td>
+<td align="right">0.40</td>
+<td align="right">0.26</td>
+<td align="right">0.54</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb35"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">fomc_pH_2</span><span class="op">)</span></span></code></pre></div>
+<p><img src="2023_mesotrione_parent_files/figure-html/unnamed-chunk-14-1.png" width="700" style="display: block; margin: auto;"></p>
+<div class="sourceCode" id="cb36"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">fomc_pH_2</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+      pH
+50% 5.75
+
+$distimes
+         DT50     DT90 DT50back
+meso 17.30248 82.91343 24.95943</code></pre>
+<div class="sourceCode" id="cb38"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">fomc_pH_2</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+     pH
+User  7
+
+$distimes
+         DT50     DT90 DT50back
+meso 6.986239 27.02927 8.136621</code></pre>
+</div>
+<div class="section level3">
+<h3 id="dfop">DFOP<a class="anchor" aria-label="anchor" href="#dfop"></a>
+</h3>
+<p>In the DFOP fits without covariate effects, random effects for two
+degradation parameters (<code>k2</code> and <code>g</code>) were
+identifiable.</p>
+<div class="sourceCode" id="cb40"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">est.</th>
+<th align="right">lower</th>
+<th align="right">upper</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">meso_0</td>
+<td align="right">93.61</td>
+<td align="right">91.58</td>
+<td align="right">95.63</td>
+</tr>
+<tr class="even">
+<td align="left">log_k1</td>
+<td align="right">-1.53</td>
+<td align="right">-2.27</td>
+<td align="right">-0.79</td>
+</tr>
+<tr class="odd">
+<td align="left">log_k2</td>
+<td align="right">-3.42</td>
+<td align="right">-3.73</td>
+<td align="right">-3.11</td>
+</tr>
+<tr class="even">
+<td align="left">g_qlogis</td>
+<td align="right">-1.67</td>
+<td align="right">-2.57</td>
+<td align="right">-0.77</td>
+</tr>
+<tr class="odd">
+<td align="left">a.1</td>
+<td align="right">4.74</td>
+<td align="right">4.02</td>
+<td align="right">5.45</td>
+</tr>
+<tr class="even">
+<td align="left">SD.log_k2</td>
+<td align="right">0.60</td>
+<td align="right">0.38</td>
+<td align="right">0.81</td>
+</tr>
+<tr class="odd">
+<td align="left">SD.g_qlogis</td>
+<td align="right">0.94</td>
+<td align="right">0.33</td>
+<td align="right">1.54</td>
+</tr>
+</tbody>
+</table>
+<p>A fit with pH dependent degradation parameters was obtained by
+excluding the same random effects as in the refined DFOP fit without
+covariate influence, and including covariate models for the two
+identifiable parameters <code>k2</code> and <code>g</code>.</p>
+<div class="sourceCode" id="cb41"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">dfop_pH</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span>, <span class="st">"log_k1"</span><span class="op">)</span>,</span>
+<span>  covariates <span class="op">=</span> <span class="va">pH</span>,</span>
+<span>  covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k2</span> <span class="op">~</span> <span class="va">pH</span>, <span class="va">g_qlogis</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<p>The corresponding parameters for the influence of soil pH are
+<code>beta_pH(log_k2)</code> for the influence of soil pH on
+<code>k2</code>, and <code>beta_pH(g_qlogis)</code> for its influence on
+<code>g</code>.</p>
+<div class="sourceCode" id="cb42"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">dfop_pH</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">est.</th>
+<th align="right">lower</th>
+<th align="right">upper</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">meso_0</td>
+<td align="right">92.84</td>
+<td align="right">90.85</td>
+<td align="right">94.84</td>
+</tr>
+<tr class="even">
+<td align="left">log_k1</td>
+<td align="right">-2.82</td>
+<td align="right">-3.09</td>
+<td align="right">-2.54</td>
+</tr>
+<tr class="odd">
+<td align="left">log_k2</td>
+<td align="right">-11.48</td>
+<td align="right">-15.32</td>
+<td align="right">-7.64</td>
+</tr>
+<tr class="even">
+<td align="left">beta_pH(log_k2)</td>
+<td align="right">1.31</td>
+<td align="right">0.69</td>
+<td align="right">1.92</td>
+</tr>
+<tr class="odd">
+<td align="left">g_qlogis</td>
+<td align="right">3.13</td>
+<td align="right">0.47</td>
+<td align="right">5.80</td>
+</tr>
+<tr class="even">
+<td align="left">beta_pH(g_qlogis)</td>
+<td align="right">-0.57</td>
+<td align="right">-1.04</td>
+<td align="right">-0.09</td>
+</tr>
+<tr class="odd">
+<td align="left">a.1</td>
+<td align="right">4.96</td>
+<td align="right">4.26</td>
+<td align="right">5.65</td>
+</tr>
+<tr class="even">
+<td align="left">SD.log_k2</td>
+<td align="right">0.76</td>
+<td align="right">0.47</td>
+<td align="right">1.05</td>
+</tr>
+<tr class="odd">
+<td align="left">SD.g_qlogis</td>
+<td align="right">0.01</td>
+<td align="right">-9.96</td>
+<td align="right">9.97</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb43"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">dfop_pH</span><span class="op">)</span></span></code></pre></div>
+<pre><code>[1] "sd(g_qlogis)"</code></pre>
+<p>Confidence intervals for neither of them include zero, indicating a
+significant difference from zero. However, the random effect for
+<code>g</code> is now ill-defined. The fit is updated without this
+ill-defined random effect.</p>
+<div class="sourceCode" id="cb45"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">dfop_pH_2</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">dfop_pH</span>,</span>
+<span>  no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span>, <span class="st">"log_k1"</span>, <span class="st">"g_qlogis"</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">dfop_pH_2</span><span class="op">)</span></span></code></pre></div>
+<pre><code>[1] "beta_pH(g_qlogis)"</code></pre>
+<p>Now, the slope parameter for the pH effect on <code>g</code> is
+ill-defined. Therefore, another attempt is made without the
+corresponding covariate model.</p>
+<div class="sourceCode" id="cb47"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">dfop_pH_3</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span>, <span class="st">"log_k1"</span><span class="op">)</span>,</span>
+<span>  covariates <span class="op">=</span> <span class="va">pH</span>,</span>
+<span>  covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k2</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">dfop_pH_3</span><span class="op">)</span></span></code></pre></div>
+<pre><code>[1] "sd(g_qlogis)"</code></pre>
+<p>As the random effect for <code>g</code> is again ill-defined, the fit
+is repeated without it.</p>
+<div class="sourceCode" id="cb49"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">dfop_pH_4</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">dfop_pH_3</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span>, <span class="st">"log_k1"</span>, <span class="st">"g_qlogis"</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">dfop_pH_4</span><span class="op">)</span></span></code></pre></div>
+<p>While no ill-defined parameters remain, model comparison suggests
+that the previous model <code>dfop_pH_2</code> with two pH dependent
+parameters is preferable, based on information criteria as well as based
+on the likelihood ratio test.</p>
+<div class="sourceCode" id="cb50"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span>, <span class="va">dfop_pH</span>, <span class="va">dfop_pH_2</span>, <span class="va">dfop_pH_3</span>, <span class="va">dfop_pH_4</span><span class="op">)</span></span></code></pre></div>
+<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
+
+                            npar    AIC    BIC     Lik
+f_saem_2[["DFOP", "const"]]    7 782.94 789.18 -384.47
+dfop_pH_4                      7 767.35 773.58 -376.68
+dfop_pH_2                      8 765.14 772.26 -374.57
+dfop_pH_3                      8 769.00 776.12 -376.50
+dfop_pH                        9 769.10 777.11 -375.55</code></pre>
+<div class="sourceCode" id="cb52"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">dfop_pH_2</span>, <span class="va">dfop_pH_4</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
+
+          npar    AIC    BIC     Lik  Chisq Df Pr(&gt;Chisq)  
+dfop_pH_4    7 767.35 773.58 -376.68                       
+dfop_pH_2    8 765.14 772.26 -374.57 4.2153  1    0.04006 *
+---
+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
+<p>When focussing on parameter identifiability using the test if the
+confidence interval includes zero, <code>dfop_pH_4</code> would still be
+the preferred model. However, it should be kept in mind that parameter
+confidence intervals are constructed using a simple linearisation of the
+likelihood. As the confidence interval of the random effect for
+<code>g</code> only marginally includes zero, it is suggested that this
+is acceptable, and that <code>dfop_pH_2</code> can be considered the
+most preferable model.</p>
+<div class="sourceCode" id="cb54"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">dfop_pH_2</span><span class="op">)</span></span></code></pre></div>
+<p><img src="2023_mesotrione_parent_files/figure-html/unnamed-chunk-19-1.png" width="700" style="display: block; margin: auto;"></p>
+<div class="sourceCode" id="cb55"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">dfop_pH_2</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+      pH
+50% 5.75
+
+$distimes
+         DT50     DT90 DT50back  DT50_k1  DT50_k2
+meso 18.36876 73.51841 22.13125 4.191901 23.98672</code></pre>
+<div class="sourceCode" id="cb57"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">dfop_pH_2</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+     pH
+User  7
+
+$distimes
+         DT50     DT90 DT50back  DT50_k1  DT50_k2
+meso 8.346428 28.34437 8.532507 4.191901 8.753618</code></pre>
+</div>
+<div class="section level3">
+<h3 id="sforb">SFORB<a class="anchor" aria-label="anchor" href="#sforb"></a>
+</h3>
+<div class="sourceCode" id="cb59"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">sforb_pH</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"SFORB"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_free_0"</span>, <span class="st">"log_k_meso_free_bound"</span><span class="op">)</span>,</span>
+<span>  covariates <span class="op">=</span> <span class="va">pH</span>,</span>
+<span>  covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k_meso_free</span> <span class="op">~</span> <span class="va">pH</span>, <span class="va">log_k_meso_bound_free</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb60"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">sforb_pH</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">est.</th>
+<th align="right">lower</th>
+<th align="right">upper</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">meso_free_0</td>
+<td align="right">93.42</td>
+<td align="right">91.32</td>
+<td align="right">95.52</td>
+</tr>
+<tr class="even">
+<td align="left">log_k_meso_free</td>
+<td align="right">-5.37</td>
+<td align="right">-6.94</td>
+<td align="right">-3.81</td>
+</tr>
+<tr class="odd">
+<td align="left">beta_pH(log_k_meso_free)</td>
+<td align="right">0.42</td>
+<td align="right">0.18</td>
+<td align="right">0.67</td>
+</tr>
+<tr class="even">
+<td align="left">log_k_meso_free_bound</td>
+<td align="right">-3.49</td>
+<td align="right">-4.92</td>
+<td align="right">-2.05</td>
+</tr>
+<tr class="odd">
+<td align="left">log_k_meso_bound_free</td>
+<td align="right">-9.98</td>
+<td align="right">-19.22</td>
+<td align="right">-0.74</td>
+</tr>
+<tr class="even">
+<td align="left">beta_pH(log_k_meso_bound_free)</td>
+<td align="right">1.23</td>
+<td align="right">-0.21</td>
+<td align="right">2.67</td>
+</tr>
+<tr class="odd">
+<td align="left">a.1</td>
+<td align="right">4.90</td>
+<td align="right">4.18</td>
+<td align="right">5.63</td>
+</tr>
+<tr class="even">
+<td align="left">SD.log_k_meso_free</td>
+<td align="right">0.35</td>
+<td align="right">0.23</td>
+<td align="right">0.47</td>
+</tr>
+<tr class="odd">
+<td align="left">SD.log_k_meso_bound_free</td>
+<td align="right">0.13</td>
+<td align="right">-1.95</td>
+<td align="right">2.20</td>
+</tr>
+</tbody>
+</table>
+<p>The confidence interval of
+<code>beta_pH(log_k_meso_bound_free)</code> includes zero, indicating
+that the influence of soil pH on <code>k_meso_bound_free</code> cannot
+reliably be quantified. Also, the confidence interval for the random
+effect on this parameter (<code>SD.log_k_meso_bound_free</code>)
+includes zero.</p>
+<p>Using the <code>illparms</code> function, these ill-defined
+parameters can be found more conveniently.</p>
+<div class="sourceCode" id="cb61"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">sforb_pH</span><span class="op">)</span></span></code></pre></div>
+<pre><code>[1] "sd(log_k_meso_bound_free)"      "beta_pH(log_k_meso_bound_free)"</code></pre>
+<p>To remove the ill-defined parameters, a second variant of the SFORB
+model with pH influence is fitted. No ill-defined parameters remain.</p>
+<div class="sourceCode" id="cb63"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">sforb_pH_2</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">sforb_pH</span>,</span>
+<span>  no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_free_0"</span>, <span class="st">"log_k_meso_free_bound"</span>, <span class="st">"log_k_meso_bound_free"</span><span class="op">)</span>,</span>
+<span>  covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k_meso_free</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">sforb_pH_2</span><span class="op">)</span></span></code></pre></div>
+<p>The model comparison of the SFORB fits includes the refined model
+without covariate effect, and both versions of the SFORB fit with
+covariate effect.</p>
+<div class="sourceCode" id="cb64"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"SFORB"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span>, <span class="va">sforb_pH</span>, <span class="va">sforb_pH_2</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
+
+                             npar    AIC    BIC     Lik   Chisq Df Pr(&gt;Chisq)  
+f_saem_2[["SFORB", "const"]]    7 783.40 789.63 -384.70                        
+sforb_pH_2                      7 770.94 777.17 -378.47 12.4616  0             
+sforb_pH                        9 768.81 776.83 -375.41  6.1258  2    0.04675 *
+---
+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
+<p>The first model including pH influence is preferable based on
+information criteria and the likelihood ratio test. However, as it is
+not fully identifiable, the second model is selected.</p>
+<div class="sourceCode" id="cb66"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">sforb_pH_2</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">est.</th>
+<th align="right">lower</th>
+<th align="right">upper</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">meso_free_0</td>
+<td align="right">93.32</td>
+<td align="right">91.16</td>
+<td align="right">95.48</td>
+</tr>
+<tr class="even">
+<td align="left">log_k_meso_free</td>
+<td align="right">-6.15</td>
+<td align="right">-7.43</td>
+<td align="right">-4.86</td>
+</tr>
+<tr class="odd">
+<td align="left">beta_pH(log_k_meso_free)</td>
+<td align="right">0.54</td>
+<td align="right">0.33</td>
+<td align="right">0.75</td>
+</tr>
+<tr class="even">
+<td align="left">log_k_meso_free_bound</td>
+<td align="right">-3.80</td>
+<td align="right">-5.20</td>
+<td align="right">-2.40</td>
+</tr>
+<tr class="odd">
+<td align="left">log_k_meso_bound_free</td>
+<td align="right">-2.95</td>
+<td align="right">-4.26</td>
+<td align="right">-1.64</td>
+</tr>
+<tr class="even">
+<td align="left">a.1</td>
+<td align="right">5.08</td>
+<td align="right">4.38</td>
+<td align="right">5.79</td>
+</tr>
+<tr class="odd">
+<td align="left">SD.log_k_meso_free</td>
+<td align="right">0.33</td>
+<td align="right">0.22</td>
+<td align="right">0.45</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb67"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">sforb_pH_2</span><span class="op">)</span></span></code></pre></div>
+<p><img src="2023_mesotrione_parent_files/figure-html/unnamed-chunk-25-1.png" width="700" style="display: block; margin: auto;"></p>
+<div class="sourceCode" id="cb68"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">sforb_pH_2</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+      pH
+50% 5.75
+
+$ff
+meso_free 
+        1 
+
+$SFORB
+   meso_b1    meso_b2     meso_g 
+0.09735824 0.02631699 0.31602120 
+
+$distimes
+         DT50     DT90 DT50back DT50_meso_b1 DT50_meso_b2
+meso 16.86549 73.15824 22.02282     7.119554     26.33839</code></pre>
+<div class="sourceCode" id="cb70"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">sforb_pH_2</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+     pH
+User  7
+
+$ff
+meso_free 
+        1 
+
+$SFORB
+   meso_b1    meso_b2     meso_g 
+0.13315233 0.03795988 0.61186191 
+
+$distimes
+         DT50     DT90 DT50back DT50_meso_b1 DT50_meso_b2
+meso 7.932495 36.93311 11.11797     5.205671        18.26</code></pre>
+</div>
+<div class="section level3">
+<h3 id="hs">HS<a class="anchor" aria-label="anchor" href="#hs"></a>
+</h3>
+<div class="sourceCode" id="cb72"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">hs_pH</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"HS"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span><span class="op">)</span>,</span>
+<span>  covariates <span class="op">=</span> <span class="va">pH</span>,</span>
+<span>  covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k1</span> <span class="op">~</span> <span class="va">pH</span>, <span class="va">log_k2</span> <span class="op">~</span> <span class="va">pH</span>, <span class="va">log_tb</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb73"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">hs_pH</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">est.</th>
+<th align="right">lower</th>
+<th align="right">upper</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">meso_0</td>
+<td align="right">93.33</td>
+<td align="right">91.47</td>
+<td align="right">95.19</td>
+</tr>
+<tr class="even">
+<td align="left">log_k1</td>
+<td align="right">-5.81</td>
+<td align="right">-7.27</td>
+<td align="right">-4.36</td>
+</tr>
+<tr class="odd">
+<td align="left">beta_pH(log_k1)</td>
+<td align="right">0.47</td>
+<td align="right">0.23</td>
+<td align="right">0.72</td>
+</tr>
+<tr class="even">
+<td align="left">log_k2</td>
+<td align="right">-6.80</td>
+<td align="right">-8.76</td>
+<td align="right">-4.83</td>
+</tr>
+<tr class="odd">
+<td align="left">beta_pH(log_k2)</td>
+<td align="right">0.54</td>
+<td align="right">0.21</td>
+<td align="right">0.87</td>
+</tr>
+<tr class="even">
+<td align="left">log_tb</td>
+<td align="right">3.25</td>
+<td align="right">1.25</td>
+<td align="right">5.25</td>
+</tr>
+<tr class="odd">
+<td align="left">beta_pH(log_tb)</td>
+<td align="right">-0.10</td>
+<td align="right">-0.43</td>
+<td align="right">0.23</td>
+</tr>
+<tr class="even">
+<td align="left">a.1</td>
+<td align="right">4.49</td>
+<td align="right">3.78</td>
+<td align="right">5.21</td>
+</tr>
+<tr class="odd">
+<td align="left">SD.log_k1</td>
+<td align="right">0.37</td>
+<td align="right">0.24</td>
+<td align="right">0.51</td>
+</tr>
+<tr class="even">
+<td align="left">SD.log_k2</td>
+<td align="right">0.29</td>
+<td align="right">0.10</td>
+<td align="right">0.48</td>
+</tr>
+<tr class="odd">
+<td align="left">SD.log_tb</td>
+<td align="right">0.25</td>
+<td align="right">-0.07</td>
+<td align="right">0.57</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb74"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">hs_pH</span><span class="op">)</span></span></code></pre></div>
+<pre><code>[1] "sd(log_tb)"      "beta_pH(log_tb)"</code></pre>
+<p>According to the output of the <code>illparms</code> function, the
+random effect on the break time <code>tb</code> cannot reliably be
+quantified, neither can the influence of soil pH on <code>tb</code>. The
+fit is repeated without the corresponding covariate model, and no
+ill-defined parameters remain.</p>
+<div class="sourceCode" id="cb76"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">hs_pH_2</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">hs_pH</span>, covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k1</span> <span class="op">~</span> <span class="va">pH</span>, <span class="va">log_k2</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">hs_pH_2</span><span class="op">)</span></span></code></pre></div>
+<p>Model comparison confirms that this model is preferable to the fit
+without covariate influence, and also to the first version with
+covariate influence.</p>
+<div class="sourceCode" id="cb77"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"HS"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span>, <span class="va">hs_pH</span>, <span class="va">hs_pH_2</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
+
+                          npar    AIC    BIC     Lik  Chisq Df Pr(&gt;Chisq)    
+f_saem_2[["HS", "const"]]    8 780.08 787.20 -382.04                         
+hs_pH_2                     10 766.47 775.37 -373.23 17.606  2  0.0001503 ***
+hs_pH                       11 769.80 779.59 -373.90  0.000  1  1.0000000    
+---
+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
+<div class="sourceCode" id="cb79"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">hs_pH_2</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">est.</th>
+<th align="right">lower</th>
+<th align="right">upper</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">meso_0</td>
+<td align="right">93.33</td>
+<td align="right">91.50</td>
+<td align="right">95.15</td>
+</tr>
+<tr class="even">
+<td align="left">log_k1</td>
+<td align="right">-5.68</td>
+<td align="right">-7.09</td>
+<td align="right">-4.27</td>
+</tr>
+<tr class="odd">
+<td align="left">beta_pH(log_k1)</td>
+<td align="right">0.46</td>
+<td align="right">0.22</td>
+<td align="right">0.69</td>
+</tr>
+<tr class="even">
+<td align="left">log_k2</td>
+<td align="right">-6.61</td>
+<td align="right">-8.34</td>
+<td align="right">-4.88</td>
+</tr>
+<tr class="odd">
+<td align="left">beta_pH(log_k2)</td>
+<td align="right">0.50</td>
+<td align="right">0.21</td>
+<td align="right">0.79</td>
+</tr>
+<tr class="even">
+<td align="left">log_tb</td>
+<td align="right">2.70</td>
+<td align="right">2.33</td>
+<td align="right">3.08</td>
+</tr>
+<tr class="odd">
+<td align="left">a.1</td>
+<td align="right">4.45</td>
+<td align="right">3.74</td>
+<td align="right">5.16</td>
+</tr>
+<tr class="even">
+<td align="left">SD.log_k1</td>
+<td align="right">0.36</td>
+<td align="right">0.22</td>
+<td align="right">0.49</td>
+</tr>
+<tr class="odd">
+<td align="left">SD.log_k2</td>
+<td align="right">0.23</td>
+<td align="right">0.02</td>
+<td align="right">0.43</td>
+</tr>
+<tr class="even">
+<td align="left">SD.log_tb</td>
+<td align="right">0.55</td>
+<td align="right">0.25</td>
+<td align="right">0.85</td>
+</tr>
+</tbody>
+</table>
+<div class="sourceCode" id="cb80"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">hs_pH_2</span><span class="op">)</span></span></code></pre></div>
+<p><img src="2023_mesotrione_parent_files/figure-html/unnamed-chunk-30-1.png" width="700" style="display: block; margin: auto;"></p>
+<div class="sourceCode" id="cb81"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">hs_pH_2</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+      pH
+50% 5.75
+
+$distimes
+         DT50     DT90 DT50back  DT50_k1  DT50_k2
+meso 14.68725 82.45287 24.82079 14.68725 29.29299</code></pre>
+<div class="sourceCode" id="cb83"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">hs_pH_2</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+     pH
+User  7
+
+$distimes
+         DT50     DT90 DT50back  DT50_k1  DT50_k2
+meso 8.298536 38.85371 11.69613 8.298536 15.71561</code></pre>
+</div>
+<div class="section level3">
+<h3 id="comparison-across-parent-models">Comparison across parent models<a class="anchor" aria-label="anchor" href="#comparison-across-parent-models"></a>
+</h3>
+<p>After model reduction for all models with pH influence, they are
+compared with each other.</p>
+<div class="sourceCode" id="cb85"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">sfo_pH</span>, <span class="va">fomc_pH_2</span>, <span class="va">dfop_pH_2</span>, <span class="va">dfop_pH_4</span>, <span class="va">sforb_pH_2</span>, <span class="va">hs_pH_2</span><span class="op">)</span></span></code></pre></div>
+<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
+
+           npar    AIC    BIC     Lik
+sfo_pH        5 783.09 787.54 -386.54
+fomc_pH_2     6 767.49 772.83 -377.75
+dfop_pH_4     7 767.35 773.58 -376.68
+sforb_pH_2    7 770.94 777.17 -378.47
+dfop_pH_2     8 765.14 772.26 -374.57
+hs_pH_2      10 766.47 775.37 -373.23</code></pre>
+<p>The DFOP model with pH influence on <code>k2</code> and
+<code>g</code> and a random effect only on <code>k2</code> is finally
+selected as the best fit.</p>
+<p>The endpoints resulting from this model are listed below. Please
+refer to the Appendix for a detailed listing.</p>
+<div class="sourceCode" id="cb87"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">dfop_pH_2</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+      pH
+50% 5.75
+
+$distimes
+         DT50     DT90 DT50back  DT50_k1  DT50_k2
+meso 18.36876 73.51841 22.13125 4.191901 23.98672</code></pre>
+<div class="sourceCode" id="cb89"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">dfop_pH_2</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code>$covariates
+     pH
+User  7
+
+$distimes
+         DT50     DT90 DT50back  DT50_k1  DT50_k2
+meso 8.346428 28.34437 8.532507 4.191901 8.753618</code></pre>
+</div>
+</div>
+<div class="section level2">
+<h2 id="conclusions">Conclusions<a class="anchor" aria-label="anchor" href="#conclusions"></a>
+</h2>
+<p>These evaluations demonstrate that covariate effects can be included
+for all types of parent degradation models. These models can then be
+further refined to make them fully identifiable.</p>
+</div>
+<div class="section level2">
+<h2 id="appendix">Appendix<a class="anchor" aria-label="anchor" href="#appendix"></a>
+</h2>
+<div class="section level3">
+<h3 id="hierarchical-fit-listings">Hierarchical fit listings<a class="anchor" aria-label="anchor" href="#hierarchical-fit-listings"></a>
+</h3>
+<div class="section level4">
+<h4 id="fits-without-covariate-effects">Fits without covariate effects<a class="anchor" aria-label="anchor" href="#fits-without-covariate-effects"></a>
+</h4>
+
+</div>
+<div class="section level4">
+<h4 id="fits-with-covariate-effects">Fits with covariate effects<a class="anchor" aria-label="anchor" href="#fits-with-covariate-effects"></a>
+</h4>
+
+</div>
+</div>
+<div class="section level3">
+<h3 id="session-info">Session info<a class="anchor" aria-label="anchor" href="#session-info"></a>
+</h3>
+<pre><code>R version 4.4.2 (2024-10-31)
+Platform: x86_64-pc-linux-gnu
+Running under: Debian GNU/Linux 12 (bookworm)
+
+Matrix products: default
+BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.11.0 
+LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.11.0
+
+locale:
+ [1] LC_CTYPE=de_DE.UTF-8       LC_NUMERIC=C              
+ [3] LC_TIME=de_DE.UTF-8        LC_COLLATE=de_DE.UTF-8    
+ [5] LC_MONETARY=de_DE.UTF-8    LC_MESSAGES=de_DE.UTF-8   
+ [7] LC_PAPER=de_DE.UTF-8       LC_NAME=C                 
+ [9] LC_ADDRESS=C               LC_TELEPHONE=C            
+[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C       
+
+time zone: Europe/Berlin
+tzcode source: system (glibc)
+
+attached base packages:
+[1] parallel  stats     graphics  grDevices utils     datasets  methods  
+[8] base     
+
+other attached packages:
+[1] rmarkdown_2.29  nvimcom_0.9-167 saemix_3.3      npde_3.5       
+[5] knitr_1.49      mkin_1.2.10    
+
+loaded via a namespace (and not attached):
+ [1] sass_0.4.9        utf8_1.2.4        generics_0.1.3    lattice_0.22-6   
+ [5] digest_0.6.37     magrittr_2.0.3    evaluate_1.0.1    grid_4.4.2       
+ [9] fastmap_1.2.0     cellranger_1.1.0  jsonlite_1.8.9    mclust_6.1.1     
+[13] gridExtra_2.3     fansi_1.0.6       scales_1.3.0      codetools_0.2-20 
+[17] textshaping_0.4.1 jquerylib_0.1.4   cli_3.6.3         rlang_1.1.4      
+[21] munsell_0.5.1     cachem_1.1.0      yaml_2.3.10       tools_4.4.2      
+[25] dplyr_1.1.4       colorspace_2.1-1  ggplot2_3.5.1     vctrs_0.6.5      
+[29] R6_2.5.1          zoo_1.8-12        lifecycle_1.0.4   fs_1.6.5         
+[33] htmlwidgets_1.6.4 MASS_7.3-61       ragg_1.3.3        pkgconfig_2.0.3  
+[37] desc_1.4.3        pkgdown_2.1.1     pillar_1.9.0      bslib_0.8.0      
+[41] gtable_0.3.6      glue_1.8.0        systemfonts_1.1.0 xfun_0.49        
+[45] tibble_3.2.1      lmtest_0.9-40     tidyselect_1.2.1  htmltools_0.5.8.1
+[49] nlme_3.1-166      compiler_4.4.2    readxl_1.4.3     </code></pre>
+</div>
+<div class="section level3">
+<h3 id="hardware-info">Hardware info<a class="anchor" aria-label="anchor" href="#hardware-info"></a>
+</h3>
+<pre><code>CPU model: AMD Ryzen 9 7950X 16-Core Processor</code></pre>
+<pre><code>MemTotal:       64927788 kB</code></pre>
+</div>
+</div>
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+      <h1>Calculation of time weighted average concentrations with mkin</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change 18 September 2019
+(rebuilt 2025-02-14)</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/twa.rmd" class="external-link"><code>vignettes/twa.rmd</code></a></small>
+      <div class="d-none name"><code>twa.rmd</code></div>
+    </div>
+
+    
+    
+<p>Since version 0.9.45.1 of the ‘mkin’ package, a function for
+calculating time weighted average concentrations for decline kinetics
+(<em>i.e.</em> only for the compound applied in the experiment) is
+included. Strictly speaking, they are maximum moving window time
+weighted average concentrations, <em>i.e.</em> the maximum time weighted
+average concentration that can be found when moving a time window of a
+specified width over the decline curve.</p>
+<p>Time weighted average concentrations for the SFO, FOMC and the DFOP
+model are calculated using the formulas given in the FOCUS kinetics
+guidance <span class="citation">(FOCUS Work Group on Degradation
+Kinetics 2014, 251)</span>:</p>
+<p>SFO:</p>
+<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><msub><mi>c</mi><mn>0</mn></msub><mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>k</mi><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mrow><mi>k</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">c_\textrm{twa} = c_0 \frac{\left( 1 - e^{- k t} \right)}{ k t} </annotation></semantics></math></p>
+<p>FOMC:</p>
+<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><msub><mi>c</mi><mn>0</mn></msub><mfrac><mi>β</mi><mrow><mi>t</mi><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><msup><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mi>t</mi><mi>β</mi></mfrac><mo>+</mo><mn>1</mn><mo stretchy="true" form="postfix">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msup><mo>−</mo><mn>1</mn><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">c_\textrm{twa} = c_0 \frac{\beta}{t (1 - \alpha)}
+                   \left( \left(\frac{t}{\beta} + 1 \right)^{1 - \alpha} - 1 \right) </annotation></semantics></math></p>
+<p>DFOP:</p>
+<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><msub><mi>c</mi><mn>0</mn></msub><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mi>g</mi><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>g</mi></mrow><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">c_\textrm{twa} = \frac{c_0}{t} \left(
+  \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) +
+  \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right)  \right) </annotation></semantics></math></p>
+<p>HS for
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>&gt;</mo><msub><mi>t</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">t &gt; t_b</annotation></semantics></math>:</p>
+<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><msub><mi>c</mi><mn>0</mn></msub><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mn>1</mn><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mrow><mo stretchy="true" form="prefix">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mi>b</mi></msub><mo stretchy="true" form="postfix">)</mo></mrow></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">c_\textrm{twa} = \frac{c_0}{t} \left(
+  \frac{1}{k_1} \left( 1 - e^{- k_1 t_b} \right) +
+  \frac{e^{- k_1 t_b}}{k_2} \left( 1 - e^{- k_2 (t - t_b)} \right)  \right) </annotation></semantics></math></p>
+<p>Often, the ratio between the time weighted average concentration
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><annotation encoding="application/x-tex">c_\textrm{twa}</annotation></semantics></math>
+and the initial concentration
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>c</mi><mn>0</mn></msub><annotation encoding="application/x-tex">c_0</annotation></semantics></math></p>
+<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><msub><mi>c</mi><mn>0</mn></msub></mfrac></mrow><annotation encoding="application/x-tex">f_\textrm{twa} = \frac{c_\textrm{twa}}{c_0}</annotation></semantics></math></p>
+<p>is needed. This can be calculated from the fitted initial
+concentration
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>c</mi><mn>0</mn></msub><annotation encoding="application/x-tex">c_0</annotation></semantics></math>
+and the time weighted average concentration
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><annotation encoding="application/x-tex">c_\textrm{twa}</annotation></semantics></math>,
+or directly from the model parameters using the following formulas:</p>
+<p>SFO:</p>
+<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>k</mi><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mrow><mi>k</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">f_\textrm{twa} = \frac{\left( 1 - e^{- k t} \right)}{k t} </annotation></semantics></math></p>
+<p>FOMC:</p>
+<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><mi>β</mi><mrow><mi>t</mi><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><msup><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mi>t</mi><mi>β</mi></mfrac><mo>+</mo><mn>1</mn><mo stretchy="true" form="postfix">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msup><mo>−</mo><mn>1</mn><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_\textrm{twa} = \frac{\beta}{t (1 - \alpha)}
+                   \left( \left(\frac{t}{\beta} + 1 \right)^{1 - \alpha} - 1 \right) </annotation></semantics></math></p>
+<p>DFOP:</p>
+<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mi>g</mi><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>g</mi></mrow><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_\textrm{twa} = \frac{1}{t} \left(
+  \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) +
+  \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right)  \right) </annotation></semantics></math></p>
+<p>HS for
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>&gt;</mo><msub><mi>t</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">t &gt; t_b</annotation></semantics></math>:</p>
+<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mn>1</mn><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mrow><mo stretchy="true" form="prefix">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mi>b</mi></msub><mo stretchy="true" form="postfix">)</mo></mrow></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_\textrm{twa} = \frac{1}{t} \left(
+  \frac{1}{k_1} \left( 1 - e^{- k_1 t_b} \right) +
+  \frac{e^{- k_1 t_b}}{k_2} \left( 1 - e^{- k_2 (t - t_b)} \right)  \right) </annotation></semantics></math></p>
+<p>Note that a method for calculating maximum moving window time
+weighted average concentrations for a model fitted by ‘mkinfit’ or from
+parent decline model parameters is included in the
+<code><a href="../reference/max_twa_parent.html">max_twa_parent()</a></code> function. If the same is needed for
+metabolites, the function <code><a href="https://pkgdown.jrwb.de/pfm/reference/max_twa.html" class="external-link">pfm::max_twa()</a></code> from the ‘pfm’
+package can be used.</p>
+<div id="refs" class="references csl-bib-body hanging-indent">
+<div id="ref-FOCUSkinetics2014" class="csl-entry">
+FOCUS Work Group on Degradation Kinetics. 2014. <em>Generic Guidance for
+Estimating Persistence and Degradation Kinetics from Environmental Fate
+Studies on Pesticides in EU Registration</em>. 1.1 ed. <a href="http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics" class="external-link">http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics</a>.
+</div>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
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+  <p>Developed by Johannes Ranke.</p>
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new file mode 100644
index 00000000..40ce9e01
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+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
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+
+      <h1>Example evaluation of FOCUS dataset Z</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change 16 January 2018
+(rebuilt 2025-02-14)</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/web_only/FOCUS_Z.rmd" class="external-link"><code>vignettes/web_only/FOCUS_Z.rmd</code></a></small>
+      <div class="d-none name"><code>FOCUS_Z.rmd</code></div>
+    </div>
+
+    
+    
+<p><a href="http://www.jrwb.de" class="external-link">Wissenschaftlicher Berater, Kronacher
+Str. 12, 79639 Grenzach-Wyhlen, Germany</a><br><a href="http://chem.uft.uni-bremen.de/ranke" class="external-link">Privatdozent at the
+University of Bremen</a></p>
+<div class="section level2">
+<h2 id="the-data">The data<a class="anchor" aria-label="anchor" href="#the-data"></a>
+</h2>
+<p>The following code defines the example dataset from Appendix 7 to the
+FOCUS kinetics report <span class="citation">(FOCUS Work Group on
+Degradation Kinetics 2014, 354)</span>.</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://pkgdown.jrwb.de/mkin/">mkin</a></span>, quietly <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="va">LOD</span> <span class="op">=</span> <span class="fl">0.5</span></span>
+<span><span class="va">FOCUS_2006_Z</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html" class="external-link">data.frame</a></span><span class="op">(</span></span>
+<span>  t <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">0.04</span>, <span class="fl">0.125</span>, <span class="fl">0.29</span>, <span class="fl">0.54</span>, <span class="fl">1</span>, <span class="fl">2</span>, <span class="fl">3</span>, <span class="fl">4</span>, <span class="fl">7</span>, <span class="fl">10</span>, <span class="fl">14</span>, <span class="fl">21</span>,</span>
+<span>        <span class="fl">42</span>, <span class="fl">61</span>, <span class="fl">96</span>, <span class="fl">124</span><span class="op">)</span>,</span>
+<span>  Z0 <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">100</span>, <span class="fl">81.7</span>, <span class="fl">70.4</span>, <span class="fl">51.1</span>, <span class="fl">41.2</span>, <span class="fl">6.6</span>, <span class="fl">4.6</span>, <span class="fl">3.9</span>, <span class="fl">4.6</span>, <span class="fl">4.3</span>, <span class="fl">6.8</span>,</span>
+<span>         <span class="fl">2.9</span>, <span class="fl">3.5</span>, <span class="fl">5.3</span>, <span class="fl">4.4</span>, <span class="fl">1.2</span>, <span class="fl">0.7</span><span class="op">)</span>,</span>
+<span>  Z1 <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">18.3</span>, <span class="fl">29.6</span>, <span class="fl">46.3</span>, <span class="fl">55.1</span>, <span class="fl">65.7</span>, <span class="fl">39.1</span>, <span class="fl">36</span>, <span class="fl">15.3</span>, <span class="fl">5.6</span>, <span class="fl">1.1</span>,</span>
+<span>         <span class="fl">1.6</span>, <span class="fl">0.6</span>, <span class="fl">0.5</span> <span class="op">*</span> <span class="va">LOD</span>, <span class="cn">NA</span>, <span class="cn">NA</span>, <span class="cn">NA</span><span class="op">)</span>,</span>
+<span>  Z2 <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="cn">NA</span>, <span class="fl">0.5</span> <span class="op">*</span> <span class="va">LOD</span>, <span class="fl">2.6</span>, <span class="fl">3.8</span>, <span class="fl">15.3</span>, <span class="fl">37.2</span>, <span class="fl">31.7</span>, <span class="fl">35.6</span>, <span class="fl">14.5</span>,</span>
+<span>         <span class="fl">0.8</span>, <span class="fl">2.1</span>, <span class="fl">1.9</span>, <span class="fl">0.5</span> <span class="op">*</span> <span class="va">LOD</span>, <span class="cn">NA</span>, <span class="cn">NA</span>, <span class="cn">NA</span><span class="op">)</span>,</span>
+<span>  Z3 <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="cn">NA</span>, <span class="cn">NA</span>, <span class="cn">NA</span>, <span class="cn">NA</span>, <span class="fl">0.5</span> <span class="op">*</span> <span class="va">LOD</span>, <span class="fl">9.2</span>, <span class="fl">13.1</span>, <span class="fl">22.3</span>, <span class="fl">28.4</span>, <span class="fl">32.5</span>,</span>
+<span>         <span class="fl">25.2</span>, <span class="fl">17.2</span>, <span class="fl">4.8</span>, <span class="fl">4.5</span>, <span class="fl">2.8</span>, <span class="fl">4.4</span><span class="op">)</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">FOCUS_2006_Z_mkin</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkin_wide_to_long.html">mkin_wide_to_long</a></span><span class="op">(</span><span class="va">FOCUS_2006_Z</span><span class="op">)</span></span></code></pre></div>
+</div>
+<div class="section level2">
+<h2 id="parent-and-one-metabolite">Parent and one metabolite<a class="anchor" aria-label="anchor" href="#parent-and-one-metabolite"></a>
+</h2>
+<p>The next step is to set up the models used for the kinetic analysis.
+As the simultaneous fit of parent and the first metabolite is usually
+straightforward, Step 1 (SFO for parent only) is skipped here. We start
+with the model 2a, with formation and decline of metabolite Z1 and the
+pathway from parent directly to sink included (default in mkin).</p>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">Z.2a</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>Z0 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z1"</span><span class="op">)</span>,</span>
+<span>                Z1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.Z.2a</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">Z.2a</span>, <span class="va">FOCUS_2006_Z_mkin</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit(Z.2a, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with</span></span>
+<span><span class="co">## value of zero were removed from the data</span></span></code></pre>
+<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">m.Z.2a</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_1-1.png" width="700"></p>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">m.Z.2a</span>, data <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span><span class="op">$</span><span class="va">bpar</span></span></code></pre></div>
+<pre><code><span><span class="co">##            Estimate se_notrans t value     Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## Z0_0       97.01488   3.301084 29.3888 3.2971e-21 91.66556 102.3642</span></span>
+<span><span class="co">## k_Z0        2.23601   0.207078 10.7979 3.3309e-11  1.95303   2.5600</span></span>
+<span><span class="co">## k_Z1        0.48212   0.063265  7.6207 2.8154e-08  0.40341   0.5762</span></span>
+<span><span class="co">## f_Z0_to_Z1  1.00000   0.094764 10.5525 5.3560e-11  0.00000   1.0000</span></span>
+<span><span class="co">## sigma       4.80411   0.635638  7.5579 3.2592e-08  3.52677   6.0815</span></span></code></pre>
+<p>As obvious from the parameter summary (the component of the summary),
+the kinetic rate constant from parent compound Z to sink is very small
+and the t-test for this parameter suggests that it is not significantly
+different from zero. This suggests, in agreement with the analysis in
+the FOCUS kinetics report, to simplify the model by removing the pathway
+to sink.</p>
+<p>A similar result can be obtained when formation fractions are used in
+the model formulation:</p>
+<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">Z.2a.ff</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>Z0 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z1"</span><span class="op">)</span>,</span>
+<span>                   Z1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>                   use_of_ff <span class="op">=</span> <span class="st">"max"</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<div class="sourceCode" id="cb11"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.Z.2a.ff</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">Z.2a.ff</span>, <span class="va">FOCUS_2006_Z_mkin</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with</span></span>
+<span><span class="co">## value of zero were removed from the data</span></span></code></pre>
+<div class="sourceCode" id="cb13"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">m.Z.2a.ff</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_2-1.png" width="700"></p>
+<div class="sourceCode" id="cb14"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">m.Z.2a.ff</span>, data <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span><span class="op">$</span><span class="va">bpar</span></span></code></pre></div>
+<pre><code><span><span class="co">##            Estimate se_notrans t value     Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## Z0_0       97.01488   3.301084 29.3888 3.2971e-21 91.66556 102.3642</span></span>
+<span><span class="co">## k_Z0        2.23601   0.207078 10.7979 3.3309e-11  1.95303   2.5600</span></span>
+<span><span class="co">## k_Z1        0.48212   0.063265  7.6207 2.8154e-08  0.40341   0.5762</span></span>
+<span><span class="co">## f_Z0_to_Z1  1.00000   0.094764 10.5525 5.3560e-11  0.00000   1.0000</span></span>
+<span><span class="co">## sigma       4.80411   0.635638  7.5579 3.2592e-08  3.52677   6.0815</span></span></code></pre>
+<p>Here, the ilr transformed formation fraction fitted in the model
+takes a very large value, and the backtransformed formation fraction
+from parent Z to Z1 is practically unity. Here, the covariance matrix
+used for the calculation of confidence intervals is not returned as the
+model is overparameterised.</p>
+<p>A simplified model is obtained by removing the pathway to the sink.
+</p>
+<p>In the following, we use the parameterisation with formation
+fractions in order to be able to compare with the results in the FOCUS
+guidance, and as it makes it easier to use parameters obtained in a
+previous fit when adding a further metabolite.</p>
+<div class="sourceCode" id="cb16"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">Z.3</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>Z0 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z1"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>               Z1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>, use_of_ff <span class="op">=</span> <span class="st">"max"</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<div class="sourceCode" id="cb18"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.Z.3</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">Z.3</span>, <span class="va">FOCUS_2006_Z_mkin</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit(Z.3, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with</span></span>
+<span><span class="co">## value of zero were removed from the data</span></span></code></pre>
+<div class="sourceCode" id="cb20"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">m.Z.3</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_3-1.png" width="700"></p>
+<div class="sourceCode" id="cb21"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">m.Z.3</span>, data <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span><span class="op">$</span><span class="va">bpar</span></span></code></pre></div>
+<pre><code><span><span class="co">##       Estimate se_notrans t value     Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## Z0_0  97.01488   2.597342  37.352 2.0106e-24 91.67597 102.3538</span></span>
+<span><span class="co">## k_Z0   2.23601   0.146904  15.221 9.1477e-15  1.95354   2.5593</span></span>
+<span><span class="co">## k_Z1   0.48212   0.041727  11.554 4.8268e-12  0.40355   0.5760</span></span>
+<span><span class="co">## sigma  4.80411   0.620208   7.746 1.6110e-08  3.52925   6.0790</span></span></code></pre>
+<p>As there is only one transformation product for Z0 and no pathway to
+sink, the formation fraction is internally fixed to unity.</p>
+</div>
+<div class="section level2">
+<h2 id="metabolites-z2-and-z3">Metabolites Z2 and Z3<a class="anchor" aria-label="anchor" href="#metabolites-z2-and-z3"></a>
+</h2>
+<p>As suggested in the FOCUS report, the pathway to sink was removed for
+metabolite Z1 as well in the next step. While this step appears
+questionable on the basis of the above results, it is followed here for
+the purpose of comparison. Also, in the FOCUS report, it is assumed that
+there is additional empirical evidence that Z1 quickly and exclusively
+hydrolyses to Z2.</p>
+<div class="sourceCode" id="cb23"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">Z.5</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>Z0 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z1"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>               Z1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z2"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>               Z2 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>, use_of_ff <span class="op">=</span> <span class="st">"max"</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<div class="sourceCode" id="cb25"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.Z.5</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">Z.5</span>, <span class="va">FOCUS_2006_Z_mkin</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit(Z.5, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with</span></span>
+<span><span class="co">## value of zero were removed from the data</span></span></code></pre>
+<div class="sourceCode" id="cb27"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">m.Z.5</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_5-1.png" width="700"></p>
+<p>Finally, metabolite Z3 is added to the model. We use the optimised
+differential equation parameter values from the previous fit in order to
+accelerate the optimization.</p>
+<div class="sourceCode" id="cb28"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">Z.FOCUS</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>Z0 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z1"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>                   Z1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z2"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>                   Z2 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z3"</span><span class="op">)</span>,</span>
+<span>                   Z3 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>                   use_of_ff <span class="op">=</span> <span class="st">"max"</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<div class="sourceCode" id="cb30"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.Z.FOCUS</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">Z.FOCUS</span>, <span class="va">FOCUS_2006_Z_mkin</span>,</span>
+<span>                     parms.ini <span class="op">=</span> <span class="va">m.Z.5</span><span class="op">$</span><span class="va">bparms.ode</span>,</span>
+<span>                     quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit(Z.FOCUS, FOCUS_2006_Z_mkin, parms.ini = m.Z.5$bparms.ode, :</span></span>
+<span><span class="co">## Observations with value of zero were removed from the data</span></span></code></pre>
+<pre><code><span><span class="co">## Warning in mkinfit(Z.FOCUS, FOCUS_2006_Z_mkin, parms.ini = m.Z.5$bparms.ode, : Optimisation did not converge:</span></span>
+<span><span class="co">## false convergence (8)</span></span></code></pre>
+<div class="sourceCode" id="cb33"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">m.Z.FOCUS</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_6-1.png" width="700"></p>
+<div class="sourceCode" id="cb34"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">m.Z.FOCUS</span>, data <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span><span class="op">$</span><span class="va">bpar</span></span></code></pre></div>
+<pre><code><span><span class="co">##             Estimate se_notrans t value     Pr(&gt;t)     Lower      Upper</span></span>
+<span><span class="co">## Z0_0       96.842440   1.994291 48.5598 4.0226e-42 92.830421 100.854459</span></span>
+<span><span class="co">## k_Z0        2.215425   0.118457 18.7023 1.0404e-23  1.989490   2.467019</span></span>
+<span><span class="co">## k_Z1        0.478307   0.028257 16.9272 6.2332e-22  0.424709   0.538669</span></span>
+<span><span class="co">## k_Z2        0.451642   0.042139 10.7178 1.6304e-14  0.374348   0.544894</span></span>
+<span><span class="co">## k_Z3        0.058692   0.015245  3.8499 1.7803e-04  0.034804   0.098975</span></span>
+<span><span class="co">## f_Z2_to_Z3  0.471483   0.058348  8.0806 9.6585e-11  0.357720   0.588287</span></span>
+<span><span class="co">## sigma       3.984431   0.383402 10.3923 4.5576e-14  3.213126   4.755737</span></span></code></pre>
+<div class="sourceCode" id="cb36"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">m.Z.FOCUS</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## $ff</span></span>
+<span><span class="co">##   Z2_Z3 Z2_sink </span></span>
+<span><span class="co">## 0.47148 0.52852 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $distimes</span></span>
+<span><span class="co">##        DT50    DT90</span></span>
+<span><span class="co">## Z0  0.31287  1.0393</span></span>
+<span><span class="co">## Z1  1.44917  4.8140</span></span>
+<span><span class="co">## Z2  1.53473  5.0983</span></span>
+<span><span class="co">## Z3 11.80991 39.2317</span></span></code></pre>
+<p>This fit corresponds to the final result chosen in Appendix 7 of the
+FOCUS report. Confidence intervals returned by mkin are based on
+internally transformed parameters, however.</p>
+</div>
+<div class="section level2">
+<h2 id="using-the-sforb-model">Using the SFORB model<a class="anchor" aria-label="anchor" href="#using-the-sforb-model"></a>
+</h2>
+<p>As the FOCUS report states, there is a certain tailing of the time
+course of metabolite Z3. Also, the time course of the parent compound is
+not fitted very well using the SFO model, as residues at a certain low
+level remain.</p>
+<p>Therefore, an additional model is offered here, using the single
+first-order reversible binding (SFORB) model for metabolite Z3. As
+expected, the
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level is lower for metabolite Z3 using this model and the
+graphical fit for Z3 is improved. However, the covariance matrix is not
+returned.</p>
+<div class="sourceCode" id="cb38"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">Z.mkin.1</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>Z0 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z1"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>                    Z1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z2"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>                    Z2 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z3"</span><span class="op">)</span>,</span>
+<span>                    Z3 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFORB"</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<div class="sourceCode" id="cb40"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.Z.mkin.1</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">Z.mkin.1</span>, <span class="va">FOCUS_2006_Z_mkin</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, quiet = TRUE): Observations</span></span>
+<span><span class="co">## with value of zero were removed from the data</span></span></code></pre>
+<div class="sourceCode" id="cb42"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">m.Z.mkin.1</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_7-1.png" width="700"></p>
+<div class="sourceCode" id="cb43"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">m.Z.mkin.1</span>, data <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span><span class="op">$</span><span class="va">cov.unscaled</span></span></code></pre></div>
+<pre><code><span><span class="co">## NULL</span></span></code></pre>
+<p>Therefore, a further stepwise model building is performed starting
+from the stage of parent and two metabolites, starting from the
+assumption that the model fit for the parent compound can be improved by
+using the SFORB model.</p>
+<div class="sourceCode" id="cb45"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">Z.mkin.3</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>Z0 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFORB"</span>, <span class="st">"Z1"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>                    Z1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z2"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>                    Z2 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<div class="sourceCode" id="cb47"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.Z.mkin.3</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">Z.mkin.3</span>, <span class="va">FOCUS_2006_Z_mkin</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE): Observations</span></span>
+<span><span class="co">## with value of zero were removed from the data</span></span></code></pre>
+<div class="sourceCode" id="cb49"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">m.Z.mkin.3</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_9-1.png" width="700"></p>
+<p>This results in a much better representation of the behaviour of the
+parent compound Z0.</p>
+<p>Finally, Z3 is added as well. These models appear overparameterised
+(no covariance matrix returned) if the sink for Z1 is left in the
+models.</p>
+<div class="sourceCode" id="cb50"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">Z.mkin.4</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>Z0 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFORB"</span>, <span class="st">"Z1"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>                    Z1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z2"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>                    Z2 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z3"</span><span class="op">)</span>,</span>
+<span>                    Z3 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<div class="sourceCode" id="cb52"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.Z.mkin.4</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">Z.mkin.4</span>, <span class="va">FOCUS_2006_Z_mkin</span>,</span>
+<span>                      parms.ini <span class="op">=</span> <span class="va">m.Z.mkin.3</span><span class="op">$</span><span class="va">bparms.ode</span>,</span>
+<span>                      quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit(Z.mkin.4, FOCUS_2006_Z_mkin, parms.ini =</span></span>
+<span><span class="co">## m.Z.mkin.3$bparms.ode, : Observations with value of zero were removed from the</span></span>
+<span><span class="co">## data</span></span></code></pre>
+<div class="sourceCode" id="cb54"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">m.Z.mkin.4</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_10-1.png" width="700"></p>
+<p>The error level of the fit, but especially of metabolite Z3, can be
+improved if the SFORB model is chosen for this metabolite, as this model
+is capable of representing the tailing of the metabolite decline
+phase.</p>
+<div class="sourceCode" id="cb55"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">Z.mkin.5</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>Z0 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFORB"</span>, <span class="st">"Z1"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>                    Z1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z2"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span>,</span>
+<span>                    Z2 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"Z3"</span><span class="op">)</span>,</span>
+<span>                    Z3 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFORB"</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<div class="sourceCode" id="cb57"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.Z.mkin.5</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">Z.mkin.5</span>, <span class="va">FOCUS_2006_Z_mkin</span>,</span>
+<span>                      parms.ini <span class="op">=</span> <span class="va">m.Z.mkin.4</span><span class="op">$</span><span class="va">bparms.ode</span><span class="op">[</span><span class="fl">1</span><span class="op">:</span><span class="fl">4</span><span class="op">]</span>,</span>
+<span>                      quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin, parms.ini =</span></span>
+<span><span class="co">## m.Z.mkin.4$bparms.ode[1:4], : Observations with value of zero were removed from</span></span>
+<span><span class="co">## the data</span></span></code></pre>
+<div class="sourceCode" id="cb59"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">m.Z.mkin.5</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_11-1.png" width="700"></p>
+<p>The summary view of the backtransformed parameters shows that we get
+no confidence intervals due to overparameterisation. As the optimized is
+excessively small, it seems reasonable to fix it to zero.</p>
+<div class="sourceCode" id="cb60"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m.Z.mkin.5a</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">Z.mkin.5</span>, <span class="va">FOCUS_2006_Z_mkin</span>,</span>
+<span>                       parms.ini <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="va">m.Z.mkin.5</span><span class="op">$</span><span class="va">bparms.ode</span><span class="op">[</span><span class="fl">1</span><span class="op">:</span><span class="fl">7</span><span class="op">]</span>,</span>
+<span>                                     k_Z3_bound_free <span class="op">=</span> <span class="fl">0</span><span class="op">)</span>,</span>
+<span>                       fixed_parms <span class="op">=</span> <span class="st">"k_Z3_bound_free"</span>,</span>
+<span>                       quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin, parms.ini =</span></span>
+<span><span class="co">## c(m.Z.mkin.5$bparms.ode[1:7], : Observations with value of zero were removed</span></span>
+<span><span class="co">## from the data</span></span></code></pre>
+<div class="sourceCode" id="cb62"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">m.Z.mkin.5a</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_11a-1.png" width="700"></p>
+<p>As expected, the residual plots for Z0 and Z3 are more random than in
+the case of the all SFO model for which they were shown above. In
+conclusion, the model is proposed as the best-fit model for the dataset
+from Appendix 7 of the FOCUS report.</p>
+<p>A graphical representation of the confidence intervals can finally be
+obtained.</p>
+<div class="sourceCode" id="cb63"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/mkinparplot.html">mkinparplot</a></span><span class="op">(</span><span class="va">m.Z.mkin.5a</span><span class="op">)</span></span></code></pre></div>
+<p><img src="FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_11b-1.png" width="700"></p>
+<p>The endpoints obtained with this model are</p>
+<div class="sourceCode" id="cb64"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">m.Z.mkin.5a</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## $ff</span></span>
+<span><span class="co">## Z0_free   Z2_Z3 Z2_sink Z3_free </span></span>
+<span><span class="co">## 1.00000 0.53656 0.46344 1.00000 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $SFORB</span></span>
+<span><span class="co">##     Z0_b1     Z0_b2      Z0_g     Z3_b1     Z3_b2      Z3_g </span></span>
+<span><span class="co">## 2.4471342 0.0075124 0.9519866 0.0800071 0.0000000 0.9347816 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $distimes</span></span>
+<span><span class="co">##      DT50   DT90 DT50back DT50_Z0_b1 DT50_Z0_b2 DT50_Z3_b1 DT50_Z3_b2</span></span>
+<span><span class="co">## Z0 0.3043 1.1848  0.35666    0.28325     92.267         NA         NA</span></span>
+<span><span class="co">## Z1 1.5148 5.0320       NA         NA         NA         NA         NA</span></span>
+<span><span class="co">## Z2 1.6414 5.4526       NA         NA         NA         NA         NA</span></span>
+<span><span class="co">## Z3     NA     NA       NA         NA         NA     8.6636        Inf</span></span></code></pre>
+<p>It is clear the degradation rate of Z3 towards the end of the
+experiment is very low as DT50_Z3_b2 (the second Eigenvalue of the
+system of two differential equations representing the SFORB system for
+Z3, corresponding to the slower rate constant of the DFOP model) is
+reported to be infinity. However, this appears to be a feature of the
+data.</p>
+</div>
+<div class="section level2">
+<h2 id="references">References<a class="anchor" aria-label="anchor" href="#references"></a>
+</h2>
+<!-- vim: set foldmethod=syntax: -->
+<div id="refs" class="references csl-bib-body hanging-indent">
+<div id="ref-FOCUSkinetics2014" class="csl-entry">
+FOCUS Work Group on Degradation Kinetics. 2014. <em>Generic Guidance for
+Estimating Persistence and Degradation Kinetics from Environmental Fate
+Studies on Pesticides in EU Registration</em>. 1.1 ed. <a href="http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics" class="external-link">http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics</a>.
+</div>
+</div>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
+</div>
+
+    </footer>
+</div>
+
+
+
+
+
+  </body>
+</html>
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+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with (generalised) nonlinear least squares</h6></li>
+    <li><a class="dropdown-item" href="../../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a></li>
+    <li><a class="dropdown-item" href="../../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/FOCUS_Z.html">Example evaluation of FOCUS Example Dataset Z</a></li>
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+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with hierarchical models (nonlinear mixed-effects models)</h6></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_pathway.html">Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2023_mesotrione_parent.html">Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_cyan_pathway.html">Testing hierarchical pathway kinetics with residue data on cyantraniliprole</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/dimethenamid_2018.html">Comparison of saemix and nlme evaluations of dimethenamid data from 2018</a></li>
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+
+      <h1>Evaluation of example datasets from Attachment 1 to the US EPA SOP for the NAFTA guidance</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">26 February 2019 (rebuilt
+2025-02-14)</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/web_only/NAFTA_examples.rmd" class="external-link"><code>vignettes/web_only/NAFTA_examples.rmd</code></a></small>
+      <div class="d-none name"><code>NAFTA_examples.rmd</code></div>
+    </div>
+
+    
+    
+<div class="section level2">
+<h2 id="introduction">Introduction<a class="anchor" aria-label="anchor" href="#introduction"></a>
+</h2>
+<p>In this document, the example evaluations provided in Attachment 1 to
+the SOP of US EPA for using the NAFTA guidance <span class="citation">(US EPA 2015)</span> are repeated using mkin. The
+original evaluations reported in the attachment were performed using
+PestDF in version 0.8.4. Note that PestDF 0.8.13 is the version
+distributed at the US EPA website today (2019-02-26).</p>
+<p>The datasets are now distributed with the mkin package.</p>
+</div>
+<div class="section level2">
+<h2 id="examples-where-dfop-did-not-converge-with-pestdf-0-8-4">Examples where DFOP did not converge with PestDF 0.8.4<a class="anchor" aria-label="anchor" href="#examples-where-dfop-did-not-converge-with-pestdf-0-8-4"></a>
+</h2>
+<p>In attachment 1, it is reported that the DFOP model does not converge
+for these datasets when PestDF 0.8.4 was used. For all four datasets,
+the DFOP model can be fitted with mkin (see below). The negative
+half-life given by PestDF 0.8.4 for these fits appears to be the result
+of a bug. The results for the other two models (SFO and IORE) are the
+same.</p>
+<div class="section level3">
+<h3 id="example-on-page-5-upper-panel">Example on page 5, upper panel<a class="anchor" aria-label="anchor" href="#example-on-page-5-upper-panel"></a>
+</h3>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p5a</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p5a"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p5a</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p5a-1.png" width="700"></p>
+<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p5a</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##       SFO      IORE      DFOP </span></span>
+<span><span class="co">## 465.21753  56.27506  32.06401 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 64.4304</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)  Lower   Upper</span></span>
+<span><span class="co">## parent_0  95.8401 4.67e-21 92.245 99.4357</span></span>
+<span><span class="co">## k_parent   0.0102 3.92e-12  0.009  0.0117</span></span>
+<span><span class="co">## sigma      4.8230 3.81e-06  3.214  6.4318</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0       1.01e+02     NA 9.91e+01 1.02e+02</span></span>
+<span><span class="co">## k__iore_parent 1.54e-05     NA 4.08e-06 5.84e-05</span></span>
+<span><span class="co">## N_parent       2.57e+00     NA 2.25e+00 2.89e+00</span></span>
+<span><span class="co">## sigma          1.68e+00     NA 1.12e+00 2.24e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)   Lower    Upper</span></span>
+<span><span class="co">## parent_0 9.99e+01 1.41e-26 98.8116 101.0810</span></span>
+<span><span class="co">## k1       2.67e-02 5.05e-06  0.0243   0.0295</span></span>
+<span><span class="co">## k2       3.41e-12 5.00e-01  0.0000      Inf</span></span>
+<span><span class="co">## g        6.47e-01 3.67e-06  0.6248   0.6677</span></span>
+<span><span class="co">## sigma    1.27e+00 8.91e-06  0.8395   1.6929</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50     DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  67.7 2.25e+02 6.77e+01</span></span>
+<span><span class="co">## IORE 58.2 1.07e+03 3.22e+02</span></span>
+<span><span class="co">## DFOP 55.5 3.70e+11 2.03e+11</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 321.51</span></span></code></pre>
+</div>
+<div class="section level3">
+<h3 id="example-on-page-5-lower-panel">Example on page 5, lower panel<a class="anchor" aria-label="anchor" href="#example-on-page-5-lower-panel"></a>
+</h3>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p5b</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p5b"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p5b</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p5b-1.png" width="700"></p>
+<div class="sourceCode" id="cb11"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p5b</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##      SFO     IORE     DFOP </span></span>
+<span><span class="co">## 94.81123 10.10936  7.55871 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 11.77879</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0   96.497 2.32e-24 94.85271 98.14155</span></span>
+<span><span class="co">## k_parent    0.008 3.42e-14  0.00737  0.00869</span></span>
+<span><span class="co">## sigma       2.295 1.22e-05  1.47976  3.11036</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0       9.85e+01 1.17e-28 9.79e+01 9.92e+01</span></span>
+<span><span class="co">## k__iore_parent 1.53e-04 6.50e-03 7.21e-05 3.26e-04</span></span>
+<span><span class="co">## N_parent       1.94e+00 5.88e-13 1.76e+00 2.12e+00</span></span>
+<span><span class="co">## sigma          7.49e-01 1.63e-05 4.82e-01 1.02e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)   Lower   Upper</span></span>
+<span><span class="co">## parent_0 9.84e+01 1.24e-27 97.8078 98.9187</span></span>
+<span><span class="co">## k1       1.55e-02 4.10e-04  0.0143  0.0167</span></span>
+<span><span class="co">## k2       9.07e-12 5.00e-01  0.0000     Inf</span></span>
+<span><span class="co">## g        6.89e-01 2.92e-03  0.6626  0.7142</span></span>
+<span><span class="co">## sigma    6.48e-01 2.38e-05  0.4147  0.8813</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50     DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  86.6 2.88e+02 8.66e+01</span></span>
+<span><span class="co">## IORE 85.5 7.17e+02 2.16e+02</span></span>
+<span><span class="co">## DFOP 83.6 1.25e+11 7.64e+10</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 215.87</span></span></code></pre>
+</div>
+<div class="section level3">
+<h3 id="example-on-page-6">Example on page 6<a class="anchor" aria-label="anchor" href="#example-on-page-6"></a>
+</h3>
+<div class="sourceCode" id="cb13"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p6</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p6"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb16"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p6</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p6-1.png" width="700"></p>
+<div class="sourceCode" id="cb17"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p6</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##       SFO      IORE      DFOP </span></span>
+<span><span class="co">## 188.45361  51.00699  42.46931 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 58.39888</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)   Lower   Upper</span></span>
+<span><span class="co">## parent_0  94.7759 7.29e-24 92.3478 97.2039</span></span>
+<span><span class="co">## k_parent   0.0179 8.02e-16  0.0166  0.0194</span></span>
+<span><span class="co">## sigma      3.0696 3.81e-06  2.0456  4.0936</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0       97.12446 2.63e-26 95.62461 98.62431</span></span>
+<span><span class="co">## k__iore_parent  0.00252 1.95e-03  0.00134  0.00472</span></span>
+<span><span class="co">## N_parent        1.49587 4.07e-13  1.33896  1.65279</span></span>
+<span><span class="co">## sigma           1.59698 5.05e-06  1.06169  2.13227</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)   Lower   Upper</span></span>
+<span><span class="co">## parent_0 9.66e+01 1.57e-25 95.3476 97.8979</span></span>
+<span><span class="co">## k1       2.55e-02 7.33e-06  0.0233  0.0278</span></span>
+<span><span class="co">## k2       3.84e-11 5.00e-01  0.0000     Inf</span></span>
+<span><span class="co">## g        8.61e-01 7.55e-06  0.8314  0.8867</span></span>
+<span><span class="co">## sigma    1.46e+00 6.93e-06  0.9661  1.9483</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50     DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  38.6 1.28e+02 3.86e+01</span></span>
+<span><span class="co">## IORE 34.0 1.77e+02 5.32e+01</span></span>
+<span><span class="co">## DFOP 34.1 8.50e+09 1.80e+10</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 53.17</span></span></code></pre>
+</div>
+<div class="section level3">
+<h3 id="example-on-page-7">Example on page 7<a class="anchor" aria-label="anchor" href="#example-on-page-7"></a>
+</h3>
+<div class="sourceCode" id="cb19"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p7</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p7"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb22"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p7</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p7-1.png" width="700"></p>
+<div class="sourceCode" id="cb23"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p7</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##      SFO     IORE     DFOP </span></span>
+<span><span class="co">## 3661.661 3195.030 3174.145 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 3334.194</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0 96.41796 4.80e-53 93.32245 99.51347</span></span>
+<span><span class="co">## k_parent  0.00735 7.64e-21  0.00641  0.00843</span></span>
+<span><span class="co">## sigma     7.94557 1.83e-15  6.46713  9.42401</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0       9.92e+01     NA 9.55e+01 1.03e+02</span></span>
+<span><span class="co">## k__iore_parent 1.60e-05     NA 1.45e-07 1.77e-03</span></span>
+<span><span class="co">## N_parent       2.45e+00     NA 1.35e+00 3.54e+00</span></span>
+<span><span class="co">## sigma          7.42e+00     NA 6.04e+00 8.80e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)   Lower    Upper</span></span>
+<span><span class="co">## parent_0 9.89e+01 9.44e-49 95.4640 102.2573</span></span>
+<span><span class="co">## k1       1.81e-02 1.75e-01  0.0116   0.0281</span></span>
+<span><span class="co">## k2       3.62e-10 5.00e-01  0.0000      Inf</span></span>
+<span><span class="co">## g        6.06e-01 2.19e-01  0.4826   0.7178</span></span>
+<span><span class="co">## sigma    7.40e+00 2.97e-15  6.0201   8.7754</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50     DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  94.3 3.13e+02 9.43e+01</span></span>
+<span><span class="co">## IORE 96.7 1.51e+03 4.55e+02</span></span>
+<span><span class="co">## DFOP 96.4 3.79e+09 1.92e+09</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 454.55</span></span></code></pre>
+</div>
+</div>
+<div class="section level2">
+<h2 id="examples-where-the-representative-half-life-deviates-from-the-observed-dt50">Examples where the representative half-life deviates from the
+observed DT50<a class="anchor" aria-label="anchor" href="#examples-where-the-representative-half-life-deviates-from-the-observed-dt50"></a>
+</h2>
+<div class="section level3">
+<h3 id="example-on-page-8">Example on page 8<a class="anchor" aria-label="anchor" href="#example-on-page-8"></a>
+</h3>
+<p>For this dataset, the IORE fit does not converge when the default
+starting values used by mkin for the IORE model are used. Therefore, a
+lower value for the rate constant is used here.</p>
+<div class="sourceCode" id="cb25"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p8</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p8"</span><span class="op">]</span><span class="op">]</span>, parms.ini <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>k__iore_parent <span class="op">=</span> <span class="fl">1e-3</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb28"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p8</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p8-1.png" width="700"></p>
+<div class="sourceCode" id="cb29"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p8</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##       SFO      IORE      DFOP </span></span>
+<span><span class="co">## 1996.9408  444.9237  547.5616 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 477.4924</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0 88.16549 6.53e-29 83.37344 92.95754</span></span>
+<span><span class="co">## k_parent  0.00803 1.67e-13  0.00674  0.00957</span></span>
+<span><span class="co">## sigma     7.44786 4.17e-10  5.66209  9.23363</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0       9.77e+01 7.03e-35 9.44e+01 1.01e+02</span></span>
+<span><span class="co">## k__iore_parent 6.14e-05 3.20e-02 2.12e-05 1.78e-04</span></span>
+<span><span class="co">## N_parent       2.27e+00 4.23e-18 2.00e+00 2.54e+00</span></span>
+<span><span class="co">## sigma          3.52e+00 5.36e-10 2.67e+00 4.36e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0 95.70619 8.99e-32 91.87941 99.53298</span></span>
+<span><span class="co">## k1        0.02500 5.25e-04  0.01422  0.04394</span></span>
+<span><span class="co">## k2        0.00273 6.84e-03  0.00125  0.00597</span></span>
+<span><span class="co">## g         0.58835 2.84e-06  0.36595  0.77970</span></span>
+<span><span class="co">## sigma     3.90001 6.94e-10  2.96260  4.83741</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50 DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  86.3  287     86.3</span></span>
+<span><span class="co">## IORE 53.4  668    201.0</span></span>
+<span><span class="co">## DFOP 55.6  517    253.0</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 201.03</span></span></code></pre>
+</div>
+</div>
+<div class="section level2">
+<h2 id="examples-where-sfo-was-not-selected-for-an-abiotic-study">Examples where SFO was not selected for an abiotic study<a class="anchor" aria-label="anchor" href="#examples-where-sfo-was-not-selected-for-an-abiotic-study"></a>
+</h2>
+<div class="section level3">
+<h3 id="example-on-page-9-upper-panel">Example on page 9, upper panel<a class="anchor" aria-label="anchor" href="#example-on-page-9-upper-panel"></a>
+</h3>
+<div class="sourceCode" id="cb31"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p9a</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p9a"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb34"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p9a</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p9a-1.png" width="700"></p>
+<div class="sourceCode" id="cb35"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p9a</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##       SFO      IORE      DFOP </span></span>
+<span><span class="co">## 839.35238  88.57064   9.93363 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 105.5678</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)   Lower   Upper</span></span>
+<span><span class="co">## parent_0  88.1933 3.06e-12 79.9447 96.4419</span></span>
+<span><span class="co">## k_parent   0.0409 2.07e-07  0.0324  0.0516</span></span>
+<span><span class="co">## sigma      7.2429 3.92e-05  4.4768 10.0090</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0       9.89e+01 1.12e-16 9.54e+01 1.02e+02</span></span>
+<span><span class="co">## k__iore_parent 1.93e-05 1.13e-01 3.49e-06 1.06e-04</span></span>
+<span><span class="co">## N_parent       2.91e+00 1.45e-09 2.50e+00 3.32e+00</span></span>
+<span><span class="co">## sigma          2.35e+00 5.31e-05 1.45e+00 3.26e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)  Lower  Upper</span></span>
+<span><span class="co">## parent_0 9.85e+01 2.54e-20 97.390 99.672</span></span>
+<span><span class="co">## k1       1.38e-01 3.52e-05  0.131  0.146</span></span>
+<span><span class="co">## k2       9.02e-13 5.00e-01  0.000    Inf</span></span>
+<span><span class="co">## g        6.52e-01 8.13e-06  0.642  0.661</span></span>
+<span><span class="co">## sigma    7.88e-01 6.13e-02  0.481  1.095</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50     DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  16.9 5.63e+01 1.69e+01</span></span>
+<span><span class="co">## IORE 11.6 3.37e+02 1.01e+02</span></span>
+<span><span class="co">## DFOP 10.5 1.38e+12 7.68e+11</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 101.43</span></span></code></pre>
+<p>In this example, the residuals of the SFO indicate a lack of fit of
+this model, so even if it was an abiotic experiment, the data do not
+suggest a simple exponential decline.</p>
+</div>
+<div class="section level3">
+<h3 id="example-on-page-9-lower-panel">Example on page 9, lower panel<a class="anchor" aria-label="anchor" href="#example-on-page-9-lower-panel"></a>
+</h3>
+<div class="sourceCode" id="cb37"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p9b</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p9b"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb40"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p9b</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p9b-1.png" width="700"></p>
+<div class="sourceCode" id="cb41"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p9b</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##      SFO     IORE     DFOP </span></span>
+<span><span class="co">## 35.64867 23.22334 35.64867 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 28.54188</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)  Lower   Upper</span></span>
+<span><span class="co">## parent_0  94.7123 2.15e-19 93.178 96.2464</span></span>
+<span><span class="co">## k_parent   0.0389 4.47e-14  0.037  0.0408</span></span>
+<span><span class="co">## sigma      1.5957 1.28e-04  0.932  2.2595</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate   Pr(&gt;t)   Lower  Upper</span></span>
+<span><span class="co">## parent_0         93.863 2.32e-18 92.4565 95.269</span></span>
+<span><span class="co">## k__iore_parent    0.127 1.85e-02  0.0504  0.321</span></span>
+<span><span class="co">## N_parent          0.711 1.88e-05  0.4843  0.937</span></span>
+<span><span class="co">## sigma             1.288 1.76e-04  0.7456  1.830</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)   Lower   Upper</span></span>
+<span><span class="co">## parent_0  94.7123 1.61e-16 93.1355 96.2891</span></span>
+<span><span class="co">## k1         0.0389 1.08e-04  0.0266  0.0569</span></span>
+<span><span class="co">## k2         0.0389 2.24e-04  0.0255  0.0592</span></span>
+<span><span class="co">## g          0.5256 5.00e-01  0.0000  1.0000</span></span>
+<span><span class="co">## sigma      1.5957 2.50e-04  0.9135  2.2779</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50 DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  17.8 59.2     17.8</span></span>
+<span><span class="co">## IORE 18.4 49.2     14.8</span></span>
+<span><span class="co">## DFOP 17.8 59.2     17.8</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 14.8</span></span></code></pre>
+<p>Here, mkin gives a longer slow DT50 for the DFOP model (17.8 days)
+than PestDF (13.5 days). Presumably, this is related to the fact that
+PestDF gives a negative value for the proportion of the fast degradation
+which should be between 0 and 1, inclusive. This parameter is called f
+in PestDF and g in mkin. In mkin, it is restricted to the interval from
+0 to 1.</p>
+</div>
+<div class="section level3">
+<h3 id="example-on-page-10">Example on page 10<a class="anchor" aria-label="anchor" href="#example-on-page-10"></a>
+</h3>
+<div class="sourceCode" id="cb43"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p10</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p10"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in sqrt(diag(covar_notrans)): NaNs produced</span></span></code></pre>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb47"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p10</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p10-1.png" width="700"></p>
+<div class="sourceCode" id="cb48"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p10</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##      SFO     IORE     DFOP </span></span>
+<span><span class="co">## 899.4089 336.4348 899.4089 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 413.4841</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)   Lower    Upper</span></span>
+<span><span class="co">## parent_0 101.7315 6.42e-11 91.9259 111.5371</span></span>
+<span><span class="co">## k_parent   0.0495 1.70e-07  0.0404   0.0607</span></span>
+<span><span class="co">## sigma      8.0152 1.28e-04  4.6813  11.3491</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate   Pr(&gt;t)  Lower   Upper</span></span>
+<span><span class="co">## parent_0          96.86 3.32e-12 90.848 102.863</span></span>
+<span><span class="co">## k__iore_parent     2.96 7.91e-02  0.687  12.761</span></span>
+<span><span class="co">## N_parent           0.00 5.00e-01 -0.372   0.372</span></span>
+<span><span class="co">## sigma              4.90 1.77e-04  2.837   6.968</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)   Lower   Upper</span></span>
+<span><span class="co">## parent_0 101.7315 1.41e-09 91.6534 111.810</span></span>
+<span><span class="co">## k1         0.0495 3.04e-03  0.0188   0.131</span></span>
+<span><span class="co">## k2         0.0495 4.92e-04  0.0197   0.124</span></span>
+<span><span class="co">## g          0.4487      NaN  0.0000   1.000</span></span>
+<span><span class="co">## sigma      8.0152 2.50e-04  4.5886  11.442</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50 DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  14.0 46.5    14.00</span></span>
+<span><span class="co">## IORE 16.4 29.4     8.86</span></span>
+<span><span class="co">## DFOP 14.0 46.5    14.00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 8.86</span></span></code></pre>
+<p>Here, a value below N is given for the IORE model, because the data
+suggests a faster decline towards the end of the experiment, which
+appears physically rather unlikely in the case of a photolysis study. It
+seems PestDF does not constrain N to values above zero, thus the slight
+difference in IORE model parameters between PestDF and mkin.</p>
+</div>
+</div>
+<div class="section level2">
+<h2 id="the-dt50-was-not-observed-during-the-study">The DT50 was not observed during the study<a class="anchor" aria-label="anchor" href="#the-dt50-was-not-observed-during-the-study"></a>
+</h2>
+<div class="section level3">
+<h3 id="example-on-page-11">Example on page 11<a class="anchor" aria-label="anchor" href="#example-on-page-11"></a>
+</h3>
+<div class="sourceCode" id="cb50"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p11</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p11"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb53"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p11</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p11-1.png" width="700"></p>
+<div class="sourceCode" id="cb54"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p11</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##      SFO     IORE     DFOP </span></span>
+<span><span class="co">## 579.6805 204.7932 144.7783 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 251.6944</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0 96.15820 4.83e-13 90.24934 1.02e+02</span></span>
+<span><span class="co">## k_parent  0.00321 4.71e-05  0.00222 4.64e-03</span></span>
+<span><span class="co">## sigma     6.43473 1.28e-04  3.75822 9.11e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0       1.05e+02     NA 9.90e+01 1.10e+02</span></span>
+<span><span class="co">## k__iore_parent 3.11e-17     NA 1.35e-20 7.18e-14</span></span>
+<span><span class="co">## N_parent       8.36e+00     NA 6.62e+00 1.01e+01</span></span>
+<span><span class="co">## sigma          3.82e+00     NA 2.21e+00 5.44e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0 1.05e+02 9.47e-13  99.9990 109.1224</span></span>
+<span><span class="co">## k1       4.41e-02 5.95e-03   0.0296   0.0658</span></span>
+<span><span class="co">## k2       9.94e-13 5.00e-01   0.0000      Inf</span></span>
+<span><span class="co">## g        3.22e-01 1.45e-03   0.2814   0.3650</span></span>
+<span><span class="co">## sigma    3.22e+00 3.52e-04   1.8410   4.5906</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##          DT50     DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  2.16e+02 7.18e+02 2.16e+02</span></span>
+<span><span class="co">## IORE 9.73e+02 1.37e+08 4.11e+07</span></span>
+<span><span class="co">## DFOP 3.07e+11 1.93e+12 6.98e+11</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 41148169</span></span></code></pre>
+<p>In this case, the DFOP fit reported for PestDF resulted in a negative
+value for the slower rate constant, which is not possible in mkin. The
+other results are in agreement.</p>
+</div>
+</div>
+<div class="section level2">
+<h2 id="n-is-less-than-1-and-the-dfop-rate-constants-are-like-the-sfo-rate-constant">N is less than 1 and the DFOP rate constants are like the SFO rate
+constant<a class="anchor" aria-label="anchor" href="#n-is-less-than-1-and-the-dfop-rate-constants-are-like-the-sfo-rate-constant"></a>
+</h2>
+<p>In the following three examples, the same results are obtained with
+mkin as reported for PestDF. As in the case on page 10, the N values
+below 1 are deemed unrealistic and appear to be the result of an
+overparameterisation.</p>
+<div class="section level3">
+<h3 id="example-on-page-12-upper-panel">Example on page 12, upper panel<a class="anchor" aria-label="anchor" href="#example-on-page-12-upper-panel"></a>
+</h3>
+<div class="sourceCode" id="cb56"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p12a</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p12a"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in summary.mkinfit(x): Could not calculate correlation; no covariance</span></span>
+<span><span class="co">## matrix</span></span></code></pre>
+<pre><code><span><span class="co">## Warning in sqrt(diag(covar_notrans)): NaNs produced</span></span></code></pre>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb61"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p12a</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p12a-1.png" width="700"></p>
+<div class="sourceCode" id="cb62"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p12a</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##      SFO     IORE     DFOP </span></span>
+<span><span class="co">## 695.4440 220.0685 695.4440 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 270.4679</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)  Lower   Upper</span></span>
+<span><span class="co">## parent_0  100.521 8.75e-12 92.461 108.581</span></span>
+<span><span class="co">## k_parent    0.124 3.61e-08  0.104   0.148</span></span>
+<span><span class="co">## sigma       7.048 1.28e-04  4.116   9.980</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate Pr(&gt;t) Lower Upper</span></span>
+<span><span class="co">## parent_0         96.823     NA    NA    NA</span></span>
+<span><span class="co">## k__iore_parent    2.436     NA    NA    NA</span></span>
+<span><span class="co">## N_parent          0.263     NA    NA    NA</span></span>
+<span><span class="co">## sigma             3.965     NA    NA    NA</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)   Lower   Upper</span></span>
+<span><span class="co">## parent_0  100.521 2.74e-10 92.2366 108.805</span></span>
+<span><span class="co">## k1          0.124 2.53e-05  0.0908   0.170</span></span>
+<span><span class="co">## k2          0.124 2.52e-02  0.0456   0.339</span></span>
+<span><span class="co">## g           0.793      NaN  0.0000   1.000</span></span>
+<span><span class="co">## sigma       7.048 2.50e-04  4.0349  10.061</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50 DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  5.58 18.5     5.58</span></span>
+<span><span class="co">## IORE 6.49 13.2     3.99</span></span>
+<span><span class="co">## DFOP 5.58 18.5     5.58</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 3.99</span></span></code></pre>
+</div>
+<div class="section level3">
+<h3 id="example-on-page-12-lower-panel">Example on page 12, lower panel<a class="anchor" aria-label="anchor" href="#example-on-page-12-lower-panel"></a>
+</h3>
+<div class="sourceCode" id="cb64"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p12b</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p12b"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in sqrt(diag(covar)): NaNs produced</span></span></code></pre>
+<pre><code><span><span class="co">## Warning in qt(alpha/2, rdf): NaNs produced</span></span></code></pre>
+<pre><code><span><span class="co">## Warning in qt(1 - alpha/2, rdf): NaNs produced</span></span></code></pre>
+<pre><code><span><span class="co">## Warning in pt(abs(tval), rdf, lower.tail = FALSE): NaNs produced</span></span></code></pre>
+<pre><code><span><span class="co">## Warning in cov2cor(ans$covar): diag(V) had non-positive or NA entries; the</span></span>
+<span><span class="co">## non-finite result may be dubious</span></span></code></pre>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb72"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p12b</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p12b-1.png" width="700"></p>
+<div class="sourceCode" id="cb73"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p12b</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##      SFO     IORE     DFOP </span></span>
+<span><span class="co">## 58.90242 19.06353 58.90242 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 51.51756</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate  Pr(&gt;t)   Lower    Upper</span></span>
+<span><span class="co">## parent_0  97.6840 0.00039 85.9388 109.4292</span></span>
+<span><span class="co">## k_parent   0.0589 0.00261  0.0431   0.0805</span></span>
+<span><span class="co">## sigma      3.4323 0.04356 -1.2377   8.1023</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate Pr(&gt;t)     Lower  Upper</span></span>
+<span><span class="co">## parent_0         95.523 0.0055 74.539157 116.51</span></span>
+<span><span class="co">## k__iore_parent    0.333 0.1433  0.000717 154.57</span></span>
+<span><span class="co">## N_parent          0.568 0.0677 -0.989464   2.13</span></span>
+<span><span class="co">## sigma             1.953 0.0975 -5.893100   9.80</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate Pr(&gt;t) Lower Upper</span></span>
+<span><span class="co">## parent_0  97.6840    NaN   NaN   NaN</span></span>
+<span><span class="co">## k1         0.0589    NaN    NA    NA</span></span>
+<span><span class="co">## k2         0.0589    NaN    NA    NA</span></span>
+<span><span class="co">## g          0.6473    NaN    NA    NA</span></span>
+<span><span class="co">## sigma      3.4323    NaN   NaN   NaN</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50 DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  11.8 39.1    11.80</span></span>
+<span><span class="co">## IORE 12.9 31.4     9.46</span></span>
+<span><span class="co">## DFOP 11.8 39.1    11.80</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 9.46</span></span></code></pre>
+</div>
+<div class="section level3">
+<h3 id="example-on-page-13">Example on page 13<a class="anchor" aria-label="anchor" href="#example-on-page-13"></a>
+</h3>
+<div class="sourceCode" id="cb75"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p13</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p13"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb78"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p13</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p13-1.png" width="700"></p>
+<div class="sourceCode" id="cb79"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p13</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##      SFO     IORE     DFOP </span></span>
+<span><span class="co">## 174.5971 142.3951 174.5971 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 172.131</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0 92.73500 5.99e-17 89.61936 95.85065</span></span>
+<span><span class="co">## k_parent  0.00258 2.42e-09  0.00223  0.00299</span></span>
+<span><span class="co">## sigma     3.41172 7.07e-05  2.05455  4.76888</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate   Pr(&gt;t)    Lower  Upper</span></span>
+<span><span class="co">## parent_0        91.6016 6.34e-16 88.53086 94.672</span></span>
+<span><span class="co">## k__iore_parent   0.0396 2.36e-01  0.00207  0.759</span></span>
+<span><span class="co">## N_parent         0.3541 1.46e-01 -0.35153  1.060</span></span>
+<span><span class="co">## sigma            3.0811 9.64e-05  1.84296  4.319</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0 92.73500     NA 8.95e+01 95.92118</span></span>
+<span><span class="co">## k1        0.00258     NA 4.18e-04  0.01592</span></span>
+<span><span class="co">## k2        0.00258     NA 1.75e-03  0.00381</span></span>
+<span><span class="co">## g         0.16452     NA 0.00e+00  1.00000</span></span>
+<span><span class="co">## sigma     3.41172     NA 2.02e+00  4.79960</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50 DT90 DT50_rep</span></span>
+<span><span class="co">## SFO   269  892      269</span></span>
+<span><span class="co">## IORE  261  560      169</span></span>
+<span><span class="co">## DFOP  269  892      269</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 168.51</span></span></code></pre>
+</div>
+</div>
+<div class="section level2">
+<h2 id="dt50-not-observed-in-the-study-and-dfop-problems-in-pestdf">DT50 not observed in the study and DFOP problems in PestDF<a class="anchor" aria-label="anchor" href="#dt50-not-observed-in-the-study-and-dfop-problems-in-pestdf"></a>
+</h2>
+<div class="sourceCode" id="cb81"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p14</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p14"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in sqrt(diag(covar)): NaNs produced</span></span></code></pre>
+<pre><code><span><span class="co">## Warning in cov2cor(ans$covar): diag(V) had non-positive or NA entries; the</span></span>
+<span><span class="co">## non-finite result may be dubious</span></span></code></pre>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb86"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p14</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p14-1.png" width="700"></p>
+<div class="sourceCode" id="cb87"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p14</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##      SFO     IORE     DFOP </span></span>
+<span><span class="co">## 48.43249 28.67746 27.26248 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 32.83337</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0 99.47124 2.06e-30 98.42254 1.01e+02</span></span>
+<span><span class="co">## k_parent  0.00279 3.75e-15  0.00256 3.04e-03</span></span>
+<span><span class="co">## sigma     1.55616 3.81e-06  1.03704 2.08e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate Pr(&gt;t) Lower Upper</span></span>
+<span><span class="co">## parent_0       1.00e+02     NA   NaN   NaN</span></span>
+<span><span class="co">## k__iore_parent 9.44e-08     NA   NaN   NaN</span></span>
+<span><span class="co">## N_parent       3.31e+00     NA   NaN   NaN</span></span>
+<span><span class="co">## sigma          1.20e+00     NA 0.796   1.6</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0 1.00e+02 2.96e-28 99.40280 101.2768</span></span>
+<span><span class="co">## k1       9.53e-03 1.20e-01  0.00638   0.0143</span></span>
+<span><span class="co">## k2       5.21e-12 5.00e-01  0.00000      Inf</span></span>
+<span><span class="co">## g        3.98e-01 2.19e-01  0.30481   0.4998</span></span>
+<span><span class="co">## sigma    1.17e+00 7.68e-06  0.77406   1.5610</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##          DT50     DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  2.48e+02 8.25e+02 2.48e+02</span></span>
+<span><span class="co">## IORE 4.34e+02 2.22e+04 6.70e+03</span></span>
+<span><span class="co">## DFOP 3.55e+10 3.44e+11 1.33e+11</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 6697.44</span></span></code></pre>
+<p>The slower rate constant reported by PestDF is negative, which is not
+physically realistic, and not possible in mkin. The other fits give the
+same results in mkin and PestDF.</p>
+</div>
+<div class="section level2">
+<h2 id="n-is-less-than-1-and-dfop-fraction-parameter-is-below-zero">N is less than 1 and DFOP fraction parameter is below zero<a class="anchor" aria-label="anchor" href="#n-is-less-than-1-and-dfop-fraction-parameter-is-below-zero"></a>
+</h2>
+<div class="sourceCode" id="cb89"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p15a</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p15a"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in sqrt(diag(covar)): NaNs produced</span></span></code></pre>
+<pre><code><span><span class="co">## Warning in cov2cor(ans$covar): diag(V) had non-positive or NA entries; the</span></span>
+<span><span class="co">## non-finite result may be dubious</span></span></code></pre>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb94"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p15a</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p15a-1.png" width="700"></p>
+<div class="sourceCode" id="cb95"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p15a</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##      SFO     IORE     DFOP </span></span>
+<span><span class="co">## 245.5248 135.0132 245.5248 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 165.9335</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower   Upper</span></span>
+<span><span class="co">## parent_0 97.96751 2.00e-15 94.32049 101.615</span></span>
+<span><span class="co">## k_parent  0.00952 4.93e-09  0.00824   0.011</span></span>
+<span><span class="co">## sigma     4.18778 1.28e-04  2.44588   5.930</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate   Pr(&gt;t)  Lower  Upper</span></span>
+<span><span class="co">## parent_0         95.874 2.94e-15 92.937 98.811</span></span>
+<span><span class="co">## k__iore_parent    0.629 2.11e-01  0.044  8.982</span></span>
+<span><span class="co">## N_parent          0.000 5.00e-01 -0.642  0.642</span></span>
+<span><span class="co">## sigma             3.105 1.78e-04  1.795  4.416</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0 97.96751 2.85e-13 94.21913 101.7159</span></span>
+<span><span class="co">## k1        0.00952 6.28e-02  0.00260   0.0349</span></span>
+<span><span class="co">## k2        0.00952 1.27e-04  0.00652   0.0139</span></span>
+<span><span class="co">## g         0.21241 5.00e-01       NA       NA</span></span>
+<span><span class="co">## sigma     4.18778 2.50e-04  2.39747   5.9781</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50 DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  72.8  242     72.8</span></span>
+<span><span class="co">## IORE 76.3  137     41.3</span></span>
+<span><span class="co">## DFOP 72.8  242     72.8</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 41.33</span></span></code></pre>
+<div class="sourceCode" id="cb97"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p15b</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p15b"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Warning in summary.mkinfit(x): Could not calculate correlation; no covariance</span></span>
+<span><span class="co">## matrix</span></span></code></pre>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The half-life obtained from the IORE model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb101"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p15b</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p15b-1.png" width="700"></p>
+<div class="sourceCode" id="cb102"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p15b</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##       SFO      IORE      DFOP </span></span>
+<span><span class="co">## 106.91629  68.55574 106.91629 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 84.25618</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0 1.01e+02 3.06e-17 98.31594 1.03e+02</span></span>
+<span><span class="co">## k_parent 4.86e-03 2.48e-10  0.00435 5.42e-03</span></span>
+<span><span class="co">## sigma    2.76e+00 1.28e-04  1.61402 3.91e+00</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate   Pr(&gt;t)    Lower  Upper</span></span>
+<span><span class="co">## parent_0          99.83 1.81e-16 97.51348 102.14</span></span>
+<span><span class="co">## k__iore_parent     0.38 3.22e-01  0.00352  41.05</span></span>
+<span><span class="co">## N_parent           0.00 5.00e-01 -1.07696   1.08</span></span>
+<span><span class="co">## sigma              2.21 2.57e-04  1.23245   3.19</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate Pr(&gt;t) Lower Upper</span></span>
+<span><span class="co">## parent_0 1.01e+02     NA    NA    NA</span></span>
+<span><span class="co">## k1       4.86e-03     NA    NA    NA</span></span>
+<span><span class="co">## k2       4.86e-03     NA    NA    NA</span></span>
+<span><span class="co">## g        1.88e-01     NA    NA    NA</span></span>
+<span><span class="co">## sigma    2.76e+00     NA    NA    NA</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50 DT90 DT50_rep</span></span>
+<span><span class="co">## SFO   143  474    143.0</span></span>
+<span><span class="co">## IORE  131  236     71.2</span></span>
+<span><span class="co">## DFOP  143  474    143.0</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 71.18</span></span></code></pre>
+<p>In mkin, only the IORE fit is affected (deemed unrealistic), as the
+fraction parameter of the DFOP model is restricted to the interval
+between 0 and 1 in mkin. The SFO fits give the same results for both
+mkin and PestDF.</p>
+</div>
+<div class="section level2">
+<h2 id="the-dfop-fraction-parameter-is-greater-than-1">The DFOP fraction parameter is greater than 1<a class="anchor" aria-label="anchor" href="#the-dfop-fraction-parameter-is-greater-than-1"></a>
+</h2>
+<div class="sourceCode" id="cb104"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">p16</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/nafta.html">nafta</a></span><span class="op">(</span><span class="va">NAFTA_SOP_Attachment</span><span class="op">[[</span><span class="st">"p16"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c</span></span></code></pre>
+<pre><code><span><span class="co">## The representative half-life of the IORE model is longer than the one corresponding</span></span></code></pre>
+<pre><code><span><span class="co">## to the terminal degradation rate found with the DFOP model.</span></span></code></pre>
+<pre><code><span><span class="co">## The representative half-life obtained from the DFOP model may be used</span></span></code></pre>
+<div class="sourceCode" id="cb109"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">p16</span><span class="op">)</span></span></code></pre></div>
+<p><img src="NAFTA_examples_files/figure-html/p16-1.png" width="700"></p>
+<div class="sourceCode" id="cb110"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">p16</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Sums of squares:</span></span>
+<span><span class="co">##      SFO     IORE     DFOP </span></span>
+<span><span class="co">## 3831.804 2062.008 1550.980 </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Critical sum of squares for checking the SFO model:</span></span>
+<span><span class="co">## [1] 2247.348</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Parameters:</span></span>
+<span><span class="co">## $SFO</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)  Lower Upper</span></span>
+<span><span class="co">## parent_0   71.953 2.33e-13 60.509 83.40</span></span>
+<span><span class="co">## k_parent    0.159 4.86e-05  0.102  0.25</span></span>
+<span><span class="co">## sigma      11.302 1.25e-08  8.308 14.30</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $IORE</span></span>
+<span><span class="co">##                Estimate   Pr(&gt;t)    Lower    Upper</span></span>
+<span><span class="co">## parent_0       8.74e+01 2.48e-16 7.72e+01 97.52972</span></span>
+<span><span class="co">## k__iore_parent 4.55e-04 2.16e-01 3.48e-05  0.00595</span></span>
+<span><span class="co">## N_parent       2.70e+00 1.21e-08 1.99e+00  3.40046</span></span>
+<span><span class="co">## sigma          8.29e+00 1.61e-08 6.09e+00 10.49062</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## $DFOP</span></span>
+<span><span class="co">##          Estimate   Pr(&gt;t)   Lower  Upper</span></span>
+<span><span class="co">## parent_0  88.5333 7.40e-18 79.9836 97.083</span></span>
+<span><span class="co">## k1        18.8461 5.00e-01  0.0000    Inf</span></span>
+<span><span class="co">## k2         0.0776 1.41e-05  0.0518  0.116</span></span>
+<span><span class="co">## g          0.4733 1.41e-09  0.3674  0.582</span></span>
+<span><span class="co">## sigma      7.1902 2.11e-08  5.2785  9.102</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## DTx values:</span></span>
+<span><span class="co">##      DT50 DT90 DT50_rep</span></span>
+<span><span class="co">## SFO  4.35 14.4     4.35</span></span>
+<span><span class="co">## IORE 1.48 32.1     9.67</span></span>
+<span><span class="co">## DFOP 0.67 21.4     8.93</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">## Representative half-life:</span></span>
+<span><span class="co">## [1] 8.93</span></span></code></pre>
+<p>In PestDF, the DFOP fit seems to have stuck in a local minimum, as
+mkin finds a solution with a much lower
+<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>χ</mi><mn>2</mn></msup><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math>
+error level. As the half-life from the slower rate constant of the DFOP
+model is larger than the IORE derived half-life, the NAFTA
+recommendation obtained with mkin is to use the DFOP representative
+half-life of 8.9 days.</p>
+</div>
+<div class="section level2">
+<h2 id="conclusions">Conclusions<a class="anchor" aria-label="anchor" href="#conclusions"></a>
+</h2>
+<p>The results obtained with mkin deviate from the results obtained with
+PestDF either in cases where one of the interpretive rules would apply,
+i.e. the IORE parameter N is less than one or the DFOP k values obtained
+with PestDF are equal to the SFO k values, or in cases where the DFOP
+model did not converge, which often lead to negative rate constants
+returned by PestDF.</p>
+<p>Therefore, mkin appears to suitable for kinetic evaluations according
+to the NAFTA guidance.</p>
+</div>
+<div class="section level2">
+<h2 class="unnumbered" id="references">References<a class="anchor" aria-label="anchor" href="#references"></a>
+</h2>
+<div id="refs" class="references csl-bib-body hanging-indent">
+<div id="ref-usepa2015" class="csl-entry">
+US EPA. 2015. <span>“Standard Operating Procedure for Using the NAFTA
+Guidance to Calculate Representative Half-Life Values and Characterizing
+Pesticide Degradation.”</span> <a href="https://www.epa.gov/pesticide-science-and-assessing-pesticide-risks/standard-operating-procedure-using-nafta-guidance" class="external-link">https://www.epa.gov/pesticide-science-and-assessing-pesticide-risks/standard-operating-procedure-using-nafta-guidance</a>.
+</div>
+</div>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
+</div>
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+</div>
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@@ -0,0 +1,1133 @@
+<!DOCTYPE html>
+<!-- Generated by pkgdown: do not edit by hand --><html lang="en">
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+<title>Benchmark timings for mkin • mkin</title>
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+
+    <small class="nav-text text-info me-auto" data-bs-toggle="tooltip" data-bs-placement="bottom" title="In-development version">1.2.10</small>
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+<li><a class="dropdown-item" href="../../articles/mkin.html">Introduction to mkin</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with (generalised) nonlinear least squares</h6></li>
+    <li><a class="dropdown-item" href="../../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a></li>
+    <li><a class="dropdown-item" href="../../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/FOCUS_Z.html">Example evaluation of FOCUS Example Dataset Z</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with hierarchical models (nonlinear mixed-effects models)</h6></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_pathway.html">Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2023_mesotrione_parent.html">Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_cyan_pathway.html">Testing hierarchical pathway kinetics with residue data on cyantraniliprole</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/dimethenamid_2018.html">Comparison of saemix and nlme evaluations of dimethenamid data from 2018</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/multistart.html">Short demo of the multistart method</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Performance</h6></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/compiled_models.html">Performance benefit by using compiled model definitions in mkin</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/benchmarks.html">Benchmark timings for mkin</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/saem_benchmarks.html">Benchmark timings for saem.mmkin</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Miscellaneous</h6></li>
+    <li><a class="dropdown-item" href="../../articles/twa.html">Calculation of time weighted average concentrations with mkin</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a></li>
+  </ul>
+</li>
+<li class="nav-item"><a class="nav-link" href="../../coverage/coverage.html">Test coverage</a></li>
+<li class="nav-item"><a class="nav-link" href="../../news/index.html">News</a></li>
+      </ul>
+<ul class="navbar-nav">
+<li class="nav-item"><form class="form-inline" role="search">
+ <input class="form-control" type="search" name="search-input" id="search-input" autocomplete="off" aria-label="Search site" placeholder="Search for" data-search-index="../../search.json">
+</form></li>
+<li class="nav-item"><a class="external-link nav-link" href="https://github.com/jranke/mkin/" aria-label="GitHub"><span class="fa fab fa-github fa-lg"></span></a></li>
+      </ul>
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+  </div>
+</nav><div class="container template-article">
+
+
+
+
+<div class="row">
+  <main id="main" class="col-md-9"><div class="page-header">
+
+      <h1>Benchmark timings for mkin</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change 17 February 2023
+(rebuilt 2025-02-14)</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/web_only/benchmarks.rmd" class="external-link"><code>vignettes/web_only/benchmarks.rmd</code></a></small>
+      <div class="d-none name"><code>benchmarks.rmd</code></div>
+    </div>
+
+    
+    
+<p>Each system is characterized by the operating system type, the CPU
+type, the mkin version, and, as in June 2022 the current R version lead
+to worse performance, the R version. A compiler was available, so if no
+analytical solution was available, compiled ODE models are used.</p>
+<p>Every fit is only performed once, so the accuracy of the benchmarks
+is limited.</p>
+<p>The following wrapper function for <code>mmkin</code> is used because
+the way the error model is specified was changed in mkin version
+0.9.49.1.</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw">if</span> <span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/utils/packageDescription.html" class="external-link">packageVersion</a></span><span class="op">(</span><span class="st">"mkin"</span><span class="op">)</span> <span class="op">&gt;</span> <span class="st">"0.9.48.1"</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="va">mmkin_bench</span> <span class="op">&lt;-</span> <span class="kw">function</span><span class="op">(</span><span class="va">models</span>, <span class="va">datasets</span>, <span class="va">error_model</span> <span class="op">=</span> <span class="st">"const"</span><span class="op">)</span> <span class="op">{</span></span>
+<span>    <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="va">models</span>, <span class="va">datasets</span>, error_model <span class="op">=</span> <span class="va">error_model</span>, cores <span class="op">=</span> <span class="fl">1</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span>  <span class="op">}</span></span>
+<span><span class="op">}</span> <span class="kw">else</span> <span class="op">{</span></span>
+<span>  <span class="va">mmkin_bench</span> <span class="op">&lt;-</span> <span class="kw">function</span><span class="op">(</span><span class="va">models</span>, <span class="va">datasets</span>, <span class="va">error_model</span> <span class="op">=</span> <span class="cn">NULL</span><span class="op">)</span> <span class="op">{</span></span>
+<span>    <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="va">models</span>, <span class="va">datasets</span>, reweight.method <span class="op">=</span> <span class="va">error_model</span>, cores <span class="op">=</span> <span class="fl">1</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span>  <span class="op">}</span></span>
+<span><span class="op">}</span></span></code></pre></div>
+<div class="section level2">
+<h2 id="test-cases">Test cases<a class="anchor" aria-label="anchor" href="#test-cases"></a>
+</h2>
+<p>Parent only:</p>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">FOCUS_C</span> <span class="op">&lt;-</span> <span class="va">FOCUS_2006_C</span></span>
+<span><span class="va">FOCUS_D</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/subset.html" class="external-link">subset</a></span><span class="op">(</span><span class="va">FOCUS_2006_D</span>, <span class="va">value</span> <span class="op">!=</span> <span class="fl">0</span><span class="op">)</span></span>
+<span><span class="va">parent_datasets</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">FOCUS_C</span>, <span class="va">FOCUS_D</span><span class="op">)</span></span>
+<span></span>
+<span></span>
+<span><span class="va">t1</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="fu">mmkin_bench</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"FOMC"</span>, <span class="st">"DFOP"</span>, <span class="st">"HS"</span><span class="op">)</span>, <span class="va">parent_datasets</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t2</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="fu">mmkin_bench</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"FOMC"</span>, <span class="st">"DFOP"</span>, <span class="st">"HS"</span><span class="op">)</span>, <span class="va">parent_datasets</span>,</span>
+<span>    error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span></code></pre></div>
+<p>One metabolite:</p>
+<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">SFO_SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>  parent <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"m1"</span><span class="op">)</span>,</span>
+<span>  m1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="va">FOMC_SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>  parent <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"FOMC"</span>, <span class="st">"m1"</span><span class="op">)</span>,</span>
+<span>  m1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="va">DFOP_SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>  parent <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"FOMC"</span>, <span class="st">"m1"</span><span class="op">)</span>, <span class="co"># erroneously used FOMC twice, not fixed for consistency</span></span>
+<span>  m1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="va">t3</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="fu">mmkin_bench</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">SFO_SFO</span>, <span class="va">FOMC_SFO</span>, <span class="va">DFOP_SFO</span><span class="op">)</span>, <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">FOCUS_D</span><span class="op">)</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t4</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="fu">mmkin_bench</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">SFO_SFO</span>, <span class="va">FOMC_SFO</span>, <span class="va">DFOP_SFO</span><span class="op">)</span>, <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">FOCUS_D</span><span class="op">)</span>,</span>
+<span>    error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t5</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="fu">mmkin_bench</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">SFO_SFO</span>, <span class="va">FOMC_SFO</span>, <span class="va">DFOP_SFO</span><span class="op">)</span>, <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">FOCUS_D</span><span class="op">)</span>,</span>
+<span>    error_model <span class="op">=</span> <span class="st">"obs"</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span></code></pre></div>
+<p>Two metabolites, synthetic data:</p>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">m_synth_SFO_lin</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M1"</span><span class="op">)</span>,</span>
+<span>                           M1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M2"</span><span class="op">)</span>,</span>
+<span>                           M2 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>                           use_of_ff <span class="op">=</span> <span class="st">"max"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">m_synth_DFOP_par</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"DFOP"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"M1"</span>, <span class="st">"M2"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>                           M1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>                           M2 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>                           use_of_ff <span class="op">=</span> <span class="st">"max"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">SFO_lin_a</span> <span class="op">&lt;-</span> <span class="va">synthetic_data_for_UBA_2014</span><span class="op">[[</span><span class="fl">1</span><span class="op">]</span><span class="op">]</span><span class="op">$</span><span class="va">data</span></span>
+<span></span>
+<span><span class="va">DFOP_par_c</span> <span class="op">&lt;-</span> <span class="va">synthetic_data_for_UBA_2014</span><span class="op">[[</span><span class="fl">12</span><span class="op">]</span><span class="op">]</span><span class="op">$</span><span class="va">data</span></span>
+<span></span>
+<span><span class="va">t6</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="fu">mmkin_bench</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">m_synth_SFO_lin</span><span class="op">)</span>, <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">SFO_lin_a</span><span class="op">)</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t7</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="fu">mmkin_bench</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">m_synth_DFOP_par</span><span class="op">)</span>, <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">DFOP_par_c</span><span class="op">)</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span></span>
+<span><span class="va">t8</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="fu">mmkin_bench</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">m_synth_SFO_lin</span><span class="op">)</span>, <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">SFO_lin_a</span><span class="op">)</span>,</span>
+<span>    error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t9</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="fu">mmkin_bench</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">m_synth_DFOP_par</span><span class="op">)</span>, <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">DFOP_par_c</span><span class="op">)</span>,</span>
+<span>    error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span></span>
+<span><span class="va">t10</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="fu">mmkin_bench</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">m_synth_SFO_lin</span><span class="op">)</span>, <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">SFO_lin_a</span><span class="op">)</span>,</span>
+<span>    error_model <span class="op">=</span> <span class="st">"obs"</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t11</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="fu">mmkin_bench</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">m_synth_DFOP_par</span><span class="op">)</span>, <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">DFOP_par_c</span><span class="op">)</span>,</span>
+<span>    error_model <span class="op">=</span> <span class="st">"obs"</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span></code></pre></div>
+</div>
+<div class="section level2">
+<h2 id="results">Results<a class="anchor" aria-label="anchor" href="#results"></a>
+</h2>
+<p>Benchmarks for all available error models are shown. They are
+intended for improving mkin, not for comparing CPUs or operating
+systems. All trademarks belong to their respective owners.</p>
+<div class="section level3">
+<h3 id="parent-only">Parent only<a class="anchor" aria-label="anchor" href="#parent-only"></a>
+</h3>
+<p>Constant variance (t1) and two-component error model (t2) for four
+models fitted to two datasets, i.e. eight fits for each test.</p>
+<table style="width:100%;" class="table">
+<colgroup>
+<col width="8%">
+<col width="54%">
+<col width="8%">
+<col width="12%">
+<col width="8%">
+<col width="9%">
+</colgroup>
+<thead><tr class="header">
+<th align="left">OS</th>
+<th align="left">CPU</th>
+<th align="left">R</th>
+<th align="left">mkin</th>
+<th align="right">t1</th>
+<th align="right">t2</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.48.1</td>
+<td align="right">3.610</td>
+<td align="right">11.019</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.1</td>
+<td align="right">8.184</td>
+<td align="right">22.889</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.2</td>
+<td align="right">7.064</td>
+<td align="right">12.558</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.3</td>
+<td align="right">7.296</td>
+<td align="right">21.239</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.4</td>
+<td align="right">5.936</td>
+<td align="right">20.545</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.50.2</td>
+<td align="right">1.714</td>
+<td align="right">3.971</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.50.3</td>
+<td align="right">1.752</td>
+<td align="right">4.156</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.50.4</td>
+<td align="right">1.786</td>
+<td align="right">3.729</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">1.0.3</td>
+<td align="right">1.881</td>
+<td align="right">3.504</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">1.0.4</td>
+<td align="right">1.867</td>
+<td align="right">3.450</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.1.3</td>
+<td align="left">1.1.0</td>
+<td align="right">1.791</td>
+<td align="right">3.289</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.0</td>
+<td align="right">1.842</td>
+<td align="right">3.453</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">i7-4710MQ</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.0</td>
+<td align="right">1.959</td>
+<td align="right">4.116</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">i7-4710MQ</td>
+<td align="left">4.1.3</td>
+<td align="left">1.1.0</td>
+<td align="right">1.877</td>
+<td align="right">3.906</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">i7-4710MQ</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.1</td>
+<td align="right">1.644</td>
+<td align="right">3.172</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.1</td>
+<td align="right">1.770</td>
+<td align="right">3.377</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.2</td>
+<td align="right">1.957</td>
+<td align="right">3.633</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.0</td>
+<td align="right">2.140</td>
+<td align="right">3.774</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.2</td>
+<td align="right">2.187</td>
+<td align="right">3.851</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.0</td>
+<td align="right">1.288</td>
+<td align="right">1.794</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.2</td>
+<td align="right">1.276</td>
+<td align="right">1.804</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.3</td>
+<td align="right">1.370</td>
+<td align="right">1.883</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.3</td>
+<td align="left">1.2.3</td>
+<td align="right">1.406</td>
+<td align="right">1.948</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.3.0</td>
+<td align="left">1.2.4</td>
+<td align="right">1.386</td>
+<td align="right">1.960</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Intel(R) Xeon(R) Gold 6134 CPU @ 3.20GHz</td>
+<td align="left">4.3.1</td>
+<td align="left">1.2.5</td>
+<td align="right">2.369</td>
+<td align="right">3.632</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Intel(R) Xeon(R) Gold 6134 CPU @ 3.20GHz</td>
+<td align="left">4.3.1</td>
+<td align="left">1.2.6</td>
+<td align="right">2.856</td>
+<td align="right">4.960</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.3.2</td>
+<td align="left">1.2.6</td>
+<td align="right">1.408</td>
+<td align="right">2.041</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.4.2</td>
+<td align="left">1.2.9</td>
+<td align="right">1.323</td>
+<td align="right">1.925</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.4.2</td>
+<td align="left">1.2.10</td>
+<td align="right">1.371</td>
+<td align="right">1.980</td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="section level3">
+<h3 id="one-metabolite">One metabolite<a class="anchor" aria-label="anchor" href="#one-metabolite"></a>
+</h3>
+<p>Constant variance (t3), two-component error model (t4), and variance
+by variable (t5) for three models fitted to one dataset, i.e. three fits
+for each test.</p>
+<table class="table">
+<colgroup>
+<col width="7%">
+<col width="50%">
+<col width="7%">
+<col width="11%">
+<col width="7%">
+<col width="8%">
+<col width="7%">
+</colgroup>
+<thead><tr class="header">
+<th align="left">OS</th>
+<th align="left">CPU</th>
+<th align="left">R</th>
+<th align="left">mkin</th>
+<th align="right">t3</th>
+<th align="right">t4</th>
+<th align="right">t5</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.48.1</td>
+<td align="right">3.764</td>
+<td align="right">14.347</td>
+<td align="right">9.495</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.1</td>
+<td align="right">4.649</td>
+<td align="right">13.789</td>
+<td align="right">6.395</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.2</td>
+<td align="right">4.786</td>
+<td align="right">8.461</td>
+<td align="right">5.675</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.3</td>
+<td align="right">4.510</td>
+<td align="right">13.805</td>
+<td align="right">7.386</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.4</td>
+<td align="right">4.446</td>
+<td align="right">15.335</td>
+<td align="right">6.002</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.50.2</td>
+<td align="right">1.402</td>
+<td align="right">6.174</td>
+<td align="right">2.764</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.50.3</td>
+<td align="right">1.430</td>
+<td align="right">6.615</td>
+<td align="right">2.878</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.50.4</td>
+<td align="right">1.397</td>
+<td align="right">7.251</td>
+<td align="right">2.810</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">1.0.3</td>
+<td align="right">1.430</td>
+<td align="right">6.344</td>
+<td align="right">2.798</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">1.0.4</td>
+<td align="right">1.415</td>
+<td align="right">6.364</td>
+<td align="right">2.820</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.1.3</td>
+<td align="left">1.1.0</td>
+<td align="right">1.310</td>
+<td align="right">6.279</td>
+<td align="right">2.681</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.0</td>
+<td align="right">3.802</td>
+<td align="right">21.247</td>
+<td align="right">8.461</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">i7-4710MQ</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.0</td>
+<td align="right">3.334</td>
+<td align="right">19.521</td>
+<td align="right">7.565</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">i7-4710MQ</td>
+<td align="left">4.1.3</td>
+<td align="left">1.1.0</td>
+<td align="right">1.578</td>
+<td align="right">8.058</td>
+<td align="right">3.339</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">i7-4710MQ</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.1</td>
+<td align="right">1.230</td>
+<td align="right">5.839</td>
+<td align="right">2.444</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.1</td>
+<td align="right">1.308</td>
+<td align="right">5.758</td>
+<td align="right">2.558</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.2</td>
+<td align="right">1.503</td>
+<td align="right">6.147</td>
+<td align="right">2.803</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.0</td>
+<td align="right">1.554</td>
+<td align="right">6.193</td>
+<td align="right">2.843</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.2</td>
+<td align="right">1.585</td>
+<td align="right">6.335</td>
+<td align="right">3.003</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.0</td>
+<td align="right">0.792</td>
+<td align="right">2.378</td>
+<td align="right">1.245</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.2</td>
+<td align="right">0.784</td>
+<td align="right">2.355</td>
+<td align="right">1.233</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.3</td>
+<td align="right">0.770</td>
+<td align="right">2.011</td>
+<td align="right">1.123</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.3</td>
+<td align="left">1.2.3</td>
+<td align="right">0.793</td>
+<td align="right">2.109</td>
+<td align="right">1.178</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.3.0</td>
+<td align="left">1.2.4</td>
+<td align="right">0.779</td>
+<td align="right">2.080</td>
+<td align="right">1.106</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Intel(R) Xeon(R) Gold 6134 CPU @ 3.20GHz</td>
+<td align="left">4.3.1</td>
+<td align="left">1.2.5</td>
+<td align="right">1.823</td>
+<td align="right">5.555</td>
+<td align="right">2.404</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Intel(R) Xeon(R) Gold 6134 CPU @ 3.20GHz</td>
+<td align="left">4.3.1</td>
+<td align="left">1.2.6</td>
+<td align="right">1.761</td>
+<td align="right">5.405</td>
+<td align="right">2.462</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.3.2</td>
+<td align="left">1.2.6</td>
+<td align="right">0.795</td>
+<td align="right">2.228</td>
+<td align="right">1.178</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.4.2</td>
+<td align="left">1.2.9</td>
+<td align="right">0.754</td>
+<td align="right">2.153</td>
+<td align="right">1.139</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.4.2</td>
+<td align="left">1.2.10</td>
+<td align="right">0.762</td>
+<td align="right">2.179</td>
+<td align="right">1.131</td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="section level3">
+<h3 id="two-metabolites">Two metabolites<a class="anchor" aria-label="anchor" href="#two-metabolites"></a>
+</h3>
+<p>Constant variance (t6 and t7), two-component error model (t8 and t9),
+and variance by variable (t10 and t11) for one model fitted to one
+dataset, i.e. one fit for each test.</p>
+<table style="width:100%;" class="table">
+<colgroup>
+<col width="5%">
+<col width="40%">
+<col width="5%">
+<col width="8%">
+<col width="5%">
+<col width="5%">
+<col width="5%">
+<col width="6%">
+<col width="5%">
+<col width="6%">
+</colgroup>
+<thead><tr class="header">
+<th align="left">OS</th>
+<th align="left">CPU</th>
+<th align="left">R</th>
+<th align="left">mkin</th>
+<th align="right">t6</th>
+<th align="right">t7</th>
+<th align="right">t8</th>
+<th align="right">t9</th>
+<th align="right">t10</th>
+<th align="right">t11</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.48.1</td>
+<td align="right">2.623</td>
+<td align="right">4.587</td>
+<td align="right">7.525</td>
+<td align="right">16.621</td>
+<td align="right">8.576</td>
+<td align="right">31.267</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.1</td>
+<td align="right">2.542</td>
+<td align="right">4.128</td>
+<td align="right">4.632</td>
+<td align="right">8.171</td>
+<td align="right">3.676</td>
+<td align="right">5.636</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.2</td>
+<td align="right">2.723</td>
+<td align="right">4.478</td>
+<td align="right">4.862</td>
+<td align="right">7.618</td>
+<td align="right">3.579</td>
+<td align="right">5.574</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.3</td>
+<td align="right">2.643</td>
+<td align="right">4.374</td>
+<td align="right">7.020</td>
+<td align="right">11.124</td>
+<td align="right">5.388</td>
+<td align="right">7.365</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.49.4</td>
+<td align="right">2.635</td>
+<td align="right">4.259</td>
+<td align="right">4.737</td>
+<td align="right">7.763</td>
+<td align="right">3.427</td>
+<td align="right">5.626</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.50.2</td>
+<td align="right">0.777</td>
+<td align="right">1.236</td>
+<td align="right">1.332</td>
+<td align="right">2.872</td>
+<td align="right">2.069</td>
+<td align="right">2.987</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.50.3</td>
+<td align="right">0.858</td>
+<td align="right">1.264</td>
+<td align="right">1.333</td>
+<td align="right">2.984</td>
+<td align="right">2.113</td>
+<td align="right">3.073</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">0.9.50.4</td>
+<td align="right">0.783</td>
+<td align="right">1.282</td>
+<td align="right">1.486</td>
+<td align="right">3.815</td>
+<td align="right">1.958</td>
+<td align="right">3.105</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">1.0.3</td>
+<td align="right">0.763</td>
+<td align="right">1.244</td>
+<td align="right">1.457</td>
+<td align="right">3.054</td>
+<td align="right">1.923</td>
+<td align="right">2.839</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">NA</td>
+<td align="left">1.0.4</td>
+<td align="right">0.785</td>
+<td align="right">1.252</td>
+<td align="right">1.466</td>
+<td align="right">3.091</td>
+<td align="right">1.936</td>
+<td align="right">2.826</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.1.3</td>
+<td align="left">1.1.0</td>
+<td align="right">0.744</td>
+<td align="right">1.227</td>
+<td align="right">1.288</td>
+<td align="right">3.553</td>
+<td align="right">1.895</td>
+<td align="right">2.738</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.0</td>
+<td align="right">3.018</td>
+<td align="right">4.165</td>
+<td align="right">5.036</td>
+<td align="right">10.844</td>
+<td align="right">6.623</td>
+<td align="right">9.722</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">i7-4710MQ</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.0</td>
+<td align="right">2.522</td>
+<td align="right">3.792</td>
+<td align="right">4.143</td>
+<td align="right">11.268</td>
+<td align="right">5.935</td>
+<td align="right">8.728</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">i7-4710MQ</td>
+<td align="left">4.1.3</td>
+<td align="left">1.1.0</td>
+<td align="right">0.907</td>
+<td align="right">1.535</td>
+<td align="right">1.589</td>
+<td align="right">4.544</td>
+<td align="right">2.302</td>
+<td align="right">3.463</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">i7-4710MQ</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.1</td>
+<td align="right">0.678</td>
+<td align="right">1.095</td>
+<td align="right">1.149</td>
+<td align="right">3.247</td>
+<td align="right">1.658</td>
+<td align="right">2.472</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.1</td>
+<td align="right">0.696</td>
+<td align="right">1.124</td>
+<td align="right">1.321</td>
+<td align="right">2.786</td>
+<td align="right">1.744</td>
+<td align="right">2.566</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.1</td>
+<td align="left">1.1.2</td>
+<td align="right">0.861</td>
+<td align="right">1.295</td>
+<td align="right">1.507</td>
+<td align="right">3.102</td>
+<td align="right">1.961</td>
+<td align="right">2.852</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.0</td>
+<td align="right">0.913</td>
+<td align="right">1.345</td>
+<td align="right">1.539</td>
+<td align="right">3.011</td>
+<td align="right">1.987</td>
+<td align="right">2.802</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 7 1700</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.2</td>
+<td align="right">0.935</td>
+<td align="right">1.381</td>
+<td align="right">1.551</td>
+<td align="right">3.209</td>
+<td align="right">1.976</td>
+<td align="right">3.013</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.0</td>
+<td align="right">0.445</td>
+<td align="right">0.591</td>
+<td align="right">0.660</td>
+<td align="right">1.190</td>
+<td align="right">0.814</td>
+<td align="right">1.100</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.2</td>
+<td align="right">0.443</td>
+<td align="right">0.586</td>
+<td align="right">0.661</td>
+<td align="right">1.176</td>
+<td align="right">0.803</td>
+<td align="right">1.097</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.2</td>
+<td align="left">1.2.3</td>
+<td align="right">0.418</td>
+<td align="right">0.530</td>
+<td align="right">0.591</td>
+<td align="right">1.006</td>
+<td align="right">0.716</td>
+<td align="right">0.949</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.2.3</td>
+<td align="left">1.2.3</td>
+<td align="right">0.432</td>
+<td align="right">0.549</td>
+<td align="right">0.609</td>
+<td align="right">1.052</td>
+<td align="right">0.743</td>
+<td align="right">0.989</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.3.0</td>
+<td align="left">1.2.4</td>
+<td align="right">0.410</td>
+<td align="right">0.526</td>
+<td align="right">0.553</td>
+<td align="right">1.249</td>
+<td align="right">0.712</td>
+<td align="right">0.948</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Intel(R) Xeon(R) Gold 6134 CPU @ 3.20GHz</td>
+<td align="left">4.3.1</td>
+<td align="left">1.2.5</td>
+<td align="right">0.798</td>
+<td align="right">1.096</td>
+<td align="right">1.217</td>
+<td align="right">3.173</td>
+<td align="right">1.634</td>
+<td align="right">2.271</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Intel(R) Xeon(R) Gold 6134 CPU @ 3.20GHz</td>
+<td align="left">4.3.1</td>
+<td align="left">1.2.6</td>
+<td align="right">0.813</td>
+<td align="right">1.136</td>
+<td align="right">1.220</td>
+<td align="right">3.114</td>
+<td align="right">1.598</td>
+<td align="right">2.255</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.3.2</td>
+<td align="left">1.2.6</td>
+<td align="right">0.439</td>
+<td align="right">0.557</td>
+<td align="right">0.585</td>
+<td align="right">1.338</td>
+<td align="right">0.749</td>
+<td align="right">0.999</td>
+</tr>
+<tr class="even">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.4.2</td>
+<td align="left">1.2.9</td>
+<td align="right">0.424</td>
+<td align="right">0.534</td>
+<td align="right">0.560</td>
+<td align="right">1.298</td>
+<td align="right">0.735</td>
+<td align="right">0.981</td>
+</tr>
+<tr class="odd">
+<td align="left">Linux</td>
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">4.4.2</td>
+<td align="left">1.2.10</td>
+<td align="right">0.426</td>
+<td align="right">0.533</td>
+<td align="right">0.565</td>
+<td align="right">1.296</td>
+<td align="right">0.728</td>
+<td align="right">0.978</td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
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+<div class="pkgdown-footer-right">
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diff --git a/docs/dev/articles/web_only/compiled_models.html b/docs/dev/articles/web_only/compiled_models.html
new file mode 100644
index 00000000..113f7b67
--- /dev/null
+++ b/docs/dev/articles/web_only/compiled_models.html
@@ -0,0 +1,231 @@
+<!DOCTYPE html>
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+    <small class="nav-text text-info me-auto" data-bs-toggle="tooltip" data-bs-placement="bottom" title="In-development version">1.2.10</small>
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+    <li><a class="dropdown-item" href="../../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a></li>
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+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with hierarchical models (nonlinear mixed-effects models)</h6></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_pathway.html">Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2023_mesotrione_parent.html">Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_cyan_pathway.html">Testing hierarchical pathway kinetics with residue data on cyantraniliprole</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/dimethenamid_2018.html">Comparison of saemix and nlme evaluations of dimethenamid data from 2018</a></li>
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+
+      <h1>Performance benefit by using compiled model definitions in mkin</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">2025-02-14</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/web_only/compiled_models.rmd" class="external-link"><code>vignettes/web_only/compiled_models.rmd</code></a></small>
+      <div class="d-none name"><code>compiled_models.rmd</code></div>
+    </div>
+
+    
+    
+<div class="section level2">
+<h2 id="how-to-benefit-from-compiled-models">How to benefit from compiled models<a class="anchor" aria-label="anchor" href="#how-to-benefit-from-compiled-models"></a>
+</h2>
+<p>When using an mkin version equal to or greater than 0.9-36 and a C
+compiler is available, you will see a message that the model is being
+compiled from autogenerated C code when defining a model using mkinmod.
+Starting from version 0.9.49.9, the <code><a href="../../reference/mkinmod.html">mkinmod()</a></code> function
+checks for presence of a compiler using</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu">pkgbuild</span><span class="fu">::</span><span class="fu"><a href="https://pkgbuild.r-lib.org/reference/has_compiler.html" class="external-link">has_compiler</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<p>In previous versions, it used <code>Sys.which("gcc")</code> for this
+check.</p>
+<p>On Linux, you need to have the essential build tools like make and
+gcc or clang installed. On Debian based linux distributions, these will
+be pulled in by installing the build-essential package.</p>
+<p>On MacOS, which I do not use personally, I have had reports that a
+compiler is available by default.</p>
+<p>On Windows, you need to install Rtools and have the path to its bin
+directory in your PATH variable. You do not need to modify the PATH
+variable when installing Rtools. Instead, I would recommend to put the
+line</p>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/Sys.setenv.html" class="external-link">Sys.setenv</a></span><span class="op">(</span>PATH <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/paste.html" class="external-link">paste</a></span><span class="op">(</span><span class="st">"C:/Rtools/bin"</span>, <span class="fu"><a href="https://rdrr.io/r/base/Sys.getenv.html" class="external-link">Sys.getenv</a></span><span class="op">(</span><span class="st">"PATH"</span><span class="op">)</span>, sep<span class="op">=</span><span class="st">";"</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<p>into your .Rprofile startup file. This is just a text file with some
+R code that is executed when your R session starts. It has to be named
+.Rprofile and has to be located in your home directory, which will
+generally be your Documents folder. You can check the location of the
+home directory used by R by issuing</p>
+<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/Sys.getenv.html" class="external-link">Sys.getenv</a></span><span class="op">(</span><span class="st">"HOME"</span><span class="op">)</span></span></code></pre></div>
+</div>
+<div class="section level2">
+<h2 id="comparison-with-other-solution-methods">Comparison with other solution methods<a class="anchor" aria-label="anchor" href="#comparison-with-other-solution-methods"></a>
+</h2>
+<p>First, we build a simple degradation model for a parent compound with
+one metabolite, and we remove zero values from the dataset.</p>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="st"><a href="https://pkgdown.jrwb.de/mkin/">"mkin"</a></span>, quietly <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="va">SFO_SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>  parent <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"m1"</span><span class="op">)</span>,</span>
+<span>  m1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">FOCUS_D</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/subset.html" class="external-link">subset</a></span><span class="op">(</span><span class="va">FOCUS_2006_D</span>, <span class="va">value</span> <span class="op">!=</span> <span class="fl">0</span><span class="op">)</span></span></code></pre></div>
+<p>We can compare the performance of the Eigenvalue based solution
+against the compiled version and the R implementation of the
+differential equations using the benchmark package. In the output of
+below code, the warnings about zero being removed from the FOCUS D
+dataset are suppressed. Since mkin version 0.9.49.11, an analytical
+solution is also implemented, which is included in the tests below.</p>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw">if</span> <span class="op">(</span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">require</a></span><span class="op">(</span><span class="va"><a href="http://rbenchmark.googlecode.com" class="external-link">rbenchmark</a></span><span class="op">)</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="va">b.1</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/rbenchmark/man/benchmark.html" class="external-link">benchmark</a></span><span class="op">(</span></span>
+<span>    <span class="st">"deSolve, not compiled"</span> <span class="op">=</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">SFO_SFO</span>, <span class="va">FOCUS_D</span>,</span>
+<span>       solution_type <span class="op">=</span> <span class="st">"deSolve"</span>,</span>
+<span>       use_compiled <span class="op">=</span> <span class="cn">FALSE</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span>,</span>
+<span>    <span class="st">"Eigenvalue based"</span> <span class="op">=</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">SFO_SFO</span>, <span class="va">FOCUS_D</span>,</span>
+<span>       solution_type <span class="op">=</span> <span class="st">"eigen"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span>,</span>
+<span>    <span class="st">"deSolve, compiled"</span> <span class="op">=</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">SFO_SFO</span>, <span class="va">FOCUS_D</span>,</span>
+<span>       solution_type <span class="op">=</span> <span class="st">"deSolve"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span>,</span>
+<span>    <span class="st">"analytical"</span> <span class="op">=</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">SFO_SFO</span>, <span class="va">FOCUS_D</span>,</span>
+<span>       solution_type <span class="op">=</span> <span class="st">"analytical"</span>,</span>
+<span>       use_compiled <span class="op">=</span> <span class="cn">FALSE</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span>,</span>
+<span>    replications <span class="op">=</span> <span class="fl">1</span>, order <span class="op">=</span> <span class="st">"relative"</span>,</span>
+<span>    columns <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"test"</span>, <span class="st">"replications"</span>, <span class="st">"relative"</span>, <span class="st">"elapsed"</span><span class="op">)</span><span class="op">)</span></span>
+<span>  <span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">b.1</span><span class="op">)</span></span>
+<span><span class="op">}</span> <span class="kw">else</span> <span class="op">{</span></span>
+<span>  <span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="st">"R package rbenchmark is not available"</span><span class="op">)</span></span>
+<span><span class="op">}</span></span></code></pre></div>
+<pre><code><span><span class="co">##                    test replications relative elapsed</span></span>
+<span><span class="co">## 4            analytical            1    1.000   0.102</span></span>
+<span><span class="co">## 3     deSolve, compiled            1    1.324   0.135</span></span>
+<span><span class="co">## 2      Eigenvalue based            1    1.706   0.174</span></span>
+<span><span class="co">## 1 deSolve, not compiled            1   22.627   2.308</span></span></code></pre>
+<p>We see that using the compiled model is by more than a factor of 10
+faster than using deSolve without compiled code.</p>
+</div>
+<div class="section level2">
+<h2 id="model-without-analytical-solution">Model without analytical solution<a class="anchor" aria-label="anchor" href="#model-without-analytical-solution"></a>
+</h2>
+<p>This evaluation is also taken from the example section of mkinfit. No
+analytical solution is available for this system, and now Eigenvalue
+based solution is possible, so only deSolve using with or without
+compiled code is available.</p>
+<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw">if</span> <span class="op">(</span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">require</a></span><span class="op">(</span><span class="va"><a href="http://rbenchmark.googlecode.com" class="external-link">rbenchmark</a></span><span class="op">)</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="va">FOMC_SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    parent <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"FOMC"</span>, <span class="st">"m1"</span><span class="op">)</span>,</span>
+<span>    m1 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span> <span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span></span>
+<span></span>
+<span>  <span class="va">b.2</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/rbenchmark/man/benchmark.html" class="external-link">benchmark</a></span><span class="op">(</span></span>
+<span>    <span class="st">"deSolve, not compiled"</span> <span class="op">=</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">FOMC_SFO</span>, <span class="va">FOCUS_D</span>,</span>
+<span>                                      use_compiled <span class="op">=</span> <span class="cn">FALSE</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span>,</span>
+<span>    <span class="st">"deSolve, compiled"</span> <span class="op">=</span> <span class="fu"><a href="../../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">FOMC_SFO</span>, <span class="va">FOCUS_D</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span>,</span>
+<span>    replications <span class="op">=</span> <span class="fl">1</span>, order <span class="op">=</span> <span class="st">"relative"</span>,</span>
+<span>    columns <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"test"</span>, <span class="st">"replications"</span>, <span class="st">"relative"</span>, <span class="st">"elapsed"</span><span class="op">)</span><span class="op">)</span></span>
+<span>  <span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">b.2</span><span class="op">)</span></span>
+<span>  <span class="va">factor_FOMC_SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/Round.html" class="external-link">round</a></span><span class="op">(</span><span class="va">b.2</span><span class="op">[</span><span class="st">"1"</span>, <span class="st">"relative"</span><span class="op">]</span><span class="op">)</span></span>
+<span><span class="op">}</span> <span class="kw">else</span> <span class="op">{</span></span>
+<span>  <span class="va">factor_FOMC_SFO</span> <span class="op">&lt;-</span> <span class="cn">NA</span></span>
+<span>  <span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="st">"R package benchmark is not available"</span><span class="op">)</span></span>
+<span><span class="op">}</span></span></code></pre></div>
+<pre><code><span><span class="co">## Temporary DLL for differentials generated and loaded</span></span></code></pre>
+<pre><code><span><span class="co">##                    test replications relative elapsed</span></span>
+<span><span class="co">## 2     deSolve, compiled            1    1.000   0.170</span></span>
+<span><span class="co">## 1 deSolve, not compiled            1   23.865   4.057</span></span></code></pre>
+<p>Here we get a performance benefit of a factor of 24 using the version
+of the differential equation model compiled from C code!</p>
+<p>This vignette was built with mkin 1.2.10 on</p>
+<pre><code><span><span class="co">## R version 4.4.2 (2024-10-31)</span></span>
+<span><span class="co">## Platform: x86_64-pc-linux-gnu</span></span>
+<span><span class="co">## Running under: Debian GNU/Linux 12 (bookworm)</span></span></code></pre>
+<pre><code><span><span class="co">## CPU model: AMD Ryzen 9 7950X 16-Core Processor</span></span></code></pre>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
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+
+    </footer>
+</div>
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+</html>
diff --git a/docs/dev/articles/web_only/dimethenamid_2018.html b/docs/dev/articles/web_only/dimethenamid_2018.html
new file mode 100644
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--- /dev/null
+++ b/docs/dev/articles/web_only/dimethenamid_2018.html
@@ -0,0 +1,651 @@
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+
+      <h1>Example evaluations of the dimethenamid data from 2018</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change 1 July 2022,
+built on 14 Feb 2025</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/web_only/dimethenamid_2018.rmd" class="external-link"><code>vignettes/web_only/dimethenamid_2018.rmd</code></a></small>
+      <div class="d-none name"><code>dimethenamid_2018.rmd</code></div>
+    </div>
+
+    
+    
+<p><a href="http://www.jrwb.de" class="external-link">Wissenschaftlicher Berater, Kronacher
+Str. 12, 79639 Grenzach-Wyhlen, Germany</a></p>
+<div class="section level2">
+<h2 id="introduction">Introduction<a class="anchor" aria-label="anchor" href="#introduction"></a>
+</h2>
+<p>A first analysis of the data analysed here was presented in a recent
+journal article on nonlinear mixed-effects models in degradation
+kinetics <span class="citation">(Ranke et al. 2021)</span>. That
+analysis was based on the <code>nlme</code> package and a development
+version of the <code>saemix</code> package that was unpublished at the
+time. Meanwhile, version 3.0 of the <code>saemix</code> package is
+available from the CRAN repository. Also, it turned out that there was
+an error in the handling of the Borstel data in the mkin package at the
+time, leading to the duplication of a few data points from that soil.
+The dataset in the mkin package has been corrected, and the interface to
+<code>saemix</code> in the mkin package has been updated to use the
+released version.</p>
+<p>This vignette is intended to present an up to date analysis of the
+data, using the corrected dataset and released versions of
+<code>mkin</code> and <code>saemix</code>.</p>
+</div>
+<div class="section level2">
+<h2 id="data">Data<a class="anchor" aria-label="anchor" href="#data"></a>
+</h2>
+<p>Residue data forming the basis for the endpoints derived in the
+conclusion on the peer review of the pesticide risk assessment of
+dimethenamid-P published by the European Food Safety Authority (EFSA) in
+2018 <span class="citation">(EFSA 2018)</span> were transcribed from the
+risk assessment report <span class="citation">(Rapporteur Member State
+Germany, Co-Rapporteur Member State Bulgaria 2018)</span> which can be
+downloaded from the Open EFSA repository <a href="https://open.efsa.europa.eu" class="external-link">https://open.efsa.europa.eu/study-inventory/EFSA-Q-2014-00716</a>.</p>
+<p>The data are <a href="https://pkgdown.jrwb.de/mkin/reference/dimethenamid_2018.html">available
+in the mkin package</a>. The following code (hidden by default, please
+use the button to the right to show it) treats the data available for
+the racemic mixture dimethenamid (DMTA) and its enantiomer
+dimethenamid-P (DMTAP) in the same way, as no difference between their
+degradation behaviour was identified in the EU risk assessment. The
+observation times of each dataset are multiplied with the corresponding
+normalisation factor also available in the dataset, in order to make it
+possible to describe all datasets with a single set of parameters.</p>
+<p>Also, datasets observed in the same soil are merged, resulting in
+dimethenamid (DMTA) data from six soils.</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://pkgdown.jrwb.de/mkin/">mkin</a></span>, quietly <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="va">dmta_ds</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">lapply</a></span><span class="op">(</span><span class="fl">1</span><span class="op">:</span><span class="fl">7</span>, <span class="kw">function</span><span class="op">(</span><span class="va">i</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="va">ds_i</span> <span class="op">&lt;-</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span><span class="op">[[</span><span class="va">i</span><span class="op">]</span><span class="op">]</span><span class="op">$</span><span class="va">data</span></span>
+<span>  <span class="va">ds_i</span><span class="op">[</span><span class="va">ds_i</span><span class="op">$</span><span class="va">name</span> <span class="op">==</span> <span class="st">"DMTAP"</span>, <span class="st">"name"</span><span class="op">]</span> <span class="op">&lt;-</span>  <span class="st">"DMTA"</span></span>
+<span>  <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">&lt;-</span> <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">*</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">f_time_norm</span><span class="op">[</span><span class="va">i</span><span class="op">]</span></span>
+<span>  <span class="va">ds_i</span></span>
+<span><span class="op">}</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">)</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">sapply</a></span><span class="op">(</span><span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span>, <span class="kw">function</span><span class="op">(</span><span class="va">ds</span><span class="op">)</span> <span class="va">ds</span><span class="op">$</span><span class="va">title</span><span class="op">)</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span>, <span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="cn">NULL</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="cn">NULL</span></span></code></pre></div>
+</div>
+<div class="section level2">
+<h2 id="parent-degradation">Parent degradation<a class="anchor" aria-label="anchor" href="#parent-degradation"></a>
+</h2>
+<p>We evaluate the observed degradation of the parent compound using
+simple exponential decline (SFO) and biexponential decline (DFOP), using
+constant variance (const) and a two-component variance (tc) as error
+models.</p>
+<div class="section level3">
+<h3 id="separate-evaluations">Separate evaluations<a class="anchor" aria-label="anchor" href="#separate-evaluations"></a>
+</h3>
+<p>As a first step, to get a visual impression of the fit of the
+different models, we do separate evaluations for each soil using the
+mmkin function from the mkin package:</p>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_parent_mkin_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"DFOP"</span><span class="op">)</span>, <span class="va">dmta_ds</span>,</span>
+<span>  error_model <span class="op">=</span> <span class="st">"const"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="va">f_parent_mkin_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"DFOP"</span><span class="op">)</span>, <span class="va">dmta_ds</span>,</span>
+<span>  error_model <span class="op">=</span> <span class="st">"tc"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<p>The plot of the individual SFO fits shown below suggests that at
+least in some datasets the degradation slows down towards later time
+points, and that the scatter of the residuals error is smaller for
+smaller values (panel to the right):</p>
+<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="fu"><a href="../../reference/mixed.html">mixed</a></span><span class="op">(</span><span class="va">f_parent_mkin_const</span><span class="op">[</span><span class="st">"SFO"</span>, <span class="op">]</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<p><img src="dimethenamid_2018_files/figure-html/f_parent_mkin_sfo_const-1.png" width="700"></p>
+<p>Using biexponential decline (DFOP) results in a slightly more random
+scatter of the residuals:</p>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="fu"><a href="../../reference/mixed.html">mixed</a></span><span class="op">(</span><span class="va">f_parent_mkin_const</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<p><img src="dimethenamid_2018_files/figure-html/f_parent_mkin_dfop_const-1.png" width="700"></p>
+<p>The population curve (bold line) in the above plot results from
+taking the mean of the individual transformed parameters, i.e. of log k1
+and log k2, as well as of the logit of the g parameter of the DFOP
+model). Here, this procedure does not result in parameters that
+represent the degradation well, because in some datasets the fitted
+value for k2 is extremely close to zero, leading to a log k2 value that
+dominates the average. This is alleviated if only rate constants that
+pass the t-test for significant difference from zero (on the
+untransformed scale) are considered in the averaging:</p>
+<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="fu"><a href="../../reference/mixed.html">mixed</a></span><span class="op">(</span><span class="va">f_parent_mkin_const</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span><span class="op">)</span>, test_log_parms <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<p><img src="dimethenamid_2018_files/figure-html/f_parent_mkin_dfop_const_test-1.png" width="700"></p>
+<p>While this is visually much more satisfactory, such an average
+procedure could introduce a bias, as not all results from the individual
+fits enter the population curve with the same weight. This is where
+nonlinear mixed-effects models can help out by treating all datasets
+with equally by fitting a parameter distribution model together with the
+degradation model and the error model (see below).</p>
+<p>The remaining trend of the residuals to be higher for higher
+predicted residues is reduced by using the two-component error
+model:</p>
+<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="fu"><a href="../../reference/mixed.html">mixed</a></span><span class="op">(</span><span class="va">f_parent_mkin_tc</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span><span class="op">)</span>, test_log_parms <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<p><img src="dimethenamid_2018_files/figure-html/f_parent_mkin_dfop_tc_test-1.png" width="700"></p>
+<p>However, note that in the case of using this error model, the fits to
+the Flaach and BBA 2.3 datasets appear to be ill-defined, indicated by
+the fact that they did not converge:</p>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">f_parent_mkin_tc</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<pre><code>&lt;mmkin&gt; object
+Status of individual fits:
+
+      dataset
+model  Calke Borstel Flaach BBA 2.2 BBA 2.3 Elliot
+  DFOP OK    OK      OK     OK      C       OK    
+
+C: Optimisation did not converge:
+iteration limit reached without convergence (10)
+OK: No warnings</code></pre>
+</div>
+<div class="section level3">
+<h3 id="nonlinear-mixed-effects-models">Nonlinear mixed-effects models<a class="anchor" aria-label="anchor" href="#nonlinear-mixed-effects-models"></a>
+</h3>
+<p>Instead of taking a model selection decision for each of the
+individual fits, we fit nonlinear mixed-effects models (using different
+fitting algorithms as implemented in different packages) and do model
+selection using all available data at the same time. In order to make
+sure that these decisions are not unduly influenced by the type of
+algorithm used, by implementation details or by the use of wrong control
+parameters, we compare the model selection results obtained with
+different R packages, with different algorithms and checking control
+parameters.</p>
+<div class="section level4">
+<h4 id="nlme">nlme<a class="anchor" aria-label="anchor" href="#nlme"></a>
+</h4>
+<p>The nlme package was the first R extension providing facilities to
+fit nonlinear mixed-effects models. We would like to do model selection
+from all four combinations of degradation models and error models based
+on the AIC. However, fitting the DFOP model with constant variance and
+using default control parameters results in an error, signalling that
+the maximum number of 50 iterations was reached, potentially indicating
+overparameterisation. Nevertheless, the algorithm converges when the
+two-component error model is used in combination with the DFOP model.
+This can be explained by the fact that the smaller residues observed at
+later sampling times get more weight when using the two-component error
+model which will counteract the tendency of the algorithm to try
+parameter combinations unsuitable for fitting these data.</p>
+<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://svn.r-project.org/R-packages/trunk/nlme/" class="external-link">nlme</a></span><span class="op">)</span></span>
+<span><span class="va">f_parent_nlme_sfo_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/nlme/man/nlme.html" class="external-link">nlme</a></span><span class="op">(</span><span class="va">f_parent_mkin_const</span><span class="op">[</span><span class="st">"SFO"</span>, <span class="op">]</span><span class="op">)</span></span>
+<span><span class="co"># f_parent_nlme_dfop_const &lt;- nlme(f_parent_mkin_const["DFOP", ])</span></span>
+<span><span class="va">f_parent_nlme_sfo_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/nlme/man/nlme.html" class="external-link">nlme</a></span><span class="op">(</span><span class="va">f_parent_mkin_tc</span><span class="op">[</span><span class="st">"SFO"</span>, <span class="op">]</span><span class="op">)</span></span>
+<span><span class="va">f_parent_nlme_dfop_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/nlme/man/nlme.html" class="external-link">nlme</a></span><span class="op">(</span><span class="va">f_parent_mkin_tc</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span><span class="op">)</span></span></code></pre></div>
+<p>Note that a certain degree of overparameterisation is also indicated
+by a warning obtained when fitting DFOP with the two-component error
+model (‘false convergence’ in the ‘LME step’ in iteration 3). However,
+as this warning does not occur in later iterations, and specifically not
+in the last of the 5 iterations, we can ignore this warning.</p>
+<p>The model comparison function of the nlme package can directly be
+applied to these fits showing a much lower AIC for the DFOP model fitted
+with the two-component error model. Also, the likelihood ratio test
+indicates that this difference is significant as the p-value is below
+0.0001.</p>
+<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span></span>
+<span>  <span class="va">f_parent_nlme_sfo_const</span>, <span class="va">f_parent_nlme_sfo_tc</span>, <span class="va">f_parent_nlme_dfop_tc</span></span>
+<span><span class="op">)</span></span></code></pre></div>
+<pre><code>                        Model df    AIC    BIC  logLik   Test L.Ratio p-value
+f_parent_nlme_sfo_const     1  5 796.60 811.82 -393.30                       
+f_parent_nlme_sfo_tc        2  6 798.60 816.86 -393.30 1 vs 2    0.00   0.998
+f_parent_nlme_dfop_tc       3 10 671.91 702.34 -325.95 2 vs 3  134.69  &lt;.0001</code></pre>
+<p>In addition to these fits, attempts were also made to include
+correlations between random effects by using the log Cholesky
+parameterisation of the matrix specifying them. The code used for these
+attempts can be made visible below.</p>
+<div class="sourceCode" id="cb12"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_parent_nlme_sfo_const_logchol</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/nlme/man/nlme.html" class="external-link">nlme</a></span><span class="op">(</span><span class="va">f_parent_mkin_const</span><span class="op">[</span><span class="st">"SFO"</span>, <span class="op">]</span>,</span>
+<span>  random <span class="op">=</span> <span class="fu">nlme</span><span class="fu">::</span><span class="fu"><a href="https://rdrr.io/pkg/nlme/man/pdLogChol.html" class="external-link">pdLogChol</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">DMTA_0</span> <span class="op">~</span> <span class="fl">1</span>, <span class="va">log_k_DMTA</span> <span class="op">~</span> <span class="fl">1</span><span class="op">)</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_parent_nlme_sfo_const</span>, <span class="va">f_parent_nlme_sfo_const_logchol</span><span class="op">)</span></span>
+<span><span class="va">f_parent_nlme_sfo_tc_logchol</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/nlme/man/nlme.html" class="external-link">nlme</a></span><span class="op">(</span><span class="va">f_parent_mkin_tc</span><span class="op">[</span><span class="st">"SFO"</span>, <span class="op">]</span>,</span>
+<span>  random <span class="op">=</span> <span class="fu">nlme</span><span class="fu">::</span><span class="fu"><a href="https://rdrr.io/pkg/nlme/man/pdLogChol.html" class="external-link">pdLogChol</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">DMTA_0</span> <span class="op">~</span> <span class="fl">1</span>, <span class="va">log_k_DMTA</span> <span class="op">~</span> <span class="fl">1</span><span class="op">)</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_parent_nlme_sfo_tc</span>, <span class="va">f_parent_nlme_sfo_tc_logchol</span><span class="op">)</span></span>
+<span><span class="va">f_parent_nlme_dfop_tc_logchol</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/nlme/man/nlme.html" class="external-link">nlme</a></span><span class="op">(</span><span class="va">f_parent_mkin_const</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span>,</span>
+<span>  random <span class="op">=</span> <span class="fu">nlme</span><span class="fu">::</span><span class="fu"><a href="https://rdrr.io/pkg/nlme/man/pdLogChol.html" class="external-link">pdLogChol</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">DMTA_0</span> <span class="op">~</span> <span class="fl">1</span>, <span class="va">log_k1</span> <span class="op">~</span> <span class="fl">1</span>, <span class="va">log_k2</span> <span class="op">~</span> <span class="fl">1</span>, <span class="va">g_qlogis</span> <span class="op">~</span> <span class="fl">1</span><span class="op">)</span><span class="op">)</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_parent_nlme_dfop_tc</span>, <span class="va">f_parent_nlme_dfop_tc_logchol</span><span class="op">)</span></span></code></pre></div>
+<p>While the SFO variants converge fast, the additional parameters
+introduced by this lead to convergence warnings for the DFOP model. The
+model comparison clearly show that adding correlations between random
+effects does not improve the fits.</p>
+<p>The selected model (DFOP with two-component error) fitted to the data
+assuming no correlations between random effects is shown below.</p>
+<div class="sourceCode" id="cb13"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_parent_nlme_dfop_tc</span><span class="op">)</span></span></code></pre></div>
+<p><img src="dimethenamid_2018_files/figure-html/plot_parent_nlme-1.png" width="700"></p>
+</div>
+<div class="section level4">
+<h4 id="saemix">saemix<a class="anchor" aria-label="anchor" href="#saemix"></a>
+</h4>
+<p>The saemix package provided the first Open Source implementation of
+the Stochastic Approximation to the Expectation Maximisation (SAEM)
+algorithm. SAEM fits of degradation models can be conveniently performed
+using an interface to the saemix package available in current
+development versions of the mkin package.</p>
+<p>The corresponding SAEM fits of the four combinations of degradation
+and error models are fitted below. As there is no convergence criterion
+implemented in the saemix package, the convergence plots need to be
+manually checked for every fit. We define control settings that work
+well for all the parent data fits shown in this vignette.</p>
+<div class="sourceCode" id="cb14"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">saemix</span><span class="op">)</span></span>
+<span><span class="va">saemix_control</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/saemix/man/saemixControl.html" class="external-link">saemixControl</a></span><span class="op">(</span>nbiter.saemix <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">800</span>, <span class="fl">300</span><span class="op">)</span>, nb.chains <span class="op">=</span> <span class="fl">15</span>,</span>
+<span>    print <span class="op">=</span> <span class="cn">FALSE</span>, save <span class="op">=</span> <span class="cn">FALSE</span>, save.graphs <span class="op">=</span> <span class="cn">FALSE</span>, displayProgress <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span></span>
+<span><span class="va">saemix_control_moreiter</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/saemix/man/saemixControl.html" class="external-link">saemixControl</a></span><span class="op">(</span>nbiter.saemix <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">1600</span>, <span class="fl">300</span><span class="op">)</span>, nb.chains <span class="op">=</span> <span class="fl">15</span>,</span>
+<span>    print <span class="op">=</span> <span class="cn">FALSE</span>, save <span class="op">=</span> <span class="cn">FALSE</span>, save.graphs <span class="op">=</span> <span class="cn">FALSE</span>, displayProgress <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span></span>
+<span><span class="va">saemix_control_10k</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/saemix/man/saemixControl.html" class="external-link">saemixControl</a></span><span class="op">(</span>nbiter.saemix <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">10000</span>, <span class="fl">300</span><span class="op">)</span>, nb.chains <span class="op">=</span> <span class="fl">15</span>,</span>
+<span>    print <span class="op">=</span> <span class="cn">FALSE</span>, save <span class="op">=</span> <span class="cn">FALSE</span>, save.graphs <span class="op">=</span> <span class="cn">FALSE</span>, displayProgress <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span></span></code></pre></div>
+<p>The convergence plot for the SFO model using constant variance is
+shown below.</p>
+<div class="sourceCode" id="cb15"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_parent_saemix_sfo_const</span> <span class="op">&lt;-</span> <span class="fu">mkin</span><span class="fu">::</span><span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_parent_mkin_const</span><span class="op">[</span><span class="st">"SFO"</span>, <span class="op">]</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>  control <span class="op">=</span> <span class="va">saemix_control</span>, transformations <span class="op">=</span> <span class="st">"saemix"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_parent_saemix_sfo_const</span><span class="op">$</span><span class="va">so</span>, plot.type <span class="op">=</span> <span class="st">"convergence"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="dimethenamid_2018_files/figure-html/f_parent_saemix_sfo_const-1.png" width="700"></p>
+<p>Obviously the selected number of iterations is sufficient to reach
+convergence. This can also be said for the SFO fit using the
+two-component error model.</p>
+<div class="sourceCode" id="cb16"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_parent_saemix_sfo_tc</span> <span class="op">&lt;-</span> <span class="fu">mkin</span><span class="fu">::</span><span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_parent_mkin_tc</span><span class="op">[</span><span class="st">"SFO"</span>, <span class="op">]</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>  control <span class="op">=</span> <span class="va">saemix_control</span>, transformations <span class="op">=</span> <span class="st">"saemix"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_parent_saemix_sfo_tc</span><span class="op">$</span><span class="va">so</span>, plot.type <span class="op">=</span> <span class="st">"convergence"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="dimethenamid_2018_files/figure-html/f_parent_saemix_sfo_tc-1.png" width="700"></p>
+<p>When fitting the DFOP model with constant variance (see below),
+parameter convergence is not as unambiguous.</p>
+<div class="sourceCode" id="cb17"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_parent_saemix_dfop_const</span> <span class="op">&lt;-</span> <span class="fu">mkin</span><span class="fu">::</span><span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_parent_mkin_const</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>  control <span class="op">=</span> <span class="va">saemix_control</span>, transformations <span class="op">=</span> <span class="st">"saemix"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_const</span><span class="op">$</span><span class="va">so</span>, plot.type <span class="op">=</span> <span class="st">"convergence"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="dimethenamid_2018_files/figure-html/f_parent_saemix_dfop_const-1.png" width="700"></p>
+<div class="sourceCode" id="cb18"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_const</span><span class="op">)</span></span></code></pre></div>
+<pre><code>Kinetic nonlinear mixed-effects model fit by SAEM
+Structural model:
+d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Likelihood computed by importance sampling
+  AIC BIC logLik
+  706 704   -344
+
+Fitted parameters:
+          estimate    lower   upper
+DMTA_0    97.99583 96.50079 99.4909
+k1         0.06377  0.03432  0.0932
+k2         0.00848  0.00444  0.0125
+g          0.95701  0.91313  1.0009
+a.1        1.82141  1.60516  2.0377
+SD.DMTA_0  1.64787  0.45729  2.8384
+SD.k1      0.57439  0.24731  0.9015
+SD.k2      0.03296 -2.50524  2.5712
+SD.g       1.10266  0.32354  1.8818</code></pre>
+<p>While the other parameters converge to credible values, the variance
+of k2 (<code>omega2.k2</code>) converges to a very small value. The
+printout of the <code>saem.mmkin</code> model shows that the estimated
+standard deviation of k2 across the population of soils
+(<code>SD.k2</code>) is ill-defined, indicating overparameterisation of
+this model.</p>
+<p>When the DFOP model is fitted with the two-component error model, we
+also observe that the estimated variance of k2 becomes very small, while
+being ill-defined, as illustrated by the excessive confidence interval
+of <code>SD.k2</code>.</p>
+<div class="sourceCode" id="cb20"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_parent_saemix_dfop_tc</span> <span class="op">&lt;-</span> <span class="fu">mkin</span><span class="fu">::</span><span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_parent_mkin_tc</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>  control <span class="op">=</span> <span class="va">saemix_control</span>, transformations <span class="op">=</span> <span class="st">"saemix"</span><span class="op">)</span></span>
+<span><span class="va">f_parent_saemix_dfop_tc_moreiter</span> <span class="op">&lt;-</span> <span class="fu">mkin</span><span class="fu">::</span><span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_parent_mkin_tc</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>,</span>
+<span>  control <span class="op">=</span> <span class="va">saemix_control_moreiter</span>, transformations <span class="op">=</span> <span class="st">"saemix"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_tc</span><span class="op">$</span><span class="va">so</span>, plot.type <span class="op">=</span> <span class="st">"convergence"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="dimethenamid_2018_files/figure-html/f_parent_saemix_dfop_tc-1.png" width="700"></p>
+<div class="sourceCode" id="cb21"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_tc</span><span class="op">)</span></span></code></pre></div>
+<pre><code>Kinetic nonlinear mixed-effects model fit by SAEM
+Structural model:
+d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Likelihood computed by importance sampling
+  AIC BIC logLik
+  666 664   -323
+
+Fitted parameters:
+          estimate     lower    upper
+DMTA_0    98.24165  96.29190 100.1914
+k1         0.06421   0.03352   0.0949
+k2         0.00866   0.00617   0.0111
+g          0.95340   0.91218   0.9946
+a.1        1.06463   0.87979   1.2495
+b.1        0.02964   0.02266   0.0366
+SD.DMTA_0  2.03611   0.40361   3.6686
+SD.k1      0.59534   0.25692   0.9338
+SD.k2      0.00042 -73.00540  73.0062
+SD.g       1.04234   0.37189   1.7128</code></pre>
+<p>Doubling the number of iterations in the first phase of the algorithm
+leads to a slightly lower likelihood, and therefore to slightly higher
+AIC and BIC values. With even more iterations, the algorithm stops with
+an error message. This is related to the variance of k2 approximating
+zero and has been submitted as a <a href="https://github.com/saemixdevelopment/saemixextension/issues/29" class="external-link">bug
+to the saemix package</a>, as the algorithm does not converge in this
+case.</p>
+<p>An alternative way to fit DFOP in combination with the two-component
+error model is to use the model formulation with transformed parameters
+as used per default in mkin. When using this option, convergence is
+slower, but eventually the algorithm stops as well with the same error
+message.</p>
+<p>The four combinations (SFO/const, SFO/tc, DFOP/const and DFOP/tc) and
+the version with increased iterations can be compared using the model
+comparison function of the saemix package:</p>
+<div class="sourceCode" id="cb23"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">AIC_parent_saemix</span> <span class="op">&lt;-</span> <span class="fu">saemix</span><span class="fu">::</span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/compare.saemix.html" class="external-link">compare.saemix</a></span><span class="op">(</span></span>
+<span>  <span class="va">f_parent_saemix_sfo_const</span><span class="op">$</span><span class="va">so</span>,</span>
+<span>  <span class="va">f_parent_saemix_sfo_tc</span><span class="op">$</span><span class="va">so</span>,</span>
+<span>  <span class="va">f_parent_saemix_dfop_const</span><span class="op">$</span><span class="va">so</span>,</span>
+<span>  <span class="va">f_parent_saemix_dfop_tc</span><span class="op">$</span><span class="va">so</span>,</span>
+<span>  <span class="va">f_parent_saemix_dfop_tc_moreiter</span><span class="op">$</span><span class="va">so</span><span class="op">)</span></span></code></pre></div>
+<pre><code>Likelihoods calculated by importance sampling</code></pre>
+<div class="sourceCode" id="cb25"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/colnames.html" class="external-link">rownames</a></span><span class="op">(</span><span class="va">AIC_parent_saemix</span><span class="op">)</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span></span>
+<span>  <span class="st">"SFO const"</span>, <span class="st">"SFO tc"</span>, <span class="st">"DFOP const"</span>, <span class="st">"DFOP tc"</span>, <span class="st">"DFOP tc more iterations"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">AIC_parent_saemix</span><span class="op">)</span></span></code></pre></div>
+<pre><code>                           AIC    BIC
+SFO const               796.38 795.34
+SFO tc                  798.38 797.13
+DFOP const              705.75 703.88
+DFOP tc                 665.67 663.59
+DFOP tc more iterations 665.85 663.76</code></pre>
+<p>In order to check the influence of the likelihood calculation
+algorithms implemented in saemix, the likelihood from Gaussian
+quadrature is added to the best fit, and the AIC values obtained from
+the three methods are compared.</p>
+<div class="sourceCode" id="cb27"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_parent_saemix_dfop_tc</span><span class="op">$</span><span class="va">so</span> <span class="op">&lt;-</span></span>
+<span>  <span class="fu">saemix</span><span class="fu">::</span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/llgq.saemix.html" class="external-link">llgq.saemix</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_tc</span><span class="op">$</span><span class="va">so</span><span class="op">)</span></span>
+<span><span class="va">AIC_parent_saemix_methods</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span></span>
+<span>  is <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/stats/AIC.html" class="external-link">AIC</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_tc</span><span class="op">$</span><span class="va">so</span>, method <span class="op">=</span> <span class="st">"is"</span><span class="op">)</span>,</span>
+<span>  gq <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/stats/AIC.html" class="external-link">AIC</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_tc</span><span class="op">$</span><span class="va">so</span>, method <span class="op">=</span> <span class="st">"gq"</span><span class="op">)</span>,</span>
+<span>  lin <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/stats/AIC.html" class="external-link">AIC</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_tc</span><span class="op">$</span><span class="va">so</span>, method <span class="op">=</span> <span class="st">"lin"</span><span class="op">)</span></span>
+<span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">AIC_parent_saemix_methods</span><span class="op">)</span></span></code></pre></div>
+<pre><code>    is     gq    lin 
+665.67 665.74 665.13 </code></pre>
+<p>The AIC values based on importance sampling and Gaussian quadrature
+are very similar. Using linearisation is known to be less accurate, but
+still gives a similar value.</p>
+<p>In order to illustrate that the comparison of the three method
+depends on the degree of convergence obtained in the fit, the same
+comparison is shown below for the fit using the defaults for the number
+of iterations and the number of MCMC chains.</p>
+<p>When using OpenBlas for linear algebra, there is a large difference
+in the values obtained with Gaussian quadrature, so the larger number of
+iterations makes a lot of difference. When using the LAPACK version
+coming with Debian Bullseye, the AIC based on Gaussian quadrature is
+almost the same as the one obtained with the other methods, also when
+using defaults for the fit.</p>
+<div class="sourceCode" id="cb29"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_parent_saemix_dfop_tc_defaults</span> <span class="op">&lt;-</span> <span class="fu">mkin</span><span class="fu">::</span><span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_parent_mkin_tc</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span><span class="op">)</span></span>
+<span><span class="va">f_parent_saemix_dfop_tc_defaults</span><span class="op">$</span><span class="va">so</span> <span class="op">&lt;-</span></span>
+<span>  <span class="fu">saemix</span><span class="fu">::</span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/llgq.saemix.html" class="external-link">llgq.saemix</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_tc_defaults</span><span class="op">$</span><span class="va">so</span><span class="op">)</span></span>
+<span><span class="va">AIC_parent_saemix_methods_defaults</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span></span>
+<span>  is <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/stats/AIC.html" class="external-link">AIC</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_tc_defaults</span><span class="op">$</span><span class="va">so</span>, method <span class="op">=</span> <span class="st">"is"</span><span class="op">)</span>,</span>
+<span>  gq <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/stats/AIC.html" class="external-link">AIC</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_tc_defaults</span><span class="op">$</span><span class="va">so</span>, method <span class="op">=</span> <span class="st">"gq"</span><span class="op">)</span>,</span>
+<span>  lin <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/stats/AIC.html" class="external-link">AIC</a></span><span class="op">(</span><span class="va">f_parent_saemix_dfop_tc_defaults</span><span class="op">$</span><span class="va">so</span>, method <span class="op">=</span> <span class="st">"lin"</span><span class="op">)</span></span>
+<span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="va">AIC_parent_saemix_methods_defaults</span><span class="op">)</span></span></code></pre></div>
+<pre><code>    is     gq    lin 
+670.09 669.37 671.29 </code></pre>
+</div>
+</div>
+<div class="section level3">
+<h3 id="comparison">Comparison<a class="anchor" aria-label="anchor" href="#comparison"></a>
+</h3>
+<p>The following table gives the AIC values obtained with both backend
+packages using the same control parameters (800 iterations burn-in, 300
+iterations second phase, 15 chains).</p>
+<div class="sourceCode" id="cb31"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">AIC_all</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html" class="external-link">data.frame</a></span><span class="op">(</span></span>
+<span>  check.names <span class="op">=</span> <span class="cn">FALSE</span>,</span>
+<span>  <span class="st">"Degradation model"</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"SFO"</span>, <span class="st">"DFOP"</span>, <span class="st">"DFOP"</span><span class="op">)</span>,</span>
+<span>  <span class="st">"Error model"</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"const"</span>, <span class="st">"tc"</span>, <span class="st">"const"</span>, <span class="st">"tc"</span><span class="op">)</span>,</span>
+<span>  nlme <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/stats/AIC.html" class="external-link">AIC</a></span><span class="op">(</span><span class="va">f_parent_nlme_sfo_const</span><span class="op">)</span>, <span class="fu"><a href="https://rdrr.io/r/stats/AIC.html" class="external-link">AIC</a></span><span class="op">(</span><span class="va">f_parent_nlme_sfo_tc</span><span class="op">)</span>, <span class="cn">NA</span>, <span class="fu"><a href="https://rdrr.io/r/stats/AIC.html" class="external-link">AIC</a></span><span class="op">(</span><span class="va">f_parent_nlme_dfop_tc</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>  saemix_lin <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">sapply</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">f_parent_saemix_sfo_const</span><span class="op">$</span><span class="va">so</span>, <span class="va">f_parent_saemix_sfo_tc</span><span class="op">$</span><span class="va">so</span>,</span>
+<span>    <span class="va">f_parent_saemix_dfop_const</span><span class="op">$</span><span class="va">so</span>, <span class="va">f_parent_saemix_dfop_tc</span><span class="op">$</span><span class="va">so</span><span class="op">)</span>, <span class="va">AIC</span>, method <span class="op">=</span> <span class="st">"lin"</span><span class="op">)</span>,</span>
+<span>  saemix_is <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">sapply</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">f_parent_saemix_sfo_const</span><span class="op">$</span><span class="va">so</span>, <span class="va">f_parent_saemix_sfo_tc</span><span class="op">$</span><span class="va">so</span>,</span>
+<span>    <span class="va">f_parent_saemix_dfop_const</span><span class="op">$</span><span class="va">so</span>, <span class="va">f_parent_saemix_dfop_tc</span><span class="op">$</span><span class="va">so</span><span class="op">)</span>, <span class="va">AIC</span>, method <span class="op">=</span> <span class="st">"is"</span><span class="op">)</span></span>
+<span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="va">AIC_all</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left">Degradation model</th>
+<th align="left">Error model</th>
+<th align="right">nlme</th>
+<th align="right">saemix_lin</th>
+<th align="right">saemix_is</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">SFO</td>
+<td align="left">const</td>
+<td align="right">796.60</td>
+<td align="right">796.60</td>
+<td align="right">796.38</td>
+</tr>
+<tr class="even">
+<td align="left">SFO</td>
+<td align="left">tc</td>
+<td align="right">798.60</td>
+<td align="right">798.60</td>
+<td align="right">798.38</td>
+</tr>
+<tr class="odd">
+<td align="left">DFOP</td>
+<td align="left">const</td>
+<td align="right">NA</td>
+<td align="right">704.95</td>
+<td align="right">705.75</td>
+</tr>
+<tr class="even">
+<td align="left">DFOP</td>
+<td align="left">tc</td>
+<td align="right">671.91</td>
+<td align="right">665.13</td>
+<td align="right">665.67</td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="section level2">
+<h2 id="conclusion">Conclusion<a class="anchor" aria-label="anchor" href="#conclusion"></a>
+</h2>
+<p>A more detailed analysis of the dimethenamid dataset confirmed that
+the DFOP model provides the most appropriate description of the decline
+of the parent compound in these data. On the other hand, closer
+inspection of the results revealed that the variability of the k2
+parameter across the population of soils is ill-defined. This coincides
+with the observation that this parameter cannot robustly be quantified
+for some of the soils.</p>
+<p>Regarding the regulatory use of these data, it is claimed that an
+improved characterisation of the mean parameter values across the
+population is obtained using the nonlinear mixed-effects models
+presented here. However, attempts to quantify the variability of the
+slower rate constant of the biphasic decline of dimethenamid indicate
+that the data are not sufficient to characterise this variability to a
+satisfactory precision.</p>
+</div>
+<div class="section level2">
+<h2 id="session-info">Session Info<a class="anchor" aria-label="anchor" href="#session-info"></a>
+</h2>
+<div class="sourceCode" id="cb32"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/utils/sessionInfo.html" class="external-link">sessionInfo</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<pre><code>R version 4.4.2 (2024-10-31)
+Platform: x86_64-pc-linux-gnu
+Running under: Debian GNU/Linux 12 (bookworm)
+
+Matrix products: default
+BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.11.0 
+LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.11.0
+
+locale:
+ [1] LC_CTYPE=de_DE.UTF-8       LC_NUMERIC=C              
+ [3] LC_TIME=C                  LC_COLLATE=de_DE.UTF-8    
+ [5] LC_MONETARY=de_DE.UTF-8    LC_MESSAGES=de_DE.UTF-8   
+ [7] LC_PAPER=de_DE.UTF-8       LC_NAME=C                 
+ [9] LC_ADDRESS=C               LC_TELEPHONE=C            
+[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C       
+
+time zone: Europe/Berlin
+tzcode source: system (glibc)
+
+attached base packages:
+[1] stats     graphics  grDevices utils     datasets  methods   base     
+
+other attached packages:
+[1] saemix_3.3   npde_3.5     nlme_3.1-166 mkin_1.2.10  knitr_1.49  
+
+loaded via a namespace (and not attached):
+ [1] gtable_0.3.6      jsonlite_1.8.9    dplyr_1.1.4       compiler_4.4.2   
+ [5] tidyselect_1.2.1  parallel_4.4.2    gridExtra_2.3     jquerylib_0.1.4  
+ [9] systemfonts_1.1.0 scales_1.3.0      textshaping_0.4.1 yaml_2.3.10      
+[13] fastmap_1.2.0     lattice_0.22-6    ggplot2_3.5.1     R6_2.5.1         
+[17] generics_0.1.3    lmtest_0.9-40     MASS_7.3-61       htmlwidgets_1.6.4
+[21] tibble_3.2.1      desc_1.4.3        munsell_0.5.1     bslib_0.8.0      
+[25] pillar_1.9.0      rlang_1.1.4       utf8_1.2.4        cachem_1.1.0     
+[29] xfun_0.49         fs_1.6.5          sass_0.4.9        cli_3.6.3        
+[33] pkgdown_2.1.1     magrittr_2.0.3    digest_0.6.37     grid_4.4.2       
+[37] mclust_6.1.1      lifecycle_1.0.4   vctrs_0.6.5       evaluate_1.0.1   
+[41] glue_1.8.0        codetools_0.2-20  ragg_1.3.3        zoo_1.8-12       
+[45] fansi_1.0.6       colorspace_2.1-1  rmarkdown_2.29    pkgconfig_2.0.3  
+[49] tools_4.4.2       htmltools_0.5.8.1</code></pre>
+</div>
+<div class="section level2">
+<h2 id="references">References<a class="anchor" aria-label="anchor" href="#references"></a>
+</h2>
+<!-- vim: set foldmethod=syntax: -->
+<div id="refs" class="references csl-bib-body hanging-indent">
+<div id="ref-efsa_2018_dimethenamid" class="csl-entry">
+EFSA. 2018. <span>“Peer Review of the Pesticide Risk Assessment of the
+Active Substance Dimethenamid-p.”</span> <em>EFSA Journal</em> 16: 5211.
+</div>
+<div id="ref-ranke2021" class="csl-entry">
+Ranke, Johannes, Janina Wöltjen, Jana Schmidt, and Emmanuelle Comets.
+2021. <span>“Taking Kinetic Evaluations of Degradation Data to the Next
+Level with Nonlinear Mixed-Effects Models.”</span> <em>Environments</em>
+8 (8). <a href="https://doi.org/10.3390/environments8080071" class="external-link">https://doi.org/10.3390/environments8080071</a>.
+</div>
+<div id="ref-dimethenamid_rar_2018_b8" class="csl-entry">
+Rapporteur Member State Germany, Co-Rapporteur Member State Bulgaria.
+2018. <span>“<span class="nocase">Renewal Assessment Report
+Dimethenamid-P Volume 3 - B.8 Environmental fate and behaviour, Rev. 2 -
+November 2017</span>.”</span> <a href="https://open.efsa.europa.eu/study-inventory/EFSA-Q-2014-00716" class="external-link">https://open.efsa.europa.eu/study-inventory/EFSA-Q-2014-00716</a>.
+</div>
+</div>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
+</div>
+
+    </footer>
+</div>
+
+
+
+
+
+  </body>
+</html>
diff --git a/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/f_parent_mkin_dfop_const-1.png b/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/f_parent_mkin_dfop_const-1.png
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diff --git a/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/f_parent_mkin_sfo_const-1.png b/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/f_parent_mkin_sfo_const-1.png
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diff --git a/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/f_parent_saemix_dfop_const-1.png b/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/f_parent_saemix_dfop_const-1.png
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diff --git a/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/f_parent_saemix_dfop_tc-1.png b/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/f_parent_saemix_dfop_tc-1.png
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diff --git a/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/f_parent_saemix_sfo_const-1.png b/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/f_parent_saemix_sfo_const-1.png
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diff --git a/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/plot_parent_nlme-1.png b/docs/dev/articles/web_only/dimethenamid_2018_files/figure-html/plot_parent_nlme-1.png
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diff --git a/docs/dev/articles/web_only/multistart.html b/docs/dev/articles/web_only/multistart.html
new file mode 100644
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--- /dev/null
+++ b/docs/dev/articles/web_only/multistart.html
@@ -0,0 +1,179 @@
+<!DOCTYPE html>
+<!-- Generated by pkgdown: do not edit by hand --><html lang="en">
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+<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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+<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
+<title>Short demo of the multistart method • mkin</title>
+<script src="../../deps/jquery-3.6.0/jquery-3.6.0.min.js"></script><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
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+    <small class="nav-text text-info me-auto" data-bs-toggle="tooltip" data-bs-placement="bottom" title="In-development version">1.2.10</small>
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+<li><a class="dropdown-item" href="../../articles/mkin.html">Introduction to mkin</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with (generalised) nonlinear least squares</h6></li>
+    <li><a class="dropdown-item" href="../../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a></li>
+    <li><a class="dropdown-item" href="../../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a></li>
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+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with hierarchical models (nonlinear mixed-effects models)</h6></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_pathway.html">Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2023_mesotrione_parent.html">Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_cyan_pathway.html">Testing hierarchical pathway kinetics with residue data on cyantraniliprole</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/dimethenamid_2018.html">Comparison of saemix and nlme evaluations of dimethenamid data from 2018</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/multistart.html">Short demo of the multistart method</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Performance</h6></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/compiled_models.html">Performance benefit by using compiled model definitions in mkin</a></li>
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+    <li><a class="dropdown-item" href="../../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a></li>
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+
+      <h1>Short demo of the multistart method</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change 20 April 2023
+(rebuilt 2025-02-14)</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/web_only/multistart.rmd" class="external-link"><code>vignettes/web_only/multistart.rmd</code></a></small>
+      <div class="d-none name"><code>multistart.rmd</code></div>
+    </div>
+
+    
+    
+<p>The dimethenamid data from 2018 from seven soils is used as example
+data in this vignette.</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://pkgdown.jrwb.de/mkin/">mkin</a></span><span class="op">)</span></span>
+<span><span class="va">dmta_ds</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">lapply</a></span><span class="op">(</span><span class="fl">1</span><span class="op">:</span><span class="fl">7</span>, <span class="kw">function</span><span class="op">(</span><span class="va">i</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="va">ds_i</span> <span class="op">&lt;-</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span><span class="op">[[</span><span class="va">i</span><span class="op">]</span><span class="op">]</span><span class="op">$</span><span class="va">data</span></span>
+<span>  <span class="va">ds_i</span><span class="op">[</span><span class="va">ds_i</span><span class="op">$</span><span class="va">name</span> <span class="op">==</span> <span class="st">"DMTAP"</span>, <span class="st">"name"</span><span class="op">]</span> <span class="op">&lt;-</span>  <span class="st">"DMTA"</span></span>
+<span>  <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">&lt;-</span> <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">*</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">f_time_norm</span><span class="op">[</span><span class="va">i</span><span class="op">]</span></span>
+<span>  <span class="va">ds_i</span></span>
+<span><span class="op">}</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">)</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">sapply</a></span><span class="op">(</span><span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span>, <span class="kw">function</span><span class="op">(</span><span class="va">ds</span><span class="op">)</span> <span class="va">ds</span><span class="op">$</span><span class="va">title</span><span class="op">)</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span>, <span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="cn">NULL</span></span></code></pre></div>
+<p>First, we check the DFOP model with the two-component error model and
+random effects for all degradation parameters.</p>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_mmkin</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="st">"DFOP"</span>, <span class="va">dmta_ds</span>, error_model <span class="op">=</span> <span class="st">"tc"</span>, cores <span class="op">=</span> <span class="fl">7</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="va">f_saem_full</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_mmkin</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_full</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## [1] "sd(log_k2)"</span></span></code></pre>
+<p>We see that not all variability parameters are identifiable. The
+<code>illparms</code> function tells us that the confidence interval for
+the standard deviation of ‘log_k2’ includes zero. We check this
+assessment using multiple runs with different starting values.</p>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_saem_full_multi</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/multistart.html">multistart</a></span><span class="op">(</span><span class="va">f_saem_full</span>, n <span class="op">=</span> <span class="fl">16</span>, cores <span class="op">=</span> <span class="fl">16</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/parplot.html">parplot</a></span><span class="op">(</span><span class="va">f_saem_full_multi</span>, lpos <span class="op">=</span> <span class="st">"topleft"</span><span class="op">)</span></span></code></pre></div>
+<p><img src="multistart_files/figure-html/unnamed-chunk-3-1.png" width="700"></p>
+<p>This confirms that the variance of k2 is the most problematic
+parameter, so we reduce the parameter distribution model by removing the
+intersoil variability for k2.</p>
+<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">f_saem_reduced</span> <span class="op">&lt;-</span> <span class="fu">stats</span><span class="fu">::</span><span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_saem_full</span>, no_random_effect <span class="op">=</span> <span class="st">"log_k2"</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_reduced</span><span class="op">)</span></span>
+<span><span class="va">f_saem_reduced_multi</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/multistart.html">multistart</a></span><span class="op">(</span><span class="va">f_saem_reduced</span>, n <span class="op">=</span> <span class="fl">16</span>, cores <span class="op">=</span> <span class="fl">16</span><span class="op">)</span></span>
+<span><span class="fu"><a href="../../reference/parplot.html">parplot</a></span><span class="op">(</span><span class="va">f_saem_reduced_multi</span>, lpos <span class="op">=</span> <span class="st">"topright"</span>, ylim <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.5</span>, <span class="fl">2</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
+<p><img src="multistart_files/figure-html/unnamed-chunk-4-1.png" width="700"></p>
+<p>The results confirm that all remaining parameters can be determined
+with sufficient certainty.</p>
+<p>We can also analyse the log-likelihoods obtained in the multiple
+runs:</p>
+<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/llhist.html">llhist</a></span><span class="op">(</span><span class="va">f_saem_reduced_multi</span><span class="op">)</span></span></code></pre></div>
+<p><img src="multistart_files/figure-html/unnamed-chunk-5-1.png" width="700"></p>
+<p>We can use the <code>anova</code> method to compare the models.</p>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_full</span>, <span class="fu"><a href="../../reference/multistart.html">best</a></span><span class="op">(</span><span class="va">f_saem_full_multi</span><span class="op">)</span>,</span>
+<span>  <span class="va">f_saem_reduced</span>, <span class="fu"><a href="../../reference/multistart.html">best</a></span><span class="op">(</span><span class="va">f_saem_reduced_multi</span><span class="op">)</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## Data: 155 observations of 1 variable(s) grouped in 6 datasets</span></span>
+<span><span class="co">## </span></span>
+<span><span class="co">##                            npar    AIC    BIC     Lik  Chisq Df Pr(&gt;Chisq)</span></span>
+<span><span class="co">## f_saem_reduced                9 663.67 661.80 -322.84                     </span></span>
+<span><span class="co">## best(f_saem_reduced_multi)    9 663.65 661.78 -322.82 0.0219  0           </span></span>
+<span><span class="co">## f_saem_full                  10 670.09 668.01 -325.05 0.0000  1          1</span></span>
+<span><span class="co">## best(f_saem_full_multi)      10 665.61 663.52 -322.80 4.4870  0</span></span></code></pre>
+<p>The reduced model results in lower AIC and BIC values, so it is
+clearly preferable. Using multiple starting values gives a large
+improvement in case of the full model, because it is less well-defined,
+which impedes convergence. For the reduced model, using multiple
+starting values only results in a small improvement of the model
+fit.</p>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
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+
+    </footer>
+</div>
+
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+</html>
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@@ -0,0 +1,789 @@
+<!DOCTYPE html>
+<!-- Generated by pkgdown: do not edit by hand --><html lang="en">
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+<li><a class="dropdown-item" href="../../articles/mkin.html">Introduction to mkin</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with (generalised) nonlinear least squares</h6></li>
+    <li><a class="dropdown-item" href="../../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a></li>
+    <li><a class="dropdown-item" href="../../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/FOCUS_Z.html">Example evaluation of FOCUS Example Dataset Z</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Example evaluations with hierarchical models (nonlinear mixed-effects models)</h6></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_pathway.html">Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2023_mesotrione_parent.html">Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</a></li>
+    <li><a class="dropdown-item" href="../../articles/prebuilt/2022_cyan_pathway.html">Testing hierarchical pathway kinetics with residue data on cyantraniliprole</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/dimethenamid_2018.html">Comparison of saemix and nlme evaluations of dimethenamid data from 2018</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/multistart.html">Short demo of the multistart method</a></li>
+    <li><hr class="dropdown-divider"></li>
+    <li><h6 class="dropdown-header" data-toc-skip>Performance</h6></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/compiled_models.html">Performance benefit by using compiled model definitions in mkin</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/benchmarks.html">Benchmark timings for mkin</a></li>
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+    <li><a class="dropdown-item" href="../../articles/twa.html">Calculation of time weighted average concentrations with mkin</a></li>
+    <li><a class="dropdown-item" href="../../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a></li>
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+
+      <h1>Benchmark timings for saem.mmkin</h1>
+                        <h4 data-toc-skip class="author">Johannes
+Ranke</h4>
+            
+            <h4 data-toc-skip class="date">Last change 17 February 2023
+(rebuilt 2025-02-14)</h4>
+      
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/web_only/saem_benchmarks.rmd" class="external-link"><code>vignettes/web_only/saem_benchmarks.rmd</code></a></small>
+      <div class="d-none name"><code>saem_benchmarks.rmd</code></div>
+    </div>
+
+    
+    
+<p>Each system is characterized by operating system type, CPU type, mkin
+version, saemix version and R version. A compiler was available, so if
+no analytical solution was available, compiled ODE models are used.</p>
+<p>Every fit is only performed once, so the accuracy of the benchmarks
+is limited.</p>
+<p>For the initial mmkin fits, we use all available cores.</p>
+<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">n_cores</span> <span class="op">&lt;-</span> <span class="fu">parallel</span><span class="fu">::</span><span class="fu"><a href="https://rdrr.io/r/parallel/detectCores.html" class="external-link">detectCores</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
+<div class="section level2">
+<h2 id="test-data">Test data<a class="anchor" aria-label="anchor" href="#test-data"></a>
+</h2>
+<p>Please refer to the vignette <code>dimethenamid_2018</code> for an
+explanation of the following preprocessing.</p>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">dmta_ds</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">lapply</a></span><span class="op">(</span><span class="fl">1</span><span class="op">:</span><span class="fl">7</span>, <span class="kw">function</span><span class="op">(</span><span class="va">i</span><span class="op">)</span> <span class="op">{</span></span>
+<span>  <span class="va">ds_i</span> <span class="op">&lt;-</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span><span class="op">[[</span><span class="va">i</span><span class="op">]</span><span class="op">]</span><span class="op">$</span><span class="va">data</span></span>
+<span>  <span class="va">ds_i</span><span class="op">[</span><span class="va">ds_i</span><span class="op">$</span><span class="va">name</span> <span class="op">==</span> <span class="st">"DMTAP"</span>, <span class="st">"name"</span><span class="op">]</span> <span class="op">&lt;-</span>  <span class="st">"DMTA"</span></span>
+<span>  <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">&lt;-</span> <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">*</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">f_time_norm</span><span class="op">[</span><span class="va">i</span><span class="op">]</span></span>
+<span>  <span class="va">ds_i</span></span>
+<span><span class="op">}</span><span class="op">)</span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">)</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">sapply</a></span><span class="op">(</span><span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span>, <span class="kw">function</span><span class="op">(</span><span class="va">ds</span><span class="op">)</span> <span class="va">ds</span><span class="op">$</span><span class="va">title</span><span class="op">)</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span>, <span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="cn">NULL</span></span>
+<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="cn">NULL</span></span></code></pre></div>
+</div>
+<div class="section level2">
+<h2 id="test-cases">Test cases<a class="anchor" aria-label="anchor" href="#test-cases"></a>
+</h2>
+<div class="section level3">
+<h3 id="parent-only">Parent only<a class="anchor" aria-label="anchor" href="#parent-only"></a>
+</h3>
+<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">parent_mods</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"DFOP"</span>, <span class="st">"SFORB"</span>, <span class="st">"HS"</span><span class="op">)</span></span>
+<span><span class="va">parent_sep_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="va">parent_mods</span>, <span class="va">dmta_ds</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span>, cores <span class="op">=</span> <span class="va">n_cores</span><span class="op">)</span></span>
+<span><span class="va">parent_sep_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">parent_sep_const</span>, error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">t1</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="va">sfo_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">parent_sep_const</span><span class="op">[</span><span class="st">"SFO"</span>, <span class="op">]</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t2</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="va">dfop_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">parent_sep_const</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t3</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="va">sforb_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">parent_sep_const</span><span class="op">[</span><span class="st">"SFORB"</span>, <span class="op">]</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t4</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="va">hs_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">parent_sep_const</span><span class="op">[</span><span class="st">"HS"</span>, <span class="op">]</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t5</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="va">sfo_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">parent_sep_tc</span><span class="op">[</span><span class="st">"SFO"</span>, <span class="op">]</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t6</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="va">dfop_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">parent_sep_tc</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t7</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="va">sforb_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">parent_sep_tc</span><span class="op">[</span><span class="st">"SFORB"</span>, <span class="op">]</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t8</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="va">hs_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">parent_sep_tc</span><span class="op">[</span><span class="st">"HS"</span>, <span class="op">]</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span></code></pre></div>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span></span>
+<span>  <span class="va">sfo_const</span>, <span class="va">dfop_const</span>, <span class="va">sforb_const</span>, <span class="va">hs_const</span>,</span>
+<span>  <span class="va">sfo_tc</span>, <span class="va">dfop_tc</span>, <span class="va">sforb_tc</span>, <span class="va">hs_tc</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>, digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<table class="table">
+<thead><tr class="header">
+<th align="left"></th>
+<th align="right">npar</th>
+<th align="right">AIC</th>
+<th align="right">BIC</th>
+<th align="right">Lik</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">sfo_const</td>
+<td align="right">5</td>
+<td align="right">796.3</td>
+<td align="right">795.3</td>
+<td align="right">-393.2</td>
+</tr>
+<tr class="even">
+<td align="left">sfo_tc</td>
+<td align="right">6</td>
+<td align="right">798.3</td>
+<td align="right">797.1</td>
+<td align="right">-393.2</td>
+</tr>
+<tr class="odd">
+<td align="left">dfop_const</td>
+<td align="right">9</td>
+<td align="right">709.4</td>
+<td align="right">707.5</td>
+<td align="right">-345.7</td>
+</tr>
+<tr class="even">
+<td align="left">sforb_const</td>
+<td align="right">9</td>
+<td align="right">710.0</td>
+<td align="right">708.1</td>
+<td align="right">-346.0</td>
+</tr>
+<tr class="odd">
+<td align="left">hs_const</td>
+<td align="right">9</td>
+<td align="right">713.7</td>
+<td align="right">711.8</td>
+<td align="right">-347.8</td>
+</tr>
+<tr class="even">
+<td align="left">dfop_tc</td>
+<td align="right">10</td>
+<td align="right">670.1</td>
+<td align="right">668.0</td>
+<td align="right">-325.0</td>
+</tr>
+<tr class="odd">
+<td align="left">sforb_tc</td>
+<td align="right">10</td>
+<td align="right">662.9</td>
+<td align="right">660.8</td>
+<td align="right">-321.4</td>
+</tr>
+<tr class="even">
+<td align="left">hs_tc</td>
+<td align="right">10</td>
+<td align="right">667.2</td>
+<td align="right">665.1</td>
+<td align="right">-323.6</td>
+</tr>
+</tbody>
+</table>
+<p>The above model comparison suggests to use the SFORB model with
+two-component error. For comparison, we keep the DFOP model with
+two-component error, as it competes with SFORB for biphasic curves.</p>
+<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">dfop_tc</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## [1] "sd(log_k2)"</span></span></code></pre>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">sforb_tc</span><span class="op">)</span></span></code></pre></div>
+<pre><code><span><span class="co">## [1] "sd(log_k_DMTA_bound_free)"</span></span></code></pre>
+<p>For these two models, random effects for the transformed parameters
+<code>k2</code> and <code>k_DMTA_bound_free</code> could not be
+quantified.</p>
+</div>
+<div class="section level3">
+<h3 id="one-metabolite">One metabolite<a class="anchor" aria-label="anchor" href="#one-metabolite"></a>
+</h3>
+<p>We remove parameters that were found to be ill-defined in the parent
+only fits.</p>
+<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">one_met_mods</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span></span>
+<span>  DFOP_SFO <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    DMTA <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"DFOP"</span>, <span class="st">"M23"</span><span class="op">)</span>,</span>
+<span>    M23 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>  SFORB_SFO <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    DMTA <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFORB"</span>, <span class="st">"M23"</span><span class="op">)</span>,</span>
+<span>    M23 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">one_met_sep_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="va">one_met_mods</span>, <span class="va">dmta_ds</span>, error_model <span class="op">=</span> <span class="st">"const"</span>,</span>
+<span>  cores <span class="op">=</span> <span class="va">n_cores</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span><span class="va">one_met_sep_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="va">one_met_mods</span>, <span class="va">dmta_ds</span>, error_model <span class="op">=</span> <span class="st">"tc"</span>,</span>
+<span>  cores <span class="op">=</span> <span class="va">n_cores</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">t9</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="va">dfop_sfo_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">one_met_sep_tc</span><span class="op">[</span><span class="st">"DFOP_SFO"</span>, <span class="op">]</span>,</span>
+<span>    no_random_effect <span class="op">=</span> <span class="st">"log_k2"</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span>
+<span><span class="va">t10</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="va">sforb_sfo_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">one_met_sep_tc</span><span class="op">[</span><span class="st">"SFORB_SFO"</span>, <span class="op">]</span>,</span>
+<span>    no_random_effect <span class="op">=</span> <span class="st">"log_k_DMTA_bound_free"</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span></code></pre></div>
+</div>
+<div class="section level3">
+<h3 id="three-metabolites">Three metabolites<a class="anchor" aria-label="anchor" href="#three-metabolites"></a>
+</h3>
+<p>For the case of three metabolites, we only keep the SFORB model in
+order to limit the time for compiling this vignette, and as fitting in
+parallel may disturb the benchmark. Again, we do not include random
+effects that were ill-defined in previous fits of subsets of the
+degradation model.</p>
+<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">sforb_sfo_tc</span><span class="op">)</span></span></code></pre></div>
+<div class="sourceCode" id="cb11"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">three_met_mods</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span></span>
+<span>  SFORB_SFO3_plus <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span></span>
+<span>    DMTA <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFORB"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"M23"</span>, <span class="st">"M27"</span>, <span class="st">"M31"</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>    M23 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>    M27 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
+<span>    M31 <span class="op">=</span> <span class="fu"><a href="../../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M27"</span>, sink <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span><span class="op">)</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">three_met_sep_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span><span class="va">three_met_mods</span>, <span class="va">dmta_ds</span>, error_model <span class="op">=</span> <span class="st">"tc"</span>,</span>
+<span>  cores <span class="op">=</span> <span class="va">n_cores</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">t11</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.time.html" class="external-link">system.time</a></span><span class="op">(</span><span class="va">sforb_sfo3_plus_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">three_met_sep_tc</span><span class="op">[</span><span class="st">"SFORB_SFO3_plus"</span>, <span class="op">]</span>,</span>
+<span>    no_random_effect <span class="op">=</span> <span class="st">"log_k_DMTA_bound_free"</span><span class="op">)</span><span class="op">)</span><span class="op">[[</span><span class="st">"elapsed"</span><span class="op">]</span><span class="op">]</span></span></code></pre></div>
+</div>
+</div>
+<div class="section level2">
+<h2 id="results">Results<a class="anchor" aria-label="anchor" href="#results"></a>
+</h2>
+<p>Benchmarks for all available error models are shown. They are
+intended for improving mkin, not for comparing CPUs or operating
+systems. All trademarks belong to their respective owners.</p>
+<div class="section level3">
+<h3 id="parent-only-1">Parent only<a class="anchor" aria-label="anchor" href="#parent-only-1"></a>
+</h3>
+<p>Constant variance for SFO, DFOP, SFORB and HS.</p>
+<table class="table">
+<colgroup>
+<col width="48%">
+<col width="7%">
+<col width="8%">
+<col width="8%">
+<col width="7%">
+<col width="7%">
+<col width="7%">
+<col width="7%">
+</colgroup>
+<thead><tr class="header">
+<th align="left">CPU</th>
+<th align="left">OS</th>
+<th align="left">mkin</th>
+<th align="left">saemix</th>
+<th align="right">t1</th>
+<th align="right">t2</th>
+<th align="right">t3</th>
+<th align="right">t4</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">Ryzen 7 1700</td>
+<td align="left">Linux</td>
+<td align="left">1.2.0</td>
+<td align="left">3.2</td>
+<td align="right">2.140</td>
+<td align="right">4.626</td>
+<td align="right">4.328</td>
+<td align="right">4.998</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 7 1700</td>
+<td align="left">Linux</td>
+<td align="left">1.2.2</td>
+<td align="left">3.2</td>
+<td align="right">2.427</td>
+<td align="right">4.550</td>
+<td align="right">4.217</td>
+<td align="right">4.851</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.1</td>
+<td align="left">3.2</td>
+<td align="right">1.352</td>
+<td align="right">2.813</td>
+<td align="right">2.401</td>
+<td align="right">2.074</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.2</td>
+<td align="left">3.2</td>
+<td align="right">1.328</td>
+<td align="right">2.738</td>
+<td align="right">2.336</td>
+<td align="right">2.023</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.3</td>
+<td align="left">3.2</td>
+<td align="right">1.118</td>
+<td align="right">2.036</td>
+<td align="right">2.010</td>
+<td align="right">2.088</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.3</td>
+<td align="left">3.2</td>
+<td align="right">1.419</td>
+<td align="right">2.374</td>
+<td align="right">1.926</td>
+<td align="right">2.398</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.4</td>
+<td align="left">3.2</td>
+<td align="right">0.972</td>
+<td align="right">2.550</td>
+<td align="right">1.987</td>
+<td align="right">2.055</td>
+</tr>
+<tr class="even">
+<td align="left">Intel(R) Xeon(R) Gold 6134 CPU @ 3.20GHz</td>
+<td align="left">Linux</td>
+<td align="left">1.2.6</td>
+<td align="left">3.2</td>
+<td align="right">2.998</td>
+<td align="right">6.523</td>
+<td align="right">6.126</td>
+<td align="right">4.721</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.6</td>
+<td align="left">3.2</td>
+<td align="right">1.135</td>
+<td align="right">2.025</td>
+<td align="right">2.406</td>
+<td align="right">2.478</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.9</td>
+<td align="left">3.3</td>
+<td align="right">1.086</td>
+<td align="right">1.991</td>
+<td align="right">1.949</td>
+<td align="right">2.411</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.10</td>
+<td align="left">3.3</td>
+<td align="right">1.115</td>
+<td align="right">2.277</td>
+<td align="right">1.945</td>
+<td align="right">2.134</td>
+</tr>
+</tbody>
+</table>
+<p>Two-component error fits for SFO, DFOP, SFORB and HS.</p>
+<table class="table">
+<colgroup>
+<col width="48%">
+<col width="7%">
+<col width="8%">
+<col width="8%">
+<col width="7%">
+<col width="7%">
+<col width="7%">
+<col width="7%">
+</colgroup>
+<thead><tr class="header">
+<th align="left">CPU</th>
+<th align="left">OS</th>
+<th align="left">mkin</th>
+<th align="left">saemix</th>
+<th align="right">t5</th>
+<th align="right">t6</th>
+<th align="right">t7</th>
+<th align="right">t8</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">Ryzen 7 1700</td>
+<td align="left">Linux</td>
+<td align="left">1.2.0</td>
+<td align="left">3.2</td>
+<td align="right">5.678</td>
+<td align="right">7.441</td>
+<td align="right">8.000</td>
+<td align="right">7.980</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 7 1700</td>
+<td align="left">Linux</td>
+<td align="left">1.2.2</td>
+<td align="left">3.2</td>
+<td align="right">5.352</td>
+<td align="right">7.201</td>
+<td align="right">8.174</td>
+<td align="right">8.401</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.1</td>
+<td align="left">3.2</td>
+<td align="right">2.388</td>
+<td align="right">3.033</td>
+<td align="right">3.532</td>
+<td align="right">3.310</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.2</td>
+<td align="left">3.2</td>
+<td align="right">2.341</td>
+<td align="right">2.968</td>
+<td align="right">3.465</td>
+<td align="right">3.341</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.3</td>
+<td align="left">3.2</td>
+<td align="right">2.159</td>
+<td align="right">3.584</td>
+<td align="right">3.307</td>
+<td align="right">3.460</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.3</td>
+<td align="left">3.2</td>
+<td align="right">2.348</td>
+<td align="right">3.134</td>
+<td align="right">3.253</td>
+<td align="right">3.530</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.4</td>
+<td align="left">3.2</td>
+<td align="right">2.127</td>
+<td align="right">3.587</td>
+<td align="right">3.433</td>
+<td align="right">3.595</td>
+</tr>
+<tr class="even">
+<td align="left">Intel(R) Xeon(R) Gold 6134 CPU @ 3.20GHz</td>
+<td align="left">Linux</td>
+<td align="left">1.2.6</td>
+<td align="left">3.2</td>
+<td align="right">5.070</td>
+<td align="right">8.464</td>
+<td align="right">8.525</td>
+<td align="right">7.599</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.6</td>
+<td align="left">3.2</td>
+<td align="right">2.161</td>
+<td align="right">3.325</td>
+<td align="right">3.669</td>
+<td align="right">3.153</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.9</td>
+<td align="left">3.3</td>
+<td align="right">2.426</td>
+<td align="right">3.196</td>
+<td align="right">3.256</td>
+<td align="right">3.322</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.10</td>
+<td align="left">3.3</td>
+<td align="right">2.372</td>
+<td align="right">3.137</td>
+<td align="right">3.100</td>
+<td align="right">3.281</td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="section level3">
+<h3 id="one-metabolite-1">One metabolite<a class="anchor" aria-label="anchor" href="#one-metabolite-1"></a>
+</h3>
+<p>Two-component error for DFOP-SFO and SFORB-SFO.</p>
+<table style="width:100%;" class="table">
+<colgroup>
+<col width="53%">
+<col width="7%">
+<col width="9%">
+<col width="9%">
+<col width="9%">
+<col width="10%">
+</colgroup>
+<thead><tr class="header">
+<th align="left">CPU</th>
+<th align="left">OS</th>
+<th align="left">mkin</th>
+<th align="left">saemix</th>
+<th align="right">t9</th>
+<th align="right">t10</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">Ryzen 7 1700</td>
+<td align="left">Linux</td>
+<td align="left">1.2.0</td>
+<td align="left">3.2</td>
+<td align="right">24.465</td>
+<td align="right">800.266</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 7 1700</td>
+<td align="left">Linux</td>
+<td align="left">1.2.2</td>
+<td align="left">3.2</td>
+<td align="right">25.193</td>
+<td align="right">798.580</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.1</td>
+<td align="left">3.2</td>
+<td align="right">11.247</td>
+<td align="right">285.216</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.2</td>
+<td align="left">3.2</td>
+<td align="right">11.242</td>
+<td align="right">284.258</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.3</td>
+<td align="left">3.2</td>
+<td align="right">11.796</td>
+<td align="right">216.012</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.3</td>
+<td align="left">3.2</td>
+<td align="right">12.841</td>
+<td align="right">292.688</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.4</td>
+<td align="left">3.2</td>
+<td align="right">12.160</td>
+<td align="right">265.934</td>
+</tr>
+<tr class="even">
+<td align="left">Intel(R) Xeon(R) Gold 6134 CPU @ 3.20GHz</td>
+<td align="left">Linux</td>
+<td align="left">1.2.6</td>
+<td align="left">3.2</td>
+<td align="right">30.168</td>
+<td align="right">748.675</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.6</td>
+<td align="left">3.2</td>
+<td align="right">12.007</td>
+<td align="right">286.757</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.9</td>
+<td align="left">3.3</td>
+<td align="right">12.420</td>
+<td align="right">289.338</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.10</td>
+<td align="left">3.3</td>
+<td align="right">11.590</td>
+<td align="right">279.782</td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="section level3">
+<h3 id="three-metabolites-1">Three metabolites<a class="anchor" aria-label="anchor" href="#three-metabolites-1"></a>
+</h3>
+<p>Two-component error for SFORB-SFO3-plus</p>
+<table class="table">
+<colgroup>
+<col width="58%">
+<col width="8%">
+<col width="10%">
+<col width="10%">
+<col width="12%">
+</colgroup>
+<thead><tr class="header">
+<th align="left">CPU</th>
+<th align="left">OS</th>
+<th align="left">mkin</th>
+<th align="left">saemix</th>
+<th align="right">t11</th>
+</tr></thead>
+<tbody>
+<tr class="odd">
+<td align="left">Ryzen 7 1700</td>
+<td align="left">Linux</td>
+<td align="left">1.2.0</td>
+<td align="left">3.2</td>
+<td align="right">1289.198</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 7 1700</td>
+<td align="left">Linux</td>
+<td align="left">1.2.2</td>
+<td align="left">3.2</td>
+<td align="right">1312.445</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.1</td>
+<td align="left">3.2</td>
+<td align="right">489.939</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.2</td>
+<td align="left">3.2</td>
+<td align="right">482.970</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.3</td>
+<td align="left">3.2</td>
+<td align="right">392.364</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.3</td>
+<td align="left">3.2</td>
+<td align="right">483.027</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.4</td>
+<td align="left">3.2</td>
+<td align="right">456.252</td>
+</tr>
+<tr class="even">
+<td align="left">Intel(R) Xeon(R) Gold 6134 CPU @ 3.20GHz</td>
+<td align="left">Linux</td>
+<td align="left">1.2.6</td>
+<td align="left">3.2</td>
+<td align="right">1235.028</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.6</td>
+<td align="left">3.2</td>
+<td align="right">480.577</td>
+</tr>
+<tr class="even">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.9</td>
+<td align="left">3.3</td>
+<td align="right">485.836</td>
+</tr>
+<tr class="odd">
+<td align="left">Ryzen 9 7950X</td>
+<td align="left">Linux</td>
+<td align="left">1.2.10</td>
+<td align="left">3.3</td>
+<td align="right">469.036</td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside>
+</div>
+
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
+</div>
+
+    </footer>
+</div>
+
+
+
+
+
+  </body>
+</html>
author	Johannes Ranke <johannes.ranke@jrwb.de>	2025-02-14 09:15:20 +0100
committer	Johannes Ranke <johannes.ranke@jrwb.de>	2025-02-14 09:15:20 +0100
commit	91a5834dd701211f929fd25419dc34561ce3b4e7 (patch)
tree	860862d680fea23581d08b2948c5ac3e922a8447 /docs/dev/articles
parent	b0f08271d1dae8ffaf57f557c27eba1314ece1d5 (diff)