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<h1 data-toc-skip>Example evaluation of FOCUS Laboratory Data L1 to L3</h1>
<h4 class="author">Johannes Ranke</h4>
<h4 class="date">Last change 17 November 2016 (rebuilt 2021-02-15)</h4>
<small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/master/vignettes/FOCUS_L.rmd"><code>vignettes/FOCUS_L.rmd</code></a></small>
<div class="hidden name"><code>FOCUS_L.rmd</code></div>
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<div id="laboratory-data-l1" class="section level1">
<h1 class="hasAnchor">
<a href="#laboratory-data-l1" class="anchor"></a>Laboratory Data L1</h1>
<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 class="kw"><a href="https://rdrr.io/r/base/library.html">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 class="va">FOCUS_2006_L1</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span><span class="op">(</span>
t <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/rep.html">rep</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html">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>,
parent <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">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 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 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 class="va">FOCUS_2006_L1_mkin</span> <span class="op"><-</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></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 class="va">m.L1.SFO</span> <span class="op"><-</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 class="fu"><a href="https://rdrr.io/r/base/summary.html">summary</a></span><span class="op">(</span><span class="va">m.L1.SFO</span><span class="op">)</span></code></pre></div>
<pre><code>## mkin version used for fitting: 1.0.3
## R version used for fitting: 4.0.3
## Date of fit: Mon Feb 15 13:46:25 2021
## Date of summary: Mon Feb 15 13:46:25 2021
##
## Equations:
## d_parent/dt = - k_parent * parent
##
## Model predictions using solution type analytical
##
## Fitted using 133 model solutions performed in 0.032 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 89.85 state
## k_parent 0.10 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 89.850000 -Inf Inf
## log_k_parent -2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Results:
##
## AIC BIC logLik
## 93.88778 96.5589 -43.94389
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 92.470 1.28200 89.740 95.200
## log_k_parent -2.347 0.03763 -2.428 -2.267
## sigma 2.780 0.46330 1.792 3.767
##
## Parameter correlation:
## parent_0 log_k_parent sigma
## parent_0 1.000e+00 6.186e-01 -1.516e-09
## log_k_parent 6.186e-01 1.000e+00 -3.124e-09
## sigma -1.516e-09 -3.124e-09 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 92.47000 72.13 8.824e-21 89.74000 95.2000
## k_parent 0.09561 26.57 2.487e-14 0.08824 0.1036
## sigma 2.78000 6.00 1.216e-05 1.79200 3.7670
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 3.424 2 7
## parent 3.424 2 7
##
## Estimated disappearance times:
## DT50 DT90
## parent 7.249 24.08
##
## Data:
## time variable observed predicted residual
## 0 parent 88.3 92.471 -4.1710
## 0 parent 91.4 92.471 -1.0710
## 1 parent 85.6 84.039 1.5610
## 1 parent 84.5 84.039 0.4610
## 2 parent 78.9 76.376 2.5241
## 2 parent 77.6 76.376 1.2241
## 3 parent 72.0 69.412 2.5884
## 3 parent 71.9 69.412 2.4884
## 5 parent 50.3 57.330 -7.0301
## 5 parent 59.4 57.330 2.0699
## 7 parent 47.0 47.352 -0.3515
## 7 parent 45.1 47.352 -2.2515
## 14 parent 27.7 24.247 3.4528
## 14 parent 27.3 24.247 3.0528
## 21 parent 10.0 12.416 -2.4163
## 21 parent 10.4 12.416 -2.0163
## 30 parent 2.9 5.251 -2.3513
## 30 parent 4.0 5.251 -1.2513</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 class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html">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></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 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></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 <span class="math inline">\(\chi^2\)</span> error level is checked.</p>
<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="va">m.L1.FOMC</span> <span class="op"><-</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></code></pre></div>
<pre><code>## Warning in mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet = TRUE): Optimisation did not converge:
## false convergence (8)</code></pre>
<div class="sourceCode" id="cb8"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html">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></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 class="fu"><a href="https://rdrr.io/r/base/summary.html">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></code></pre></div>
<pre><code>## Warning in sqrt(diag(covar)): NaNs produced</code></pre>
<pre><code>## Warning in sqrt(1/diag(V)): NaNs produced</code></pre>
<pre><code>## Warning in cov2cor(ans$covar): diag(.) had 0 or NA entries; non-finite result is
## doubtful</code></pre>
<pre><code>## mkin version used for fitting: 1.0.3
## R version used for fitting: 4.0.3
## Date of fit: Mon Feb 15 13:46:26 2021
## Date of summary: Mon Feb 15 13:46:26 2021
##
## Equations:
## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
##
## Model predictions using solution type analytical
##
## Fitted using 369 model solutions performed in 0.083 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 89.85 state
## alpha 1.00 deparm
## beta 10.00 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 89.850000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
##
## Warning(s):
## Optimisation did not converge:
## false convergence (8)
##
## Results:
##
## AIC BIC logLik
## 95.88781 99.44929 -43.9439
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 92.47 1.2820 89.720 95.220
## log_alpha 13.78 NaN NaN NaN
## log_beta 16.13 NaN NaN NaN
## sigma 2.78 0.4598 1.794 3.766
##
## Parameter correlation:
## parent_0 log_alpha log_beta sigma
## parent_0 1.0000000 NaN NaN 0.0001671
## log_alpha NaN 1 NaN NaN
## log_beta NaN NaN 1 NaN
## sigma 0.0001671 NaN NaN 1.0000000
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 9.247e+01 NA NA 89.720 95.220
## alpha 9.658e+05 NA NA NA NA
## beta 1.010e+07 NA NA NA NA
## sigma 2.780e+00 NA NA 1.794 3.766
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 3.619 3 6
## parent 3.619 3 6
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 7.25 24.08 7.25</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 <span class="math inline">\(\chi^2\)</span> 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 <span class="math inline">\(\chi^2\)</span> 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 <span class="math inline">\(\chi^2\)</span> 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 <span class="math inline">\(\chi^2\)</span> 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 id="laboratory-data-l2" class="section level1">
<h1 class="hasAnchor">
<a href="#laboratory-data-l2" class="anchor"></a>Laboratory Data L2</h1>
<p>The following code defines example dataset L2 from the FOCUS kinetics report, p. 287:</p>
<div class="sourceCode" id="cb14"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="va">FOCUS_2006_L2</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span><span class="op">(</span>
t <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/rep.html">rep</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html">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>,
parent <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">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 class="fl">19.3</span>, <span class="fl">22.3</span>, <span class="fl">4.6</span>, <span class="fl">4.6</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 class="va">FOCUS_2006_L2_mkin</span> <span class="op"><-</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></code></pre></div>
<div id="sfo-fit-for-l2" class="section level2">
<h2 class="hasAnchor">
<a href="#sfo-fit-for-l2" class="anchor"></a>SFO fit for L2</h2>
<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="cb15"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="va">m.L2.SFO</span> <span class="op"><-</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 class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html">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>,
main <span class="op">=</span> <span class="st">"FOCUS L2 - SFO"</span><span class="op">)</span></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-8-1.png" width="672"></p>
<p>The <span class="math inline">\(\chi^2\)</span> 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 id="fomc-fit-for-l2" class="section level2">
<h2 class="hasAnchor">
<a href="#fomc-fit-for-l2" class="anchor"></a>FOMC fit for L2</h2>
<p>For comparison, the FOMC model is fitted as well, and the <span class="math inline">\(\chi^2\)</span> error level is checked.</p>
<div class="sourceCode" id="cb16"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="va">m.L2.FOMC</span> <span class="op"><-</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 class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html">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>,
main <span class="op">=</span> <span class="st">"FOCUS L2 - FOMC"</span><span class="op">)</span></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-9-1.png" width="672"></p>
<div class="sourceCode" id="cb17"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="fu"><a href="https://rdrr.io/r/base/summary.html">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></code></pre></div>
<pre><code>## mkin version used for fitting: 1.0.3
## R version used for fitting: 4.0.3
## Date of fit: Mon Feb 15 13:46:26 2021
## Date of summary: Mon Feb 15 13:46:26 2021
##
## Equations:
## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
##
## Model predictions using solution type analytical
##
## Fitted using 239 model solutions performed in 0.05 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 93.95 state
## alpha 1.00 deparm
## beta 10.00 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 93.950000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Results:
##
## AIC BIC logLik
## 61.78966 63.72928 -26.89483
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 93.7700 1.6130 90.05000 97.4900
## log_alpha 0.3180 0.1559 -0.04149 0.6776
## log_beta 0.2102 0.2493 -0.36460 0.7850
## sigma 2.2760 0.4645 1.20500 3.3470
##
## Parameter correlation:
## parent_0 log_alpha log_beta sigma
## parent_0 1.000e+00 -1.151e-01 -2.085e-01 -7.828e-09
## log_alpha -1.151e-01 1.000e+00 9.741e-01 -1.602e-07
## log_beta -2.085e-01 9.741e-01 1.000e+00 -1.372e-07
## sigma -7.828e-09 -1.602e-07 -1.372e-07 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 93.770 58.120 4.267e-12 90.0500 97.490
## alpha 1.374 6.414 1.030e-04 0.9594 1.969
## beta 1.234 4.012 1.942e-03 0.6945 2.192
## sigma 2.276 4.899 5.977e-04 1.2050 3.347
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 6.205 3 3
## parent 6.205 3 3
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 0.8092 5.356 1.612</code></pre>
<p>The error level at which the <span class="math inline">\(\chi^2\)</span> 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 id="dfop-fit-for-l2" class="section level2">
<h2 class="hasAnchor">
<a href="#dfop-fit-for-l2" class="anchor"></a>DFOP fit for L2</h2>
<p>Fitting the four parameter DFOP model further reduces the <span class="math inline">\(\chi^2\)</span> error level.</p>
<div class="sourceCode" id="cb19"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="va">m.L2.DFOP</span> <span class="op"><-</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 class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html">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>,
main <span class="op">=</span> <span class="st">"FOCUS L2 - DFOP"</span><span class="op">)</span></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-10-1.png" width="672"></p>
<div class="sourceCode" id="cb20"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="fu"><a href="https://rdrr.io/r/base/summary.html">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></code></pre></div>
<pre><code>## mkin version used for fitting: 1.0.3
## R version used for fitting: 4.0.3
## Date of fit: Mon Feb 15 13:46:27 2021
## Date of summary: Mon Feb 15 13:46:27 2021
##
## Equations:
## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
## time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
## * parent
##
## Model predictions using solution type analytical
##
## Fitted using 581 model solutions performed in 0.134 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 93.95 state
## k1 0.10 deparm
## k2 0.01 deparm
## g 0.50 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 93.950000 -Inf Inf
## log_k1 -2.302585 -Inf Inf
## log_k2 -4.605170 -Inf Inf
## g_qlogis 0.000000 -Inf Inf
##
## Fixed parameter values:
## None
##
## Results:
##
## AIC BIC logLik
## 52.36695 54.79148 -21.18347
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 93.950 9.998e-01 91.5900 96.3100
## log_k1 3.112 1.842e+03 -4353.0000 4359.0000
## log_k2 -1.088 6.285e-02 -1.2370 -0.9394
## g_qlogis -0.399 9.946e-02 -0.6342 -0.1638
## sigma 1.414 2.886e-01 0.7314 2.0960
##
## Parameter correlation:
## parent_0 log_k1 log_k2 g_qlogis sigma
## parent_0 1.000e+00 6.783e-07 -3.390e-10 2.665e-01 -2.967e-10
## log_k1 6.783e-07 1.000e+00 1.116e-04 -2.196e-04 -1.031e-05
## log_k2 -3.390e-10 1.116e-04 1.000e+00 -7.903e-01 2.917e-09
## g_qlogis 2.665e-01 -2.196e-04 -7.903e-01 1.000e+00 -4.408e-09
## sigma -2.967e-10 -1.031e-05 2.917e-09 -4.408e-09 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 93.9500 9.397e+01 2.036e-12 91.5900 96.3100
## k1 22.4800 5.553e-04 4.998e-01 0.0000 Inf
## k2 0.3369 1.591e+01 4.697e-07 0.2904 0.3909
## g 0.4016 1.680e+01 3.238e-07 0.3466 0.4591
## sigma 1.4140 4.899e+00 8.776e-04 0.7314 2.0960
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 2.53 4 2
## parent 2.53 4 2
##
## Estimated disappearance times:
## DT50 DT90 DT50back DT50_k1 DT50_k2
## parent 0.5335 5.311 1.599 0.03084 2.058</code></pre>
<p>Here, the DFOP model is clearly the best-fit model for dataset L2 based on the chi^2 error level criterion. However, the failure to calculate the covariance matrix indicates that the parameter estimates correlate excessively. Therefore, the FOMC model may be preferred for this dataset.</p>
</div>
</div>
<div id="laboratory-data-l3" class="section level1">
<h1 class="hasAnchor">
<a href="#laboratory-data-l3" class="anchor"></a>Laboratory Data L3</h1>
<p>The following code defines example dataset L3 from the FOCUS kinetics report, p. 290.</p>
<div class="sourceCode" id="cb22"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="va">FOCUS_2006_L3</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span><span class="op">(</span>
t <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">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>,
parent <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">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 class="va">FOCUS_2006_L3_mkin</span> <span class="op"><-</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></code></pre></div>
<div id="fit-multiple-models" class="section level2">
<h2 class="hasAnchor">
<a href="#fit-multiple-models" class="anchor"></a>Fit multiple models</h2>
<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="cb23"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="co"># Only use one core here, not to offend the CRAN checks</span>
<span class="va">mm.L3</span> <span class="op"><-</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">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 class="fu"><a href="https://rdrr.io/r/base/list.html">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 class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html">plot</a></span><span class="op">(</span><span class="va">mm.L3</span><span class="op">)</span></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-12-1.png" width="700"></p>
<p>The <span class="math inline">\(\chi^2\)</span> 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 <span class="math inline">\(\chi^2\)</span> test passes of 7%. Fitting the four parameter DFOP model further reduces the <span class="math inline">\(\chi^2\)</span> error level considerably.</p>
</div>
<div id="accessing-mmkin-objects" class="section level2">
<h2 class="hasAnchor">
<a href="#accessing-mmkin-objects" class="anchor"></a>Accessing mmkin objects</h2>
<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="cb24"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="fu"><a href="https://rdrr.io/r/base/summary.html">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></code></pre></div>
<pre><code>## mkin version used for fitting: 1.0.3
## R version used for fitting: 4.0.3
## Date of fit: Mon Feb 15 13:46:27 2021
## Date of summary: Mon Feb 15 13:46:28 2021
##
## Equations:
## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
## time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
## * parent
##
## Model predictions using solution type analytical
##
## Fitted using 376 model solutions performed in 0.081 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 97.80 state
## k1 0.10 deparm
## k2 0.01 deparm
## g 0.50 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 97.800000 -Inf Inf
## log_k1 -2.302585 -Inf Inf
## log_k2 -4.605170 -Inf Inf
## g_qlogis 0.000000 -Inf Inf
##
## Fixed parameter values:
## None
##
## Results:
##
## AIC BIC logLik
## 32.97732 33.37453 -11.48866
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 97.7500 1.01900 94.5000 101.000000
## log_k1 -0.6612 0.10050 -0.9812 -0.341300
## log_k2 -4.2860 0.04322 -4.4230 -4.148000
## g_qlogis -0.1739 0.05270 -0.3416 -0.006142
## sigma 1.0170 0.25430 0.2079 1.827000
##
## Parameter correlation:
## parent_0 log_k1 log_k2 g_qlogis sigma
## parent_0 1.000e+00 1.732e-01 2.282e-02 4.009e-01 -9.664e-08
## log_k1 1.732e-01 1.000e+00 4.945e-01 -5.809e-01 7.147e-07
## log_k2 2.282e-02 4.945e-01 1.000e+00 -6.812e-01 1.022e-06
## g_qlogis 4.009e-01 -5.809e-01 -6.812e-01 1.000e+00 -7.926e-07
## sigma -9.664e-08 7.147e-07 1.022e-06 -7.926e-07 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 97.75000 95.960 1.248e-06 94.50000 101.00000
## k1 0.51620 9.947 1.081e-03 0.37490 0.71090
## k2 0.01376 23.140 8.840e-05 0.01199 0.01579
## g 0.45660 34.920 2.581e-05 0.41540 0.49850
## sigma 1.01700 4.000 1.400e-02 0.20790 1.82700
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 2.225 4 4
## parent 2.225 4 4
##
## Estimated disappearance times:
## DT50 DT90 DT50back DT50_k1 DT50_k2
## parent 7.464 123 37.03 1.343 50.37
##
## Data:
## time variable observed predicted residual
## 0 parent 97.8 97.75 0.05396
## 3 parent 60.0 60.45 -0.44933
## 7 parent 51.0 49.44 1.56338
## 14 parent 43.0 43.84 -0.83632
## 30 parent 35.0 35.15 -0.14707
## 60 parent 22.0 23.26 -1.25919
## 91 parent 15.0 15.18 -0.18181
## 120 parent 12.0 10.19 1.81395</code></pre>
<div class="sourceCode" id="cb26"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html">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></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 <span class="math inline">\(\chi^2\)</span> 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 id="laboratory-data-l4" class="section level1">
<h1 class="hasAnchor">
<a href="#laboratory-data-l4" class="anchor"></a>Laboratory Data L4</h1>
<p>The following code defines example dataset L4 from the FOCUS kinetics report, p. 293:</p>
<div class="sourceCode" id="cb27"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="va">FOCUS_2006_L4</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span><span class="op">(</span>
t <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">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>,
parent <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">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 class="va">FOCUS_2006_L4_mkin</span> <span class="op"><-</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></code></pre></div>
<p>Fits of the SFO and FOMC models, plots and summaries are produced below:</p>
<div class="sourceCode" id="cb28"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="co"># Only use one core here, not to offend the CRAN checks</span>
<span class="va">mm.L4</span> <span class="op"><-</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">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 class="fu"><a href="https://rdrr.io/r/base/list.html">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>,
quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span>
<span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html">plot</a></span><span class="op">(</span><span class="va">mm.L4</span><span class="op">)</span></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-15-1.png" width="700"></p>
<p>The <span class="math inline">\(\chi^2\)</span> error level of 3.3% as well as the plot suggest that the SFO model fits very well. The error level at which the <span class="math inline">\(\chi^2\)</span> test passes is slightly lower for the FOMC model. However, the difference appears negligible.</p>
<div class="sourceCode" id="cb29"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="fu"><a href="https://rdrr.io/r/base/summary.html">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></code></pre></div>
<pre><code>## mkin version used for fitting: 1.0.3
## R version used for fitting: 4.0.3
## Date of fit: Mon Feb 15 13:46:28 2021
## Date of summary: Mon Feb 15 13:46:28 2021
##
## Equations:
## d_parent/dt = - k_parent * parent
##
## Model predictions using solution type analytical
##
## Fitted using 142 model solutions performed in 0.031 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 96.6 state
## k_parent 0.1 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 96.600000 -Inf Inf
## log_k_parent -2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Results:
##
## AIC BIC logLik
## 47.12133 47.35966 -20.56067
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 96.440 1.69900 92.070 100.800
## log_k_parent -5.030 0.07059 -5.211 -4.848
## sigma 3.162 0.79050 1.130 5.194
##
## Parameter correlation:
## parent_0 log_k_parent sigma
## parent_0 1.000e+00 5.938e-01 3.387e-07
## log_k_parent 5.938e-01 1.000e+00 5.830e-07
## sigma 3.387e-07 5.830e-07 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 96.440000 56.77 1.604e-08 92.070000 1.008e+02
## k_parent 0.006541 14.17 1.578e-05 0.005455 7.842e-03
## sigma 3.162000 4.00 5.162e-03 1.130000 5.194e+00
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 3.287 2 6
## parent 3.287 2 6
##
## Estimated disappearance times:
## DT50 DT90
## parent 106 352</code></pre>
<div class="sourceCode" id="cb31"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span class="fu"><a href="https://rdrr.io/r/base/summary.html">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></code></pre></div>
<pre><code>## mkin version used for fitting: 1.0.3
## R version used for fitting: 4.0.3
## Date of fit: Mon Feb 15 13:46:28 2021
## Date of summary: Mon Feb 15 13:46:28 2021
##
## Equations:
## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
##
## Model predictions using solution type analytical
##
## Fitted using 224 model solutions performed in 0.047 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 96.6 state
## alpha 1.0 deparm
## beta 10.0 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 96.600000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Results:
##
## AIC BIC logLik
## 40.37255 40.69032 -16.18628
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 99.1400 1.2670 95.6300 102.7000
## log_alpha -0.3506 0.2616 -1.0770 0.3756
## log_beta 4.1740 0.3938 3.0810 5.2670
## sigma 1.8300 0.4575 0.5598 3.1000
##
## Parameter correlation:
## parent_0 log_alpha log_beta sigma
## parent_0 1.000e+00 -4.696e-01 -5.543e-01 -2.468e-07
## log_alpha -4.696e-01 1.000e+00 9.889e-01 2.478e-08
## log_beta -5.543e-01 9.889e-01 1.000e+00 5.211e-08
## sigma -2.468e-07 2.478e-08 5.211e-08 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 99.1400 78.250 7.993e-08 95.6300 102.700
## alpha 0.7042 3.823 9.365e-03 0.3407 1.456
## beta 64.9800 2.540 3.201e-02 21.7800 193.900
## sigma 1.8300 4.000 8.065e-03 0.5598 3.100
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 2.029 3 5
## parent 2.029 3 5
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 108.9 1644 494.9</code></pre>
</div>
<div id="references" class="section level1 unnumbered">
<h1 class="hasAnchor">
<a href="#references" class="anchor"></a>References</h1>
<div id="refs" class="references hanging-indent">
<div id="ref-ranke2014">
<p>Ranke, Johannes. 2014. “Prüfung und Validierung von Modellierungssoftware als Alternative zu ModelMaker 4.0.” Umweltbundesamt Projektnummer 27452.</p>
</div>
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