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- </header><div class="row"> - <div class="col-md-9 contents"> - <div class="page-header toc-ignore"> - <h1 data-toc-skip>Example evaluation of FOCUS Laboratory Data L1 -to L3</h1> + + +<div class="row"> + <main id="main" class="col-md-9"><div class="page-header"> + + <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 2023-05-19)</h4> +(rebuilt 2025-02-16)</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="hidden name"><code>FOCUS_L.rmd</code></div> - + <div class="d-none name"><code>FOCUS_L.rmd</code></div> </div> @@ -170,10 +118,10 @@ 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"><-</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: 1.2.5 </span></span> -<span><span class="co">## R version used for fitting: 4.3.0 </span></span> -<span><span class="co">## Date of fit: Fri May 19 17:34:11 2023 </span></span> -<span><span class="co">## Date of summary: Fri May 19 17:34:11 2023 </span></span> +<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: Sun Feb 16 17:04:35 2025 </span></span> +<span><span class="co">## Date of summary: Sun Feb 16 17:04:35 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> @@ -263,7 +211,9 @@ objects.</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 <span class="math inline">\(\chi^2\)</span> error level is checked.</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"><-</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> @@ -274,20 +224,19 @@ objects.</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 sqrt(1/diag(V)): NaNs produced</span></span></code></pre> -<pre><code><span><span class="co">## Warning in cov2cor(ans$covar): diag(.) had 0 or NA entries; non-finite result</span></span> -<span><span class="co">## is doubtful</span></span></code></pre> -<pre><code><span><span class="co">## mkin version used for fitting: 1.2.5 </span></span> -<span><span class="co">## R version used for fitting: 4.3.0 </span></span> -<span><span class="co">## Date of fit: Fri May 19 17:34:12 2023 </span></span> -<span><span class="co">## Date of summary: Fri May 19 17:34:12 2023 </span></span> +<pre><code><span><span class="co">## Warning in cov2cor(ans$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: Sun Feb 16 17:04:35 2025 </span></span> +<span><span class="co">## Date of summary: Sun Feb 16 17:04:35 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 369 model solutions performed in 0.025 s</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> @@ -315,22 +264,22 @@ objects.</p> <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">## 95.88781 99.44929 -43.9439</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 13.78 NaN NaN NaN</span></span> -<span><span class="co">## log_beta 16.13 NaN NaN NaN</span></span> -<span><span class="co">## sigma 2.78 0.4598 1.794 3.766</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 1.0000000 NaN NaN 0.0001671</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 0.0001671 NaN NaN 1.0000000</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> @@ -338,9 +287,9 @@ objects.</p> <span><span class="co">## for estimators of untransformed parameters.</span></span> <span><span class="co">## Estimate t value Pr(>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 9.658e+05 NA NA NA NA</span></span> -<span><span class="co">## beta 1.010e+07 NA NA NA NA</span></span> -<span><span class="co">## sigma 2.780e+00 NA NA 1.794 3.766</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> @@ -348,43 +297,47 @@ objects.</p> <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 7.25 24.08 7.25</span></span></code></pre> +<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 <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> +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="cb14"><pre class="downlit sourceCode r"> +<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> @@ -397,15 +350,16 @@ report, p. 287:</p> <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"> +<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"><-</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 <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>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 @@ -420,18 +374,20 @@ kinetics.</p> <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 <span class="math inline">\(\chi^2\)</span> error level is checked.</p> -<div class="sourceCode" id="cb16"><pre class="downlit sourceCode r"> +<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"><-</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="cb17"><pre class="downlit sourceCode r"> +<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: 1.2.5 </span></span> -<span><span class="co">## R version used for fitting: 4.3.0 </span></span> -<span><span class="co">## Date of fit: Fri May 19 17:34:12 2023 </span></span> -<span><span class="co">## Date of summary: Fri May 19 17:34:12 2023 </span></span> +<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: Sun Feb 16 17:04:35 2025 </span></span> +<span><span class="co">## Date of summary: Sun Feb 16 17:04:35 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> @@ -473,10 +429,10 @@ kinetics.</p> <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 -7.828e-09</span></span> -<span><span class="co">## log_alpha -1.151e-01 1.000e+00 9.741e-01 -1.602e-07</span></span> -<span><span class="co">## log_beta -2.085e-01 9.741e-01 1.000e+00 -1.372e-07</span></span> -<span><span class="co">## sigma -7.828e-09 -1.602e-07 -1.372e-07 1.000e+00</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> @@ -496,26 +452,29 @@ kinetics.</p> <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 <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> +<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 <span class="math inline">\(\chi^2\)</span> error level.</p> -<div class="sourceCode" id="cb19"><pre class="downlit sourceCode r"> +<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"><-</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="cb20"><pre class="downlit sourceCode r"> +<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: 1.2.5 </span></span> -<span><span class="co">## R version used for fitting: 4.3.0 </span></span> -<span><span class="co">## Date of fit: Fri May 19 17:34:12 2023 </span></span> -<span><span class="co">## Date of summary: Fri May 19 17:34:12 2023 </span></span> +<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: Sun Feb 16 17:04:36 2025 </span></span> +<span><span class="co">## Date of summary: Sun Feb 16 17:04:36 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> @@ -524,7 +483,7 @@ the data.</p> <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 0.04 s</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> @@ -555,18 +514,18 @@ the data.</p> <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 3.112 1.842e+03 -4353.0000 4359.0000</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 6.783e-07 -3.390e-10 2.665e-01 -2.967e-10</span></span> -<span><span class="co">## log_k1 6.783e-07 1.000e+00 1.116e-04 -2.196e-04 -1.031e-05</span></span> -<span><span class="co">## log_k2 -3.390e-10 1.116e-04 1.000e+00 -7.903e-01 2.917e-09</span></span> -<span><span class="co">## g_qlogis 2.665e-01 -2.196e-04 -7.903e-01 1.000e+00 -4.408e-09</span></span> -<span><span class="co">## sigma -2.967e-10 -1.031e-05 2.917e-09 -4.408e-09 1.000e+00</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> @@ -574,7 +533,7 @@ the data.</p> <span><span class="co">## for estimators of untransformed parameters.</span></span> <span><span class="co">## Estimate t value Pr(>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 22.4800 5.553e-04 4.998e-01 0.0000 Inf</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> @@ -586,7 +545,7 @@ the data.</p> <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 0.03084 2.058</span></span></code></pre> +<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> @@ -596,7 +555,7 @@ based on the chi^2 error level criterion.</p> </h2> <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"> +<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> @@ -608,16 +567,22 @@ report, p. 290.</p> 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"> +<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"><-</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 <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> +<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> @@ -627,12 +592,12 @@ 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"> +<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: 1.2.5 </span></span> -<span><span class="co">## R version used for fitting: 4.3.0 </span></span> -<span><span class="co">## Date of fit: Fri May 19 17:34:13 2023 </span></span> -<span><span class="co">## Date of summary: Fri May 19 17:34:13 2023 </span></span> +<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: Sun Feb 16 17:04:36 2025 </span></span> +<span><span class="co">## Date of summary: Sun Feb 16 17:04:36 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> @@ -679,11 +644,11 @@ summary and plot functions working on mkinfit objects.</p> <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 -9.664e-08</span></span> -<span><span class="co">## log_k1 1.732e-01 1.000e+00 4.945e-01 -5.809e-01 7.147e-07</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 -7.926e-07</span></span> -<span><span class="co">## sigma -9.664e-08 7.147e-07 1.022e-06 -7.926e-07 1.000e+00</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> @@ -715,14 +680,15 @@ summary and plot functions working on mkinfit objects.</p> <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="cb26"><pre class="downlit sourceCode r"> +<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 <span class="math inline">\(\chi^2\)</span> error -level criterion for laboratory data L3.</p> +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 @@ -735,38 +701,40 @@ parameter <code>g</code> is quite narrow.</p> </h2> <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"> +<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"><-</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="cb28"><pre class="downlit sourceCode r"> +<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"><-</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 <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"> +<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: 1.2.5 </span></span> -<span><span class="co">## R version used for fitting: 4.3.0 </span></span> -<span><span class="co">## Date of fit: Fri May 19 17:34:13 2023 </span></span> -<span><span class="co">## Date of summary: Fri May 19 17:34:13 2023 </span></span> +<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: Sun Feb 16 17:04:36 2025 </span></span> +<span><span class="co">## Date of summary: Sun Feb 16 17:04:37 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 0.009 s</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> @@ -798,9 +766,9 @@ appears negligible.</p> <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 3.387e-07</span></span> -<span><span class="co">## log_k_parent 5.938e-01 1.000e+00 5.830e-07</span></span> -<span><span class="co">## sigma 3.387e-07 5.830e-07 1.000e+00</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> @@ -819,12 +787,12 @@ appears negligible.</p> <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="cb31"><pre class="downlit sourceCode r"> +<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: 1.2.5 </span></span> -<span><span class="co">## R version used for fitting: 4.3.0 </span></span> -<span><span class="co">## Date of fit: Fri May 19 17:34:13 2023 </span></span> -<span><span class="co">## Date of summary: Fri May 19 17:34:13 2023 </span></span> +<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: Sun Feb 16 17:04:36 2025 </span></span> +<span><span class="co">## Date of summary: Sun Feb 16 17:04:37 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> @@ -866,10 +834,10 @@ appears negligible.</p> <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 -2.468e-07</span></span> -<span><span class="co">## log_alpha -4.696e-01 1.000e+00 9.889e-01 2.478e-08</span></span> -<span><span class="co">## log_beta -5.543e-01 9.889e-01 1.000e+00 5.211e-08</span></span> -<span><span class="co">## sigma -2.468e-07 2.478e-08 5.211e-08 1.000e+00</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> @@ -901,35 +869,26 @@ Validierung von Modellierungssoftware als Alternative zu ModelMaker </div> </div> </div> - 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