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</li> - <li> - <a href="../../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a> - </li> - </ul> -</li> -<li> - <a href="../../news/index.html">News</a> -</li> - </ul> -<ul class="nav navbar-nav navbar-right"> -<li> - <a href="https://github.com/jranke/mkin/" class="external-link"> - <span class="fab fa-github fa-lg"></span> - - </a> -</li> - </ul> -</div> -<!--/.nav-collapse --> - </div> -<!--/.container --> -</div> -<!--/.navbar --> - - - - </header><div class="row"> - <div class="col-md-9 contents"> - <div class="page-header toc-ignore"> - <h1 data-toc-skip>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 16 April 2023</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="hidden 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 1.2.3. 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"><-</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"><-</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"><-</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"><-</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"><-</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">$</span><span class="va">name</span> <span class="op">==</span> <span class="st">"DMTAP"</span>, <span class="st">"name"</span><span class="op">]</span> <span class="op"><-</span> <span class="st">"DMTA"</span> <span class="co"># Rename DMTAP to DMTA</span></span> -<span> <span class="va">ds_i</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/base/subset.html" class="external-link">subset</a></span><span class="op">(</span><span class="va">ds_i</span>, <span class="va">name</span> <span class="op">==</span> <span class="st">"DMTA"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"name"</span>, <span class="st">"time"</span>, <span class="st">"value"</span><span class="op">)</span><span class="op">)</span> <span class="co"># Select data</span></span> -<span> <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op"><-</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"><-</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"><-</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"><-</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"><-</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"><-</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"><-</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">|></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"><-</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">|></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">C</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">C</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"><-</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">|></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">|></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">720.4</td> -<td align="right">718.8</td> -<td align="right">-352.2</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">665.5</td> -<td align="right">663.4</td> -<td align="right">-322.8</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">|></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"><-</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">|></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="37%"> -<col width="6%"> -<col width="8%"> -<col width="8%"> -<col width="9%"> -<col width="9%"> -<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(>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">663.8</td> -<td align="right">661.9</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">665.5</td> -<td align="right">663.4</td> -<td align="right">-322.8</td> -<td align="right">0.2809</td> -<td align="right">1</td> -<td align="right">0.5961</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: 3.2 -mkin version used for pre-fitting: 1.2.2 -R version used for fitting: 4.2.2 -Date of fit: Sat Jan 28 11:22:51 2023 -Date of summary: Sat Jan 28 11:22:52 2023 - -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 3.74 s -Using 300, 100 iterations and 9 chains - -Variance model: Two-component variance function - -Starting values for degradation parameters: - DMTA_0 k1 k2 g -98.759266 0.087034 0.009933 0.930827 - -Fixed degradation parameter values: -None - -Starting values for random effects (square root of initial entries in omega): - DMTA_0 k1 k2 g -DMTA_0 98.76 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 - 663.8 661.9 -322.9 - -Optimised parameters: - est. lower upper -DMTA_0 98.228939 96.285869 100.17201 -k1 0.064063 0.033477 0.09465 -k2 0.008297 0.005824 0.01077 -g 0.953821 0.914328 0.99331 -a.1 1.068479 0.869538 1.26742 -b.1 0.029424 0.022406 0.03644 -SD.DMTA_0 2.030437 0.404824 3.65605 -SD.k1 0.594692 0.256660 0.93272 -SD.g 1.006754 0.361327 1.65218 - -Correlation: - DMTA_0 k1 k2 -k1 0.0218 -k2 0.0556 0.0355 -g -0.0516 -0.0284 -0.2800 - -Random effects: - est. lower upper -SD.DMTA_0 2.0304 0.4048 3.6560 -SD.k1 0.5947 0.2567 0.9327 -SD.g 1.0068 0.3613 1.6522 - -Variance model: - est. lower upper -a.1 1.06848 0.86954 1.26742 -b.1 0.02942 0.02241 0.03644 - -Estimated disappearance times: - DT50 DT90 DT50back DT50_k1 DT50_k2 -DMTA 11.45 41.4 12.46 10.82 83.54</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"><-</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"><-</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: 3.2 -mkin version used for pre-fitting: 1.2.2 -R version used for fitting: 4.2.2 -Date of fit: Sat Jan 28 11:22:44 2023 -Date of summary: Sat Jan 28 11:23:57 2023 - -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 0.982 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: 3.2 -mkin version used for pre-fitting: 1.2.2 -R version used for fitting: 4.2.2 -Date of fit: Sat Jan 28 11:22:46 2023 -Date of summary: Sat Jan 28 11:23:57 2023 - -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 2.39 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 95.703157 98.84049 -k_DMTA 0.056638 0.029110 0.08417 -a.1 2.660081 2.230398 3.08976 -b.1 0.001665 -0.006911 0.01024 -SD.DMTA_0 1.545520 0.145035 2.94601 -SD.k_DMTA 0.606422 0.262274 0.95057 - -Correlation: - DMTA_0 -k_DMTA 0.0169 - -Random effects: - est. lower upper -SD.DMTA_0 1.5455 0.1450 2.9460 -SD.k_DMTA 0.6064 0.2623 0.9506 - -Variance model: - est. lower upper -a.1 2.660081 2.230398 3.08976 -b.1 0.001665 -0.006911 0.01024 - -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: 3.2 -mkin version used for pre-fitting: 1.2.2 -R version used for fitting: 4.2.2 -Date of fit: Sat Jan 28 11:22:45 2023 -Date of summary: Sat Jan 28 11:23:57 2023 - -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 1.552 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 112.8746 34.8816 190.868 -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: 3.2 -mkin version used for pre-fitting: 1.2.2 -R version used for fitting: 4.2.2 -Date of fit: Sat Jan 28 11:22:46 2023 -Date of summary: Sat Jan 28 11:23:57 2023 - -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 2.764 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 - 720.4 718.8 -352.2 - -Optimised parameters: - est. lower upper -DMTA_0 98.99136 97.26011 100.72261 -alpha 5.86312 2.57485 9.15138 -beta 88.55571 29.20889 147.90254 -a.1 1.51063 1.24384 1.77741 -b.1 0.02824 0.02040 0.03609 -SD.DMTA_0 1.57436 -0.04867 3.19739 -SD.alpha 0.59871 0.17132 1.02611 -SD.beta 0.72994 0.22849 1.23139 - -Correlation: - DMTA_0 alpha -alpha -0.1363 -beta -0.1414 0.2542 - -Random effects: - est. lower upper -SD.DMTA_0 1.5744 -0.04867 3.197 -SD.alpha 0.5987 0.17132 1.026 -SD.beta 0.7299 0.22849 1.231 - -Variance model: - est. lower upper -a.1 1.51063 1.2438 1.77741 -b.1 0.02824 0.0204 0.03609 - -Estimated disappearance times: - DT50 DT90 DT50back -DMTA 11.11 42.6 12.82 - -</code></pre> -<p></p> -<caption> -Hierarchical mkin fit of the DFOP model with error model const -</caption> -<pre><code> -saemix version used for fitting: 3.2 -mkin version used for pre-fitting: 1.2.2 -R version used for fitting: 4.2.2 -Date of fit: Sat Jan 28 11:22:45 2023 -Date of summary: Sat Jan 28 11:23:57 2023 - -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 1.649 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 96.573898 99.61106 -k1 0.062499 0.030336 0.09466 -k2 0.009065 -0.005133 0.02326 -g 0.948967 0.862079 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: 3.2 -mkin version used for pre-fitting: 1.2.2 -R version used for fitting: 4.2.2 -Date of fit: Sat Jan 28 11:22:46 2023 -Date of summary: Sat Jan 28 11:23:57 2023 - -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 3.288 s -Using 300, 100 iterations and 9 chains - -Variance model: Two-component variance function - -Starting values for degradation parameters: - DMTA_0 k1 k2 g -98.759266 0.087034 0.009933 0.930827 - -Fixed degradation parameter values: -None - -Starting values for random effects (square root of initial entries in omega): - DMTA_0 k1 k2 g -DMTA_0 98.76 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 - 665.5 663.4 -322.8 - -Optimised parameters: - est. lower upper -DMTA_0 98.377019 96.447952 100.30609 -k1 0.064843 0.034607 0.09508 -k2 0.008895 0.006368 0.01142 -g 0.949696 0.903815 0.99558 -a.1 1.065241 0.865754 1.26473 -b.1 0.029340 0.022336 0.03634 -SD.DMTA_0 2.007754 0.387982 3.62753 -SD.k1 0.580473 0.250286 0.91066 -SD.k2 0.006105 -4.920337 4.93255 -SD.g 1.097149 0.412779 1.78152 - -Correlation: - DMTA_0 k1 k2 -k1 0.0235 -k2 0.0595 0.0424 -g -0.0470 -0.0278 -0.2731 - -Random effects: - est. lower upper -SD.DMTA_0 2.007754 0.3880 3.6275 -SD.k1 0.580473 0.2503 0.9107 -SD.k2 0.006105 -4.9203 4.9325 -SD.g 1.097149 0.4128 1.7815 - -Variance model: - est. lower upper -a.1 1.06524 0.86575 1.26473 -b.1 0.02934 0.02234 0.03634 - -Estimated disappearance times: - DT50 DT90 DT50back DT50_k1 DT50_k2 -DMTA 11.36 41.32 12.44 10.69 77.92 - -</code></pre> -<p></p> -<caption> -Hierarchical mkin fit of the HS model with error model const -</caption> -<pre><code> -saemix version used for fitting: 3.2 -mkin version used for pre-fitting: 1.2.2 -R version used for fitting: 4.2.2 -Date of fit: Sat Jan 28 11:22:45 2023 -Date of summary: Sat Jan 28 11:23:57 2023 - -Equations: -d_DMTA/dt = - ifelse(time <= tb, k1, k2) * DMTA - -Data: -155 observations of 1 variable(s) grouped in 6 datasets - -Model predictions using solution type analytical - -Fitted in 2.006 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: 3.2 -mkin version used for pre-fitting: 1.2.2 -R version used for fitting: 4.2.2 -Date of fit: Sat Jan 28 11:22:46 2023 -Date of summary: Sat Jan 28 11:23:57 2023 - -Equations: -d_DMTA/dt = - ifelse(time <= tb, k1, k2) * DMTA - -Data: -155 observations of 1 variable(s) grouped in 6 datasets - -Model predictions using solution type analytical - -Fitted in 3.267 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 97.76570 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 0.88590 1.29080 -b.1 0.02964 0.02257 0.03671 -SD.DMTA_0 2.04877 0.42607 3.67147 -SD.k1 0.59166 0.25621 0.92711 -SD.k2 0.30698 0.09561 0.51835 -SD.tb 0.01274 -0.10914 0.13462 - -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 0.42607 3.6715 -SD.k1 0.59166 0.25621 0.9271 -SD.k2 0.30698 0.09561 0.5183 -SD.tb 0.01274 -0.10914 0.1346 - -Variance model: - est. lower upper -a.1 1.08835 0.88590 1.29080 -b.1 0.02964 0.02257 0.03671 - -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 4.2.2 Patched (2022-11-10 r83330) -Platform: x86_64-pc-linux-gnu (64-bit) -Running under: Debian GNU/Linux bookworm/sid - -Matrix products: default -BLAS: /usr/lib/x86_64-linux-gnu/openblas-serial/libblas.so.3 -LAPACK: /usr/lib/x86_64-linux-gnu/openblas-serial/libopenblas-r0.3.21.so - -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 - -attached base packages: -[1] parallel stats graphics grDevices utils datasets methods -[8] base - -other attached packages: -[1] saemix_3.2 npde_3.3 knitr_1.41 mkin_1.2.2 - -loaded via a namespace (and not attached): - [1] deSolve_1.34 zoo_1.8-11 tidyselect_1.2.0 xfun_0.35 - [5] bslib_0.4.2 purrr_1.0.0 lattice_0.20-45 colorspace_2.0-3 - [9] vctrs_0.5.1 generics_0.1.3 htmltools_0.5.4 yaml_2.3.6 -[13] utf8_1.2.2 rlang_1.0.6 pkgdown_2.0.7 jquerylib_0.1.4 -[17] pillar_1.8.1 glue_1.6.2 DBI_1.1.3 lifecycle_1.0.3 -[21] stringr_1.5.0 munsell_0.5.0 gtable_0.3.1 ragg_1.2.4 -[25] codetools_0.2-18 memoise_2.0.1 evaluate_0.19 fastmap_1.1.0 -[29] lmtest_0.9-40 fansi_1.0.3 highr_0.9 scales_1.2.1 -[33] cachem_1.0.6 desc_1.4.2 jsonlite_1.8.4 systemfonts_1.0.4 -[37] fs_1.5.2 textshaping_0.3.6 gridExtra_2.3 ggplot2_3.4.0 -[41] digest_0.6.31 stringi_1.7.8 dplyr_1.0.10 grid_4.2.2 -[45] rprojroot_2.0.3 cli_3.5.0 tools_4.2.2 magrittr_2.0.3 -[49] sass_0.4.4 tibble_3.1.8 pkgconfig_2.0.3 assertthat_0.2.1 -[53] rmarkdown_2.19 R6_2.5.1 mclust_6.0.0 nlme_3.1-161 -[57] compiler_4.2.2 </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: 64940452 kB</code></pre> -</div> -</div> - 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