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<h1>Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</h1>
<h4 data-toc-skip class="author">Johannes
Ranke</h4>
<h4 data-toc-skip class="date">Last change 12 September 2025
(rebuilt 2025-11-28)</h4>
<small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/web_only/mesotrione_parent_2023.rmd" class="external-link"><code>vignettes/web_only/mesotrione_parent_2023.rmd</code></a></small>
<div class="d-none name"><code>mesotrione_parent_2023.rmd</code></div>
</div>
<div class="section level2">
<h2 id="introduction">Introduction<a class="anchor" aria-label="anchor" href="#introduction"></a>
</h2>
<p>The purpose of this document is to test demonstrate how nonlinear
hierarchical models (NLHM) based on the parent degradation models SFO,
FOMC, DFOP and HS can be fitted with the mkin package, also considering
the influence of covariates like soil pH on different degradation
parameters. Because in some other case studies, the SFORB
parameterisation of biexponential decline has shown some advantages over
the DFOP parameterisation, SFORB was included in the list of tested
models as well.</p>
<p>The mkin package is used in version 1.2.10, which is contains the
functions that were used for the evaluations. The <code>saemix</code>
package is used as a backend for fitting the NLHM, but is also loaded to
make the convergence plot function available.</p>
<p>This document is processed with the <code>knitr</code> package, which
also provides the <code>kable</code> function that is used to improve
the display of tabular data in R markdown documents. For parallel
processing, the <code>parallel</code> package is used.</p>
<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://pkgdown.jrwb.de/mkin/">mkin</a></span><span class="op">)</span></span>
<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://yihui.org/knitr/" class="external-link">knitr</a></span><span class="op">)</span></span>
<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">saemix</span><span class="op">)</span></span>
<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">parallel</span><span class="op">)</span></span>
<span><span class="va">n_cores</span> <span class="op"><-</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 class="section level3">
<h3 id="test-data">Test data<a class="anchor" aria-label="anchor" href="#test-data"></a>
</h3>
<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">data_file</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/base/system.file.html" class="external-link">system.file</a></span><span class="op">(</span></span>
<span> <span class="st">"testdata"</span>, <span class="st">"mesotrione_soil_efsa_2016.xlsx"</span>, package <span class="op">=</span> <span class="st">"mkin"</span><span class="op">)</span></span>
<span><span class="va">meso_ds</span> <span class="op"><-</span> <span class="fu"><a href="../../reference/read_spreadsheet.html">read_spreadsheet</a></span><span class="op">(</span><span class="va">data_file</span>, parent_only <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
<p>The following tables show the covariate data and the 18 datasets that
were read in from the spreadsheet file.</p>
<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">pH</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/base/attr.html" class="external-link">attr</a></span><span class="op">(</span><span class="va">meso_ds</span>, <span class="st">"covariates"</span><span class="op">)</span></span>
<span><span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="va">pH</span>, caption <span class="op">=</span> <span class="st">"Covariate data"</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<caption>Covariate data</caption>
<thead><tr class="header">
<th align="left"></th>
<th align="right">pH</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">Richmond</td>
<td align="right">6.2</td>
</tr>
<tr class="even">
<td align="left">Richmond 2</td>
<td align="right">6.2</td>
</tr>
<tr class="odd">
<td align="left">ERTC</td>
<td align="right">6.4</td>
</tr>
<tr class="even">
<td align="left">Toulouse</td>
<td align="right">7.7</td>
</tr>
<tr class="odd">
<td align="left">Picket Piece</td>
<td align="right">7.1</td>
</tr>
<tr class="even">
<td align="left">721</td>
<td align="right">5.6</td>
</tr>
<tr class="odd">
<td align="left">722</td>
<td align="right">5.7</td>
</tr>
<tr class="even">
<td align="left">723</td>
<td align="right">5.4</td>
</tr>
<tr class="odd">
<td align="left">724</td>
<td align="right">4.8</td>
</tr>
<tr class="even">
<td align="left">725</td>
<td align="right">5.8</td>
</tr>
<tr class="odd">
<td align="left">727</td>
<td align="right">5.1</td>
</tr>
<tr class="even">
<td align="left">728</td>
<td align="right">5.9</td>
</tr>
<tr class="odd">
<td align="left">729</td>
<td align="right">5.6</td>
</tr>
<tr class="even">
<td align="left">730</td>
<td align="right">5.3</td>
</tr>
<tr class="odd">
<td align="left">731</td>
<td align="right">6.1</td>
</tr>
<tr class="even">
<td align="left">732</td>
<td align="right">5.0</td>
</tr>
<tr class="odd">
<td align="left">741</td>
<td align="right">5.7</td>
</tr>
<tr class="even">
<td align="left">742</td>
<td align="right">7.2</td>
</tr>
</tbody>
</table>
<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="kw">for</span> <span class="op">(</span><span class="va">ds_name</span> <span class="kw">in</span> <span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">meso_ds</span><span class="op">)</span><span class="op">)</span> <span class="op">{</span></span>
<span> <span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span></span>
<span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="fu"><a href="../../reference/mkin_long_to_wide.html">mkin_long_to_wide</a></span><span class="op">(</span><span class="va">meso_ds</span><span class="op">[[</span><span class="va">ds_name</span><span class="op">]</span><span class="op">]</span><span class="op">)</span>,</span>
<span> caption <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/paste.html" class="external-link">paste</a></span><span class="op">(</span><span class="st">"Dataset"</span>, <span class="va">ds_name</span><span class="op">)</span>,</span>
<span> booktabs <span class="op">=</span> <span class="cn">TRUE</span>, row.names <span class="op">=</span> <span class="cn">FALSE</span><span class="op">)</span><span class="op">)</span></span>
<span><span class="op">}</span></span></code></pre></div>
<table class="table">
<caption>Dataset Richmond</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.000000</td>
<td align="right">91.00</td>
</tr>
<tr class="even">
<td align="right">1.179050</td>
<td align="right">86.70</td>
</tr>
<tr class="odd">
<td align="right">3.537149</td>
<td align="right">73.60</td>
</tr>
<tr class="even">
<td align="right">7.074299</td>
<td align="right">61.50</td>
</tr>
<tr class="odd">
<td align="right">10.611448</td>
<td align="right">55.70</td>
</tr>
<tr class="even">
<td align="right">15.327647</td>
<td align="right">47.70</td>
</tr>
<tr class="odd">
<td align="right">17.685747</td>
<td align="right">39.50</td>
</tr>
<tr class="even">
<td align="right">24.760046</td>
<td align="right">29.80</td>
</tr>
<tr class="odd">
<td align="right">35.371494</td>
<td align="right">19.60</td>
</tr>
<tr class="even">
<td align="right">68.384889</td>
<td align="right">5.67</td>
</tr>
<tr class="odd">
<td align="right">0.000000</td>
<td align="right">97.90</td>
</tr>
<tr class="even">
<td align="right">1.179050</td>
<td align="right">96.40</td>
</tr>
<tr class="odd">
<td align="right">3.537149</td>
<td align="right">89.10</td>
</tr>
<tr class="even">
<td align="right">7.074299</td>
<td align="right">74.40</td>
</tr>
<tr class="odd">
<td align="right">10.611448</td>
<td align="right">57.40</td>
</tr>
<tr class="even">
<td align="right">15.327647</td>
<td align="right">46.30</td>
</tr>
<tr class="odd">
<td align="right">18.864797</td>
<td align="right">35.50</td>
</tr>
<tr class="even">
<td align="right">27.118146</td>
<td align="right">27.20</td>
</tr>
<tr class="odd">
<td align="right">35.371494</td>
<td align="right">19.10</td>
</tr>
<tr class="even">
<td align="right">74.280138</td>
<td align="right">6.50</td>
</tr>
<tr class="odd">
<td align="right">108.472582</td>
<td align="right">3.40</td>
</tr>
<tr class="even">
<td align="right">142.665027</td>
<td align="right">2.20</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset Richmond 2</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.000000</td>
<td align="right">96.0</td>
</tr>
<tr class="even">
<td align="right">2.422004</td>
<td align="right">82.4</td>
</tr>
<tr class="odd">
<td align="right">5.651343</td>
<td align="right">71.2</td>
</tr>
<tr class="even">
<td align="right">8.073348</td>
<td align="right">53.1</td>
</tr>
<tr class="odd">
<td align="right">11.302687</td>
<td align="right">48.5</td>
</tr>
<tr class="even">
<td align="right">16.954030</td>
<td align="right">33.4</td>
</tr>
<tr class="odd">
<td align="right">22.605373</td>
<td align="right">24.2</td>
</tr>
<tr class="even">
<td align="right">45.210746</td>
<td align="right">11.9</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset ERTC</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.000000</td>
<td align="right">99.9</td>
</tr>
<tr class="even">
<td align="right">2.755193</td>
<td align="right">80.0</td>
</tr>
<tr class="odd">
<td align="right">6.428782</td>
<td align="right">42.1</td>
</tr>
<tr class="even">
<td align="right">9.183975</td>
<td align="right">50.1</td>
</tr>
<tr class="odd">
<td align="right">12.857565</td>
<td align="right">28.4</td>
</tr>
<tr class="even">
<td align="right">19.286347</td>
<td align="right">39.8</td>
</tr>
<tr class="odd">
<td align="right">25.715130</td>
<td align="right">29.9</td>
</tr>
<tr class="even">
<td align="right">51.430259</td>
<td align="right">2.5</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset Toulouse</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.000000</td>
<td align="right">96.8</td>
</tr>
<tr class="even">
<td align="right">2.897983</td>
<td align="right">63.3</td>
</tr>
<tr class="odd">
<td align="right">6.761960</td>
<td align="right">22.3</td>
</tr>
<tr class="even">
<td align="right">9.659942</td>
<td align="right">16.6</td>
</tr>
<tr class="odd">
<td align="right">13.523919</td>
<td align="right">16.1</td>
</tr>
<tr class="even">
<td align="right">20.285879</td>
<td align="right">17.2</td>
</tr>
<tr class="odd">
<td align="right">27.047838</td>
<td align="right">1.8</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset Picket Piece</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.000000</td>
<td align="right">102.0</td>
</tr>
<tr class="even">
<td align="right">2.841195</td>
<td align="right">73.7</td>
</tr>
<tr class="odd">
<td align="right">6.629454</td>
<td align="right">35.5</td>
</tr>
<tr class="even">
<td align="right">9.470649</td>
<td align="right">31.8</td>
</tr>
<tr class="odd">
<td align="right">13.258909</td>
<td align="right">18.0</td>
</tr>
<tr class="even">
<td align="right">19.888364</td>
<td align="right">3.7</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 721</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">86.4</td>
</tr>
<tr class="even">
<td align="right">11.24366</td>
<td align="right">61.4</td>
</tr>
<tr class="odd">
<td align="right">22.48733</td>
<td align="right">49.8</td>
</tr>
<tr class="even">
<td align="right">33.73099</td>
<td align="right">41.0</td>
</tr>
<tr class="odd">
<td align="right">44.97466</td>
<td align="right">35.1</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 722</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">90.3</td>
</tr>
<tr class="even">
<td align="right">11.24366</td>
<td align="right">52.1</td>
</tr>
<tr class="odd">
<td align="right">22.48733</td>
<td align="right">37.4</td>
</tr>
<tr class="even">
<td align="right">33.73099</td>
<td align="right">21.2</td>
</tr>
<tr class="odd">
<td align="right">44.97466</td>
<td align="right">14.3</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 723</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">89.3</td>
</tr>
<tr class="even">
<td align="right">11.24366</td>
<td align="right">70.8</td>
</tr>
<tr class="odd">
<td align="right">22.48733</td>
<td align="right">51.1</td>
</tr>
<tr class="even">
<td align="right">33.73099</td>
<td align="right">42.7</td>
</tr>
<tr class="odd">
<td align="right">44.97466</td>
<td align="right">26.7</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 724</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.000000</td>
<td align="right">89.4</td>
</tr>
<tr class="even">
<td align="right">9.008208</td>
<td align="right">65.2</td>
</tr>
<tr class="odd">
<td align="right">18.016415</td>
<td align="right">55.8</td>
</tr>
<tr class="even">
<td align="right">27.024623</td>
<td align="right">46.0</td>
</tr>
<tr class="odd">
<td align="right">36.032831</td>
<td align="right">41.7</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 725</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">89.0</td>
</tr>
<tr class="even">
<td align="right">10.99058</td>
<td align="right">35.4</td>
</tr>
<tr class="odd">
<td align="right">21.98116</td>
<td align="right">18.6</td>
</tr>
<tr class="even">
<td align="right">32.97174</td>
<td align="right">11.6</td>
</tr>
<tr class="odd">
<td align="right">43.96232</td>
<td align="right">7.6</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 727</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">91.3</td>
</tr>
<tr class="even">
<td align="right">10.96104</td>
<td align="right">63.2</td>
</tr>
<tr class="odd">
<td align="right">21.92209</td>
<td align="right">51.1</td>
</tr>
<tr class="even">
<td align="right">32.88313</td>
<td align="right">42.0</td>
</tr>
<tr class="odd">
<td align="right">43.84417</td>
<td align="right">40.8</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 728</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">91.8</td>
</tr>
<tr class="even">
<td align="right">11.24366</td>
<td align="right">43.6</td>
</tr>
<tr class="odd">
<td align="right">22.48733</td>
<td align="right">22.0</td>
</tr>
<tr class="even">
<td align="right">33.73099</td>
<td align="right">15.9</td>
</tr>
<tr class="odd">
<td align="right">44.97466</td>
<td align="right">8.8</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 729</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">91.6</td>
</tr>
<tr class="even">
<td align="right">11.24366</td>
<td align="right">60.5</td>
</tr>
<tr class="odd">
<td align="right">22.48733</td>
<td align="right">43.5</td>
</tr>
<tr class="even">
<td align="right">33.73099</td>
<td align="right">28.4</td>
</tr>
<tr class="odd">
<td align="right">44.97466</td>
<td align="right">20.5</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 730</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">92.7</td>
</tr>
<tr class="even">
<td align="right">11.07446</td>
<td align="right">58.9</td>
</tr>
<tr class="odd">
<td align="right">22.14893</td>
<td align="right">44.0</td>
</tr>
<tr class="even">
<td align="right">33.22339</td>
<td align="right">46.0</td>
</tr>
<tr class="odd">
<td align="right">44.29785</td>
<td align="right">29.3</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 731</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">92.1</td>
</tr>
<tr class="even">
<td align="right">11.24366</td>
<td align="right">64.4</td>
</tr>
<tr class="odd">
<td align="right">22.48733</td>
<td align="right">45.3</td>
</tr>
<tr class="even">
<td align="right">33.73099</td>
<td align="right">33.6</td>
</tr>
<tr class="odd">
<td align="right">44.97466</td>
<td align="right">23.5</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 732</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">90.3</td>
</tr>
<tr class="even">
<td align="right">11.24366</td>
<td align="right">58.2</td>
</tr>
<tr class="odd">
<td align="right">22.48733</td>
<td align="right">40.1</td>
</tr>
<tr class="even">
<td align="right">33.73099</td>
<td align="right">33.1</td>
</tr>
<tr class="odd">
<td align="right">44.97466</td>
<td align="right">25.8</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 741</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">90.3</td>
</tr>
<tr class="even">
<td align="right">10.84712</td>
<td align="right">68.7</td>
</tr>
<tr class="odd">
<td align="right">21.69424</td>
<td align="right">58.0</td>
</tr>
<tr class="even">
<td align="right">32.54136</td>
<td align="right">52.2</td>
</tr>
<tr class="odd">
<td align="right">43.38848</td>
<td align="right">48.0</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset 742</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">meso</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.00000</td>
<td align="right">92.0</td>
</tr>
<tr class="even">
<td align="right">11.24366</td>
<td align="right">60.9</td>
</tr>
<tr class="odd">
<td align="right">22.48733</td>
<td align="right">36.2</td>
</tr>
<tr class="even">
<td align="right">33.73099</td>
<td align="right">18.3</td>
</tr>
<tr class="odd">
<td align="right">44.97466</td>
<td align="right">8.7</td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="section level2">
<h2 id="separate-evaluations">Separate evaluations<a class="anchor" aria-label="anchor" href="#separate-evaluations"></a>
</h2>
<p>In order to obtain suitable starting parameters for the NLHM fits,
separate fits of the five models to the data for each soil are generated
using the <code>mmkin</code> function from the mkin package. In a first
step, constant variance is assumed. Convergence is checked with the
<code>status</code> function.</p>
<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">deg_mods</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"FOMC"</span>, <span class="st">"DFOP"</span>, <span class="st">"SFORB"</span>, <span class="st">"HS"</span><span class="op">)</span></span>
<span><span class="va">f_sep_const</span> <span class="op"><-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span></span>
<span> <span class="va">deg_mods</span>,</span>
<span> <span class="va">meso_ds</span>,</span>
<span> error_model <span class="op">=</span> <span class="st">"const"</span>,</span>
<span> cluster <span class="op">=</span> <span class="va">cl</span>,</span>
<span> quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span>, <span class="fl">1</span><span class="op">:</span><span class="fl">5</span><span class="op">]</span><span class="op">)</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="left">Richmond</th>
<th align="left">Richmond 2</th>
<th align="left">ERTC</th>
<th align="left">Toulouse</th>
<th align="left">Picket Piece</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">SFORB</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">HS</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
</tbody>
</table>
<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span>, <span class="fl">6</span><span class="op">:</span><span class="fl">18</span><span class="op">]</span><span class="op">)</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<colgroup>
<col width="10%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
</colgroup>
<thead><tr class="header">
<th align="left"></th>
<th align="left">721</th>
<th align="left">722</th>
<th align="left">723</th>
<th align="left">724</th>
<th align="left">725</th>
<th align="left">727</th>
<th align="left">728</th>
<th align="left">729</th>
<th align="left">730</th>
<th align="left">731</th>
<th align="left">732</th>
<th align="left">741</th>
<th align="left">742</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">SFORB</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">HS</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
</tbody>
</table>
<p>In the tables above, OK indicates convergence and C indicates failure
to converge. Most separate fits with constant variance converged, with
the exception of two FOMC fits, one SFORB fit and one HS fit.</p>
<div class="sourceCode" id="cb8"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">f_sep_tc</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_sep_const</span>, error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span></span></code></pre></div>
<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_tc</span><span class="op">[</span>, <span class="fl">1</span><span class="op">:</span><span class="fl">5</span><span class="op">]</span><span class="op">)</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="left">Richmond</th>
<th align="left">Richmond 2</th>
<th align="left">ERTC</th>
<th align="left">Toulouse</th>
<th align="left">Picket Piece</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">SFORB</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">HS</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
</tbody>
</table>
<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_tc</span><span class="op">[</span>, <span class="fl">6</span><span class="op">:</span><span class="fl">18</span><span class="op">]</span><span class="op">)</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<colgroup>
<col width="10%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
<col width="6%">
</colgroup>
<thead><tr class="header">
<th align="left"></th>
<th align="left">721</th>
<th align="left">722</th>
<th align="left">723</th>
<th align="left">724</th>
<th align="left">725</th>
<th align="left">727</th>
<th align="left">728</th>
<th align="left">729</th>
<th align="left">730</th>
<th align="left">731</th>
<th align="left">732</th>
<th align="left">741</th>
<th align="left">742</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">SFORB</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">HS</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
</tbody>
</table>
<p>With the two-component error model, the set of fits that did not
converge is larger, with convergence problems appearing for a number of
non-SFO fits.</p>
</div>
<div class="section level2">
<h2 id="hierarchical-models-without-covariate">Hierarchical models without covariate<a class="anchor" aria-label="anchor" href="#hierarchical-models-without-covariate"></a>
</h2>
<p>The following code fits hierarchical kinetic models for the ten
combinations of the five different degradation models with the two
different error models in parallel.</p>
<div class="sourceCode" id="cb11"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">f_saem_1</span> <span class="op"><-</span> <span class="fu"><a href="../../reference/mhmkin.html">mhmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">f_sep_const</span>, <span class="va">f_sep_tc</span><span class="op">)</span>, cluster <span class="op">=</span> <span class="va">cl</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">)</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="left">const</th>
<th align="left">tc</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">SFORB</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">HS</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
</tbody>
</table>
<p>All fits terminate without errors (status OK).</p>
<div class="sourceCode" id="cb12"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">)</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="right">npar</th>
<th align="right">AIC</th>
<th align="right">BIC</th>
<th align="right">Lik</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO const</td>
<td align="right">5</td>
<td align="right">800.0</td>
<td align="right">804.5</td>
<td align="right">-395.0</td>
</tr>
<tr class="even">
<td align="left">SFO tc</td>
<td align="right">6</td>
<td align="right">801.9</td>
<td align="right">807.2</td>
<td align="right">-394.9</td>
</tr>
<tr class="odd">
<td align="left">FOMC const</td>
<td align="right">7</td>
<td align="right">787.4</td>
<td align="right">793.6</td>
<td align="right">-386.7</td>
</tr>
<tr class="even">
<td align="left">FOMC tc</td>
<td align="right">8</td>
<td align="right">788.9</td>
<td align="right">796.1</td>
<td align="right">-386.5</td>
</tr>
<tr class="odd">
<td align="left">DFOP const</td>
<td align="right">9</td>
<td align="right">787.6</td>
<td align="right">795.6</td>
<td align="right">-384.8</td>
</tr>
<tr class="even">
<td align="left">SFORB const</td>
<td align="right">9</td>
<td align="right">787.4</td>
<td align="right">795.4</td>
<td align="right">-384.7</td>
</tr>
<tr class="odd">
<td align="left">HS const</td>
<td align="right">9</td>
<td align="right">781.9</td>
<td align="right">789.9</td>
<td align="right">-382.0</td>
</tr>
<tr class="even">
<td align="left">DFOP tc</td>
<td align="right">10</td>
<td align="right">787.4</td>
<td align="right">796.3</td>
<td align="right">-383.7</td>
</tr>
<tr class="odd">
<td align="left">SFORB tc</td>
<td align="right">10</td>
<td align="right">795.8</td>
<td align="right">804.7</td>
<td align="right">-387.9</td>
</tr>
<tr class="even">
<td align="left">HS tc</td>
<td align="right">10</td>
<td align="right">783.7</td>
<td align="right">792.7</td>
<td align="right">-381.9</td>
</tr>
</tbody>
</table>
<p>The model comparisons show that the fits with constant variance are
consistently preferable to the corresponding fits with two-component
error for these data. This is confirmed by the fact that the parameter
<code>b.1</code> (the relative standard deviation in the fits obtained
with the saemix package), is ill-defined in all fits.</p>
<div class="sourceCode" id="cb13"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">)</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<colgroup>
<col width="6%">
<col width="44%">
<col width="49%">
</colgroup>
<thead><tr class="header">
<th align="left"></th>
<th align="left">const</th>
<th align="left">tc</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left">sd(meso_0)</td>
<td align="left">sd(meso_0), b.1</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left">sd(meso_0), sd(log_beta)</td>
<td align="left">sd(meso_0), sd(log_beta), b.1</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left">sd(meso_0), sd(log_k1)</td>
<td align="left">sd(meso_0), sd(g_qlogis), b.1</td>
</tr>
<tr class="even">
<td align="left">SFORB</td>
<td align="left">sd(meso_free_0), sd(log_k_meso_free_bound)</td>
<td align="left">sd(meso_free_0), sd(log_k_meso_free_bound), b.1</td>
</tr>
<tr class="odd">
<td align="left">HS</td>
<td align="left">sd(meso_0)</td>
<td align="left">sd(meso_0), b.1</td>
</tr>
</tbody>
</table>
<p>For obtaining fits with only well-defined random effects, we update
the set of fits, excluding random effects for degradation parameters
that were ill-defined according to the <code>illparms</code>
function.</p>
<div class="sourceCode" id="cb14"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">f_saem_2</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_saem_1</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_1</span><span class="op">)</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">)</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="left">const</th>
<th align="left">tc</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">SFORB</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">HS</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
</tbody>
</table>
<p>The updated fits terminate without errors, and the only
ill-defined</p>
<div class="sourceCode" id="cb15"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">)</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="left">const</th>
<th align="left">tc</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left"></td>
<td align="left">b.1</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left"></td>
<td align="left">b.1</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left"></td>
<td align="left">b.1</td>
</tr>
<tr class="even">
<td align="left">SFORB</td>
<td align="left"></td>
<td align="left">b.1</td>
</tr>
<tr class="odd">
<td align="left">HS</td>
<td align="left"></td>
<td align="left">b.1</td>
</tr>
</tbody>
</table>
<p>No ill-defined errors remain in the fits with constant variance.</p>
</div>
<div class="section level2">
<h2 id="hierarchical-models-with-covariate">Hierarchical models with covariate<a class="anchor" aria-label="anchor" href="#hierarchical-models-with-covariate"></a>
</h2>
<p>In the following sections, hierarchical fits including a model for
the influence of pH on selected degradation parameters are shown for all
parent models. Constant variance is selected as the error model based on
the fits without covariate effects. Random effects that were ill-defined
in the fits without pH influence are excluded. A potential influence of
the soil pH is only included for parameters with a well-defined random
effect, because experience has shown that only for such parameters a
significant pH effect could be found.</p>
<div class="section level3">
<h3 id="sfo">SFO<a class="anchor" aria-label="anchor" href="#sfo"></a>
</h3>
<div class="sourceCode" id="cb16"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">sfo_pH</span> <span class="op"><-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"SFO"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="st">"meso_0"</span>, covariates <span class="op">=</span> <span class="va">pH</span>,</span>
<span> covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k_meso</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span>, center_covariates <span class="op">=</span> <span class="st">"median"</span><span class="op">)</span></span></code></pre></div>
<div class="sourceCode" id="cb17"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">sfo_pH</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="right">est.</th>
<th align="right">lower</th>
<th align="right">upper</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">meso_0</td>
<td align="right">91.35</td>
<td align="right">89.27</td>
<td align="right">93.43</td>
</tr>
<tr class="even">
<td align="left">log_k_meso</td>
<td align="right">-3.29</td>
<td align="right">-3.46</td>
<td align="right">-3.11</td>
</tr>
<tr class="odd">
<td align="left">beta_pH(log_k_meso)</td>
<td align="right">0.59</td>
<td align="right">0.37</td>
<td align="right">0.81</td>
</tr>
<tr class="even">
<td align="left">a.1</td>
<td align="right">5.48</td>
<td align="right">4.71</td>
<td align="right">6.24</td>
</tr>
<tr class="odd">
<td align="left">SD.log_k_meso</td>
<td align="right">0.35</td>
<td align="right">0.23</td>
<td align="right">0.47</td>
</tr>
</tbody>
</table>
<p>The parameter showing the pH influence in the above table is
<code>beta_pH(log_k_meso)</code>. Its confidence interval does not
include zero, indicating that the influence of soil pH on the log of the
degradation rate constant is significantly greater than zero.</p>
<div class="sourceCode" id="cb18"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"SFO"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span>, <span class="va">sfo_pH</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
npar AIC BIC Lik Chisq Df Pr(>Chisq)
f_saem_2[["SFO", "const"]] 4 797.56 801.12 -394.78
sfo_pH 5 783.09 787.54 -386.54 16.473 1 4.934e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
<p>The comparison with the SFO fit without covariate effect confirms
that considering the soil pH improves the model, both by comparison of
AIC and BIC and by the likelihood ratio test.</p>
<div class="sourceCode" id="cb20"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">sfo_pH</span><span class="op">)</span></span></code></pre></div>
<p><img src="mesotrione_parent_2023_files/figure-html/unnamed-chunk-8-1.png" width="700" style="display: block; margin: auto;"></p>
<p>Endpoints for a model with covariates are by default calculated for
the median of the covariate values. This quantile can be adapted, or a
specific covariate value can be given as shown below.</p>
<div class="sourceCode" id="cb21"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">sfo_pH</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
50% 5.75
$distimes
DT50 DT90
meso 18.52069 61.52441</code></pre>
<div class="sourceCode" id="cb23"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">sfo_pH</span>, covariate_quantile <span class="op">=</span> <span class="fl">0.9</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
90% 7.13
$distimes
DT50 DT90
meso 8.237019 27.36278</code></pre>
<div class="sourceCode" id="cb25"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">sfo_pH</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7.0</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
User 7
$distimes
DT50 DT90
meso 8.89035 29.5331</code></pre>
</div>
<div class="section level3">
<h3 id="fomc">FOMC<a class="anchor" aria-label="anchor" href="#fomc"></a>
</h3>
<div class="sourceCode" id="cb27"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">fomc_pH</span> <span class="op"><-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"FOMC"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="st">"meso_0"</span>, covariates <span class="op">=</span> <span class="va">pH</span>,</span>
<span> covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_alpha</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span>, center_covariates <span class="op">=</span> <span class="st">"median"</span><span class="op">)</span></span></code></pre></div>
<div class="sourceCode" id="cb28"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">fomc_pH</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="right">est.</th>
<th align="right">lower</th>
<th align="right">upper</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">meso_0</td>
<td align="right">92.84</td>
<td align="right">90.75</td>
<td align="right">94.93</td>
</tr>
<tr class="even">
<td align="left">log_alpha</td>
<td align="right">1.11</td>
<td align="right">0.48</td>
<td align="right">1.75</td>
</tr>
<tr class="odd">
<td align="left">beta_pH(log_alpha)</td>
<td align="right">0.58</td>
<td align="right">0.37</td>
<td align="right">0.79</td>
</tr>
<tr class="even">
<td align="left">log_beta</td>
<td align="right">4.21</td>
<td align="right">3.44</td>
<td align="right">4.99</td>
</tr>
<tr class="odd">
<td align="left">a.1</td>
<td align="right">5.03</td>
<td align="right">4.32</td>
<td align="right">5.73</td>
</tr>
<tr class="even">
<td align="left">SD.log_alpha</td>
<td align="right">0.00</td>
<td align="right">-23.77</td>
<td align="right">23.78</td>
</tr>
<tr class="odd">
<td align="left">SD.log_beta</td>
<td align="right">0.37</td>
<td align="right">0.01</td>
<td align="right">0.74</td>
</tr>
</tbody>
</table>
<p>As in the case of SFO, the confidence interval of the slope parameter
(here <code>beta_pH(log_alpha)</code>) quantifying the influence of soil
pH does not include zero, and the model comparison clearly indicates
that the model with covariate influence is preferable. However, the
random effect for <code>alpha</code> is not well-defined any more after
inclusion of the covariate effect (the confidence interval of
<code>SD.log_alpha</code> includes zero).</p>
<div class="sourceCode" id="cb29"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">fomc_pH</span><span class="op">)</span></span></code></pre></div>
<pre><code>[1] "sd(log_alpha)"</code></pre>
<p>Therefore, the model is updated without this random effect, and no
ill-defined parameters remain.</p>
<div class="sourceCode" id="cb31"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">fomc_pH_2</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">fomc_pH</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span>, <span class="st">"log_alpha"</span><span class="op">)</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">fomc_pH_2</span><span class="op">)</span></span></code></pre></div>
<div class="sourceCode" id="cb32"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"FOMC"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span>, <span class="va">fomc_pH</span>, <span class="va">fomc_pH_2</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
npar AIC BIC Lik Chisq Df Pr(>Chisq)
f_saem_2[["FOMC", "const"]] 5 783.25 787.71 -386.63
fomc_pH_2 6 766.50 771.84 -377.25 18.753 1 1.488e-05 ***
fomc_pH 7 770.07 776.30 -378.04 0.000 1 1
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
<p>Model comparison indicates that including pH dependence significantly
improves the fit, and that the reduced model with covariate influence
results in the most preferable FOMC fit.</p>
<div class="sourceCode" id="cb34"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">fomc_pH_2</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="right">est.</th>
<th align="right">lower</th>
<th align="right">upper</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">meso_0</td>
<td align="right">93.20</td>
<td align="right">91.10</td>
<td align="right">95.29</td>
</tr>
<tr class="even">
<td align="left">log_alpha</td>
<td align="right">0.81</td>
<td align="right">0.33</td>
<td align="right">1.30</td>
</tr>
<tr class="odd">
<td align="left">beta_pH(log_alpha)</td>
<td align="right">0.55</td>
<td align="right">0.35</td>
<td align="right">0.75</td>
</tr>
<tr class="even">
<td align="left">log_beta</td>
<td align="right">3.85</td>
<td align="right">3.22</td>
<td align="right">4.47</td>
</tr>
<tr class="odd">
<td align="left">a.1</td>
<td align="right">5.00</td>
<td align="right">4.30</td>
<td align="right">5.70</td>
</tr>
<tr class="even">
<td align="left">SD.log_beta</td>
<td align="right">0.38</td>
<td align="right">0.25</td>
<td align="right">0.52</td>
</tr>
</tbody>
</table>
<div class="sourceCode" id="cb35"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">fomc_pH_2</span><span class="op">)</span></span></code></pre></div>
<p><img src="mesotrione_parent_2023_files/figure-html/unnamed-chunk-14-1.png" width="700" style="display: block; margin: auto;"></p>
<div class="sourceCode" id="cb36"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">fomc_pH_2</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
50% 5.75
$distimes
DT50 DT90 DT50back
meso 16.86448 83.26704 25.06588</code></pre>
<div class="sourceCode" id="cb38"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">fomc_pH_2</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
User 7
$distimes
DT50 DT90 DT50back
meso 7.835768 31.46451 9.47176</code></pre>
</div>
<div class="section level3">
<h3 id="dfop">DFOP<a class="anchor" aria-label="anchor" href="#dfop"></a>
</h3>
<p>In the DFOP fits without covariate effects, random effects for two
degradation parameters (<code>k2</code> and <code>g</code>) were
identifiable.</p>
<div class="sourceCode" id="cb40"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="right">est.</th>
<th align="right">lower</th>
<th align="right">upper</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">meso_0</td>
<td align="right">93.61</td>
<td align="right">91.58</td>
<td align="right">95.63</td>
</tr>
<tr class="even">
<td align="left">log_k1</td>
<td align="right">-1.53</td>
<td align="right">-2.27</td>
<td align="right">-0.79</td>
</tr>
<tr class="odd">
<td align="left">log_k2</td>
<td align="right">-3.42</td>
<td align="right">-3.73</td>
<td align="right">-3.11</td>
</tr>
<tr class="even">
<td align="left">g_qlogis</td>
<td align="right">-1.67</td>
<td align="right">-2.57</td>
<td align="right">-0.77</td>
</tr>
<tr class="odd">
<td align="left">a.1</td>
<td align="right">4.74</td>
<td align="right">4.02</td>
<td align="right">5.45</td>
</tr>
<tr class="even">
<td align="left">SD.log_k2</td>
<td align="right">0.60</td>
<td align="right">0.38</td>
<td align="right">0.81</td>
</tr>
<tr class="odd">
<td align="left">SD.g_qlogis</td>
<td align="right">0.94</td>
<td align="right">0.33</td>
<td align="right">1.54</td>
</tr>
</tbody>
</table>
<p>A fit with pH dependent degradation parameters was obtained by
excluding the same random effects as in the refined DFOP fit without
covariate influence, and including covariate models for the two
identifiable parameters <code>k2</code> and <code>g</code>.</p>
<div class="sourceCode" id="cb41"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">dfop_pH</span> <span class="op"><-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span>, <span class="st">"log_k1"</span><span class="op">)</span>,</span>
<span> covariates <span class="op">=</span> <span class="va">pH</span>,</span>
<span> covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k2</span> <span class="op">~</span> <span class="va">pH</span>, <span class="va">g_qlogis</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span>,</span>
<span> center_covariates <span class="op">=</span> <span class="st">"median"</span></span>
<span><span class="op">)</span></span></code></pre></div>
<p>The corresponding parameters for the influence of soil pH are
<code>beta_pH(log_k2)</code> for the influence of soil pH on
<code>k2</code>, and <code>beta_pH(g_qlogis)</code> for its influence on
<code>g</code>.</p>
<div class="sourceCode" id="cb42"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">dfop_pH</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="right">est.</th>
<th align="right">lower</th>
<th align="right">upper</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">meso_0</td>
<td align="right">92.84</td>
<td align="right">90.85</td>
<td align="right">94.84</td>
</tr>
<tr class="even">
<td align="left">log_k1</td>
<td align="right">-2.82</td>
<td align="right">-3.09</td>
<td align="right">-2.54</td>
</tr>
<tr class="odd">
<td align="left">log_k2</td>
<td align="right">-3.96</td>
<td align="right">-4.47</td>
<td align="right">-3.44</td>
</tr>
<tr class="even">
<td align="left">beta_pH(log_k2)</td>
<td align="right">1.31</td>
<td align="right">0.69</td>
<td align="right">1.92</td>
</tr>
<tr class="odd">
<td align="left">g_qlogis</td>
<td align="right">-0.12</td>
<td align="right">-0.57</td>
<td align="right">0.33</td>
</tr>
<tr class="even">
<td align="left">beta_pH(g_qlogis)</td>
<td align="right">-0.57</td>
<td align="right">-1.04</td>
<td align="right">-0.09</td>
</tr>
<tr class="odd">
<td align="left">a.1</td>
<td align="right">4.96</td>
<td align="right">4.26</td>
<td align="right">5.65</td>
</tr>
<tr class="even">
<td align="left">SD.log_k2</td>
<td align="right">0.76</td>
<td align="right">0.47</td>
<td align="right">1.05</td>
</tr>
<tr class="odd">
<td align="left">SD.g_qlogis</td>
<td align="right">0.01</td>
<td align="right">-9.96</td>
<td align="right">9.97</td>
</tr>
</tbody>
</table>
<div class="sourceCode" id="cb43"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">dfop_pH</span><span class="op">)</span></span></code></pre></div>
<pre><code>[1] "sd(g_qlogis)"</code></pre>
<p>Confidence intervals for neither of them include zero, indicating a
significant difference from zero. However, the random effect for
<code>g</code> is now ill-defined. The fit is updated without this
ill-defined random effect.</p>
<div class="sourceCode" id="cb45"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">dfop_pH_2</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">dfop_pH</span>,</span>
<span> no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span>, <span class="st">"log_k1"</span>, <span class="st">"g_qlogis"</span><span class="op">)</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">dfop_pH_2</span><span class="op">)</span></span></code></pre></div>
<pre><code>[1] "beta_pH(g_qlogis)"</code></pre>
<p>Now, the slope parameter for the pH effect on <code>g</code> is
ill-defined. Therefore, another attempt is made without the
corresponding covariate model.</p>
<div class="sourceCode" id="cb47"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">dfop_pH_3</span> <span class="op"><-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"DFOP"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span>, <span class="st">"log_k1"</span><span class="op">)</span>,</span>
<span> covariates <span class="op">=</span> <span class="va">pH</span>,</span>
<span> covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k2</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span>,</span>
<span> center_covariates <span class="op">=</span> <span class="st">"median"</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">dfop_pH_3</span><span class="op">)</span></span></code></pre></div>
<pre><code>[1] "sd(g_qlogis)"</code></pre>
<p>As the random effect for <code>g</code> is again ill-defined, the fit
is repeated without it.</p>
<div class="sourceCode" id="cb49"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">dfop_pH_4</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">dfop_pH_3</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span>, <span class="st">"log_k1"</span>, <span class="st">"g_qlogis"</span><span class="op">)</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">dfop_pH_4</span><span class="op">)</span></span></code></pre></div>
<p>While no ill-defined parameters remain, model comparison suggests
that the previous model <code>dfop_pH_2</code> with two pH dependent
parameters is preferable, based on information criteria as well as based
on the likelihood ratio test.</p>
<div class="sourceCode" id="cb50"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span>, <span class="va">dfop_pH</span>, <span class="va">dfop_pH_2</span>, <span class="va">dfop_pH_3</span>, <span class="va">dfop_pH_4</span><span class="op">)</span></span></code></pre></div>
<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
npar AIC BIC Lik
f_saem_2[["DFOP", "const"]] 7 782.94 789.18 -384.47
dfop_pH_4 7 767.35 773.58 -376.68
dfop_pH_2 8 765.06 772.18 -374.53
dfop_pH_3 8 769.00 776.12 -376.50
dfop_pH 9 769.10 777.11 -375.55</code></pre>
<div class="sourceCode" id="cb52"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">dfop_pH_2</span>, <span class="va">dfop_pH_4</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
npar AIC BIC Lik Chisq Df Pr(>Chisq)
dfop_pH_4 7 767.35 773.58 -376.68
dfop_pH_2 8 765.06 772.18 -374.53 4.2909 1 0.03832 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
<p>When focussing on parameter identifiability using the test if the
confidence interval includes zero, <code>dfop_pH_4</code> would still be
the preferred model. However, it should be kept in mind that parameter
confidence intervals are constructed using a simple linearisation of the
likelihood. As the confidence interval of the random effect for
<code>g</code> only marginally includes zero, it is suggested that this
is acceptable, and that <code>dfop_pH_2</code> can be considered the
most preferable model.</p>
<div class="sourceCode" id="cb54"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">dfop_pH_2</span><span class="op">)</span></span></code></pre></div>
<p><img src="mesotrione_parent_2023_files/figure-html/unnamed-chunk-19-1.png" width="700" style="display: block; margin: auto;"></p>
<div class="sourceCode" id="cb55"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">dfop_pH_2</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
50% 5.75
$distimes
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 18.46687 74.91602 22.55197 4.568623 24.64483</code></pre>
<div class="sourceCode" id="cb57"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">dfop_pH_2</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
User 7
$distimes
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 8.370528 28.37824 8.542701 4.568623 8.785409</code></pre>
</div>
<div class="section level3">
<h3 id="sforb">SFORB<a class="anchor" aria-label="anchor" href="#sforb"></a>
</h3>
<div class="sourceCode" id="cb59"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">sforb_pH</span> <span class="op"><-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"SFORB"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_free_0"</span>, <span class="st">"log_k_meso_free_bound"</span><span class="op">)</span>,</span>
<span> covariates <span class="op">=</span> <span class="va">pH</span>,</span>
<span> covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k_meso_free</span> <span class="op">~</span> <span class="va">pH</span>, <span class="va">log_k_meso_bound_free</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span>,</span>
<span> center_covariates <span class="op">=</span> <span class="st">"median"</span><span class="op">)</span></span></code></pre></div>
<div class="sourceCode" id="cb60"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">sforb_pH</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="right">est.</th>
<th align="right">lower</th>
<th align="right">upper</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">meso_free_0</td>
<td align="right">93.42</td>
<td align="right">91.32</td>
<td align="right">95.52</td>
</tr>
<tr class="even">
<td align="left">log_k_meso_free</td>
<td align="right">-2.94</td>
<td align="right">-3.23</td>
<td align="right">-2.65</td>
</tr>
<tr class="odd">
<td align="left">beta_pH(log_k_meso_free)</td>
<td align="right">0.42</td>
<td align="right">0.18</td>
<td align="right">0.67</td>
</tr>
<tr class="even">
<td align="left">log_k_meso_free_bound</td>
<td align="right">-3.49</td>
<td align="right">-4.92</td>
<td align="right">-2.05</td>
</tr>
<tr class="odd">
<td align="left">log_k_meso_bound_free</td>
<td align="right">-2.91</td>
<td align="right">-4.30</td>
<td align="right">-1.52</td>
</tr>
<tr class="even">
<td align="left">beta_pH(log_k_meso_bound_free)</td>
<td align="right">1.23</td>
<td align="right">-0.21</td>
<td align="right">2.67</td>
</tr>
<tr class="odd">
<td align="left">a.1</td>
<td align="right">4.90</td>
<td align="right">4.18</td>
<td align="right">5.63</td>
</tr>
<tr class="even">
<td align="left">SD.log_k_meso_free</td>
<td align="right">0.35</td>
<td align="right">0.23</td>
<td align="right">0.47</td>
</tr>
<tr class="odd">
<td align="left">SD.log_k_meso_bound_free</td>
<td align="right">0.13</td>
<td align="right">-1.95</td>
<td align="right">2.20</td>
</tr>
</tbody>
</table>
<p>The confidence interval of
<code>beta_pH(log_k_meso_bound_free)</code> includes zero, indicating
that the influence of soil pH on <code>k_meso_bound_free</code> cannot
reliably be quantified. Also, the confidence interval for the random
effect on this parameter (<code>SD.log_k_meso_bound_free</code>)
includes zero.</p>
<p>Using the <code>illparms</code> function, these ill-defined
parameters can be found more conveniently.</p>
<div class="sourceCode" id="cb61"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">sforb_pH</span><span class="op">)</span></span></code></pre></div>
<pre><code>[1] "sd(log_k_meso_bound_free)" "beta_pH(log_k_meso_bound_free)"</code></pre>
<p>To remove the ill-defined parameters, a second variant of the SFORB
model with pH influence is fitted. No ill-defined parameters remain.</p>
<div class="sourceCode" id="cb63"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">sforb_pH_2</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">sforb_pH</span>,</span>
<span> no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_free_0"</span>, <span class="st">"log_k_meso_free_bound"</span>, <span class="st">"log_k_meso_bound_free"</span><span class="op">)</span>,</span>
<span> covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k_meso_free</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">sforb_pH_2</span><span class="op">)</span></span></code></pre></div>
<p>The model comparison of the SFORB fits includes the refined model
without covariate effect, and both versions of the SFORB fit with
covariate effect.</p>
<div class="sourceCode" id="cb64"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"SFORB"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span>, <span class="va">sforb_pH</span>, <span class="va">sforb_pH_2</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
npar AIC BIC Lik Chisq Df Pr(>Chisq)
f_saem_2[["SFORB", "const"]] 7 783.40 789.63 -384.70
sforb_pH_2 7 770.94 777.17 -378.47 12.4616 0
sforb_pH 9 768.81 776.83 -375.41 6.1258 2 0.04675 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
<p>The first model including pH influence is preferable based on
information criteria and the likelihood ratio test. However, as it is
not fully identifiable, the second model is selected.</p>
<div class="sourceCode" id="cb66"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">sforb_pH_2</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="right">est.</th>
<th align="right">lower</th>
<th align="right">upper</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">meso_free_0</td>
<td align="right">93.32</td>
<td align="right">91.17</td>
<td align="right">95.47</td>
</tr>
<tr class="even">
<td align="left">log_k_meso_free</td>
<td align="right">-3.02</td>
<td align="right">-3.27</td>
<td align="right">-2.78</td>
</tr>
<tr class="odd">
<td align="left">beta_pH(log_k_meso_free)</td>
<td align="right">0.54</td>
<td align="right">0.33</td>
<td align="right">0.75</td>
</tr>
<tr class="even">
<td align="left">log_k_meso_free_bound</td>
<td align="right">-3.80</td>
<td align="right">-5.18</td>
<td align="right">-2.42</td>
</tr>
<tr class="odd">
<td align="left">log_k_meso_bound_free</td>
<td align="right">-2.95</td>
<td align="right">-4.24</td>
<td align="right">-1.65</td>
</tr>
<tr class="even">
<td align="left">a.1</td>
<td align="right">5.08</td>
<td align="right">4.37</td>
<td align="right">5.79</td>
</tr>
<tr class="odd">
<td align="left">SD.log_k_meso_free</td>
<td align="right">0.33</td>
<td align="right">0.22</td>
<td align="right">0.45</td>
</tr>
</tbody>
</table>
<div class="sourceCode" id="cb67"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">sforb_pH_2</span><span class="op">)</span></span></code></pre></div>
<p><img src="mesotrione_parent_2023_files/figure-html/unnamed-chunk-25-1.png" width="700" style="display: block; margin: auto;"></p>
<div class="sourceCode" id="cb68"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">sforb_pH_2</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
50% 5.75
$ff
meso_free
1
$SFORB
meso_b1 meso_b2 meso_g
0.09735824 0.02631699 0.31602120
$distimes
DT50 DT90 DT50back DT50_meso_b1 DT50_meso_b2
meso 16.86549 73.15824 22.02282 7.119554 26.33839</code></pre>
<div class="sourceCode" id="cb70"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">sforb_pH_2</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
User 7
$ff
meso_free
1
$SFORB
meso_b1 meso_b2 meso_g
0.13315233 0.03795988 0.61186191
$distimes
DT50 DT90 DT50back DT50_meso_b1 DT50_meso_b2
meso 7.932495 36.93311 11.11797 5.205671 18.26</code></pre>
</div>
<div class="section level3">
<h3 id="hs">HS<a class="anchor" aria-label="anchor" href="#hs"></a>
</h3>
<div class="sourceCode" id="cb72"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">hs_pH</span> <span class="op"><-</span> <span class="fu"><a href="../../reference/saem.html">saem</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">[</span><span class="st">"HS"</span>, <span class="op">]</span>, no_random_effect <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"meso_0"</span><span class="op">)</span>,</span>
<span> covariates <span class="op">=</span> <span class="va">pH</span>,</span>
<span> covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k1</span> <span class="op">~</span> <span class="va">pH</span>, <span class="va">log_k2</span> <span class="op">~</span> <span class="va">pH</span>, <span class="va">log_tb</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span>,</span>
<span> center_covariates <span class="op">=</span> <span class="st">"median"</span><span class="op">)</span></span></code></pre></div>
<div class="sourceCode" id="cb73"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">hs_pH</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="right">est.</th>
<th align="right">lower</th>
<th align="right">upper</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">meso_0</td>
<td align="right">93.33</td>
<td align="right">91.47</td>
<td align="right">95.19</td>
</tr>
<tr class="even">
<td align="left">log_k1</td>
<td align="right">-3.08</td>
<td align="right">-3.28</td>
<td align="right">-2.89</td>
</tr>
<tr class="odd">
<td align="left">beta_pH(log_k1)</td>
<td align="right">0.47</td>
<td align="right">0.23</td>
<td align="right">0.72</td>
</tr>
<tr class="even">
<td align="left">log_k2</td>
<td align="right">-3.68</td>
<td align="right">-3.91</td>
<td align="right">-3.46</td>
</tr>
<tr class="odd">
<td align="left">beta_pH(log_k2)</td>
<td align="right">0.54</td>
<td align="right">0.21</td>
<td align="right">0.87</td>
</tr>
<tr class="even">
<td align="left">log_tb</td>
<td align="right">2.68</td>
<td align="right">2.38</td>
<td align="right">2.97</td>
</tr>
<tr class="odd">
<td align="left">beta_pH(log_tb)</td>
<td align="right">-0.10</td>
<td align="right">-0.43</td>
<td align="right">0.23</td>
</tr>
<tr class="even">
<td align="left">a.1</td>
<td align="right">4.49</td>
<td align="right">3.78</td>
<td align="right">5.21</td>
</tr>
<tr class="odd">
<td align="left">SD.log_k1</td>
<td align="right">0.37</td>
<td align="right">0.24</td>
<td align="right">0.51</td>
</tr>
<tr class="even">
<td align="left">SD.log_k2</td>
<td align="right">0.29</td>
<td align="right">0.10</td>
<td align="right">0.48</td>
</tr>
<tr class="odd">
<td align="left">SD.log_tb</td>
<td align="right">0.25</td>
<td align="right">-0.07</td>
<td align="right">0.57</td>
</tr>
</tbody>
</table>
<div class="sourceCode" id="cb74"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">hs_pH</span><span class="op">)</span></span></code></pre></div>
<pre><code>[1] "sd(log_tb)" "beta_pH(log_tb)"</code></pre>
<p>According to the output of the <code>illparms</code> function, the
random effect on the break time <code>tb</code> cannot reliably be
quantified, neither can the influence of soil pH on <code>tb</code>. The
fit is repeated without the corresponding covariate model, and no
ill-defined parameters remain.</p>
<div class="sourceCode" id="cb76"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">hs_pH_2</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">hs_pH</span>, covariate_models <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">log_k1</span> <span class="op">~</span> <span class="va">pH</span>, <span class="va">log_k2</span> <span class="op">~</span> <span class="va">pH</span><span class="op">)</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">hs_pH_2</span><span class="op">)</span></span></code></pre></div>
<p>Model comparison confirms that this model is preferable to the fit
without covariate influence, and also to the first version with
covariate influence.</p>
<div class="sourceCode" id="cb77"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem_2</span><span class="op">[[</span><span class="st">"HS"</span>, <span class="st">"const"</span><span class="op">]</span><span class="op">]</span>, <span class="va">hs_pH</span>, <span class="va">hs_pH_2</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre></div>
<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
npar AIC BIC Lik Chisq Df Pr(>Chisq)
f_saem_2[["HS", "const"]] 8 780.08 787.20 -382.04
hs_pH_2 10 766.47 775.37 -373.23 17.606 2 0.0001503 ***
hs_pH 11 769.80 779.59 -373.90 0.000 1 1.0000000
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
<div class="sourceCode" id="cb79"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">hs_pH_2</span><span class="op">)</span><span class="op">$</span><span class="va">confint_trans</span> <span class="op">|></span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="right">est.</th>
<th align="right">lower</th>
<th align="right">upper</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">meso_0</td>
<td align="right">93.33</td>
<td align="right">91.50</td>
<td align="right">95.15</td>
</tr>
<tr class="even">
<td align="left">log_k1</td>
<td align="right">-3.05</td>
<td align="right">-3.25</td>
<td align="right">-2.86</td>
</tr>
<tr class="odd">
<td align="left">beta_pH(log_k1)</td>
<td align="right">0.46</td>
<td align="right">0.22</td>
<td align="right">0.69</td>
</tr>
<tr class="even">
<td align="left">log_k2</td>
<td align="right">-3.74</td>
<td align="right">-3.94</td>
<td align="right">-3.54</td>
</tr>
<tr class="odd">
<td align="left">beta_pH(log_k2)</td>
<td align="right">0.50</td>
<td align="right">0.21</td>
<td align="right">0.79</td>
</tr>
<tr class="even">
<td align="left">log_tb</td>
<td align="right">2.70</td>
<td align="right">2.33</td>
<td align="right">3.08</td>
</tr>
<tr class="odd">
<td align="left">a.1</td>
<td align="right">4.45</td>
<td align="right">3.74</td>
<td align="right">5.16</td>
</tr>
<tr class="even">
<td align="left">SD.log_k1</td>
<td align="right">0.36</td>
<td align="right">0.22</td>
<td align="right">0.49</td>
</tr>
<tr class="odd">
<td align="left">SD.log_k2</td>
<td align="right">0.23</td>
<td align="right">0.02</td>
<td align="right">0.43</td>
</tr>
<tr class="even">
<td align="left">SD.log_tb</td>
<td align="right">0.55</td>
<td align="right">0.25</td>
<td align="right">0.85</td>
</tr>
</tbody>
</table>
<div class="sourceCode" id="cb80"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">hs_pH_2</span><span class="op">)</span></span></code></pre></div>
<p><img src="mesotrione_parent_2023_files/figure-html/unnamed-chunk-30-1.png" width="700" style="display: block; margin: auto;"></p>
<div class="sourceCode" id="cb81"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">hs_pH_2</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
50% 5.75
$distimes
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 14.68725 82.45287 24.82079 14.68725 29.29299</code></pre>
<div class="sourceCode" id="cb83"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">hs_pH_2</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
User 7
$distimes
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 8.298536 38.85371 11.69613 8.298536 15.71561</code></pre>
</div>
<div class="section level3">
<h3 id="comparison-across-parent-models">Comparison across parent models<a class="anchor" aria-label="anchor" href="#comparison-across-parent-models"></a>
</h3>
<p>After model reduction for all models with pH influence, they are
compared with each other.</p>
<div class="sourceCode" id="cb85"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">sfo_pH</span>, <span class="va">fomc_pH_2</span>, <span class="va">dfop_pH_2</span>, <span class="va">dfop_pH_4</span>, <span class="va">sforb_pH_2</span>, <span class="va">hs_pH_2</span><span class="op">)</span></span></code></pre></div>
<pre><code>Data: 116 observations of 1 variable(s) grouped in 18 datasets
npar AIC BIC Lik
sfo_pH 5 783.09 787.54 -386.54
fomc_pH_2 6 766.50 771.84 -377.25
dfop_pH_4 7 767.35 773.58 -376.68
sforb_pH_2 7 770.94 777.17 -378.47
dfop_pH_2 8 765.06 772.18 -374.53
hs_pH_2 10 766.47 775.37 -373.23</code></pre>
<p>The DFOP model with pH influence on <code>k2</code> and
<code>g</code> and a random effect only on <code>k2</code> is finally
selected as the best fit.</p>
<p>The endpoints resulting from this model are listed below. Please
refer to the Appendix for a detailed listing.</p>
<div class="sourceCode" id="cb87"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">dfop_pH_2</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
50% 5.75
$distimes
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 18.46687 74.91602 22.55197 4.568623 24.64483</code></pre>
<div class="sourceCode" id="cb89"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/endpoints.html">endpoints</a></span><span class="op">(</span><span class="va">dfop_pH_2</span>, covariates <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pH <span class="op">=</span> <span class="fl">7</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
<pre><code>$covariates
pH
User 7
$distimes
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 8.370528 28.37824 8.542701 4.568623 8.785409</code></pre>
</div>
</div>
<div class="section level2">
<h2 id="conclusions">Conclusions<a class="anchor" aria-label="anchor" href="#conclusions"></a>
</h2>
<p>These evaluations demonstrate that covariate effects can be included
for all types of parent degradation models. These models can then be
further refined to make them fully identifiable.</p>
</div>
<div class="section level2">
<h2 id="appendix">Appendix<a class="anchor" aria-label="anchor" href="#appendix"></a>
</h2>
<div class="section level3">
<h3 id="hierarchical-fit-listings">Hierarchical fit listings<a class="anchor" aria-label="anchor" href="#hierarchical-fit-listings"></a>
</h3>
<div class="section level4">
<h4 id="fits-without-covariate-effects">Fits without covariate effects<a class="anchor" aria-label="anchor" href="#fits-without-covariate-effects"></a>
</h4>
<caption>
Hierarchical SFO fit with constant variance
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:01 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso/dt = - k_meso * meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 0.602 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_k_meso
90.832 -3.192
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_k_meso
meso_0 6.752 0.0000
log_k_meso 0.000 0.9155
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
800 804.5 -395
Optimised parameters:
est. lower upper
meso_0 92.0705 89.9917 94.1493
log_k_meso -3.1641 -3.4286 -2.8996
a.1 5.4628 4.6421 6.2835
SD.meso_0 0.0611 -98.3545 98.4767
SD.log_k_meso 0.5616 0.3734 0.7499
Correlation:
meso_0
log_k_meso 0.1132
Random effects:
est. lower upper
SD.meso_0 0.0611 -98.3545 98.4767
SD.log_k_meso 0.5616 0.3734 0.7499
Variance model:
est. lower upper
a.1 5.463 4.642 6.284
Backtransformed parameters:
est. lower upper
meso_0 92.07053 89.99172 94.14933
k_meso 0.04225 0.03243 0.05505
Estimated disappearance times:
DT50 DT90
meso 16.41 54.5
</code></pre>
<p></p>
<caption>
Hierarchical FOMC fit with constant variance
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:02 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso/dt = - (alpha/beta) * 1/((time/beta) + 1) * meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 0.983 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_alpha log_beta
93.0520 0.6008 3.4176
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_alpha log_beta
meso_0 6.287 0.00 0.000
log_alpha 0.000 1.53 0.000
log_beta 0.000 0.00 1.724
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
787.4 793.6 -386.7
Optimised parameters:
est. lower upper
meso_0 93.5648 91.42864 95.7009
log_alpha 0.7645 0.28068 1.2484
log_beta 3.6597 3.05999 4.2594
a.1 5.0708 4.29823 5.8435
SD.meso_0 0.1691 -34.01517 34.3535
SD.log_alpha 0.3764 0.05834 0.6945
SD.log_beta 0.3903 -0.06074 0.8414
Correlation:
meso_0 log_lph
log_alpha -0.2839
log_beta -0.3443 0.8855
Random effects:
est. lower upper
SD.meso_0 0.1691 -34.01517 34.3535
SD.log_alpha 0.3764 0.05834 0.6945
SD.log_beta 0.3903 -0.06074 0.8414
Variance model:
est. lower upper
a.1 5.071 4.298 5.843
Backtransformed parameters:
est. lower upper
meso_0 93.565 91.429 95.701
alpha 2.148 1.324 3.485
beta 38.850 21.327 70.770
Estimated disappearance times:
DT50 DT90 DT50back
meso 14.8 74.64 22.47
</code></pre>
<p></p>
<caption>
Hierarchical DFOP fit with constant variance
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:02 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
* meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 1.202 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_k1 log_k2 g_qlogis
93.14689 -2.05241 -3.53079 -0.09522
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_k1 log_k2 g_qlogis
meso_0 6.418 0.000 0.000 0.00
log_k1 0.000 1.018 0.000 0.00
log_k2 0.000 0.000 1.694 0.00
g_qlogis 0.000 0.000 0.000 2.37
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
787.6 795.6 -384.8
Optimised parameters:
est. lower upper
meso_0 93.6684 91.63599 95.7008
log_k1 -1.7354 -2.61433 -0.8565
log_k2 -3.4015 -3.73323 -3.0697
g_qlogis -1.6341 -2.66133 -0.6069
a.1 4.7803 4.01269 5.5479
SD.meso_0 0.1661 -30.97086 31.3031
SD.log_k1 0.1127 -2.59680 2.8223
SD.log_k2 0.6394 0.41499 0.8638
SD.g_qlogis 0.8166 0.09785 1.5353
Correlation:
meso_0 log_k1 log_k2
log_k1 0.1757
log_k2 0.0199 0.2990
g_qlogis 0.0813 -0.7431 -0.3826
Random effects:
est. lower upper
SD.meso_0 0.1661 -30.97086 31.3031
SD.log_k1 0.1127 -2.59680 2.8223
SD.log_k2 0.6394 0.41499 0.8638
SD.g_qlogis 0.8166 0.09785 1.5353
Variance model:
est. lower upper
a.1 4.78 4.013 5.548
Backtransformed parameters:
est. lower upper
meso_0 93.66841 91.63599 95.70082
k1 0.17633 0.07322 0.42466
k2 0.03332 0.02392 0.04643
g 0.16327 0.06529 0.35277
Estimated disappearance times:
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 16.04 63.75 19.19 3.931 20.8
</code></pre>
<p></p>
<caption>
Hierarchical SFORB fit with constant variance
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:02 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso_free/dt = - k_meso_free * meso_free - k_meso_free_bound *
meso_free + k_meso_bound_free * meso_bound
d_meso_bound/dt = + k_meso_free_bound * meso_free - k_meso_bound_free *
meso_bound
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 1.557 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_free_0 log_k_meso_free log_k_meso_free_bound
93.147 -2.305 -4.230
log_k_meso_bound_free
-3.761
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_free_0 log_k_meso_free log_k_meso_free_bound
meso_free_0 6.418 0.0000 0.000
log_k_meso_free 0.000 0.9276 0.000
log_k_meso_free_bound 0.000 0.0000 2.272
log_k_meso_bound_free 0.000 0.0000 0.000
log_k_meso_bound_free
meso_free_0 0.000
log_k_meso_free 0.000
log_k_meso_free_bound 0.000
log_k_meso_bound_free 1.447
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
787.4 795.4 -384.7
Optimised parameters:
est. lower upper
meso_free_0 93.6285 91.6262 95.631
log_k_meso_free -2.8314 -3.1375 -2.525
log_k_meso_free_bound -3.2213 -4.4695 -1.973
log_k_meso_bound_free -2.4246 -3.5668 -1.282
a.1 4.7372 3.9542 5.520
SD.meso_free_0 0.1634 -32.7769 33.104
SD.log_k_meso_free 0.4885 0.3080 0.669
SD.log_k_meso_free_bound 0.2876 -1.7955 2.371
SD.log_k_meso_bound_free 0.9942 0.2181 1.770
Correlation:
ms_fr_0 lg_k_m_ lg_k_ms_f_
log_k_meso_free 0.2332
log_k_meso_free_bound 0.1100 0.5964
log_k_meso_bound_free -0.0413 0.3697 0.8025
Random effects:
est. lower upper
SD.meso_free_0 0.1634 -32.7769 33.104
SD.log_k_meso_free 0.4885 0.3080 0.669
SD.log_k_meso_free_bound 0.2876 -1.7955 2.371
SD.log_k_meso_bound_free 0.9942 0.2181 1.770
Variance model:
est. lower upper
a.1 4.737 3.954 5.52
Backtransformed parameters:
est. lower upper
meso_free_0 93.62849 91.62622 95.63075
k_meso_free 0.05893 0.04339 0.08004
k_meso_free_bound 0.03990 0.01145 0.13903
k_meso_bound_free 0.08851 0.02825 0.27736
Estimated Eigenvalues of SFORB model(s):
meso_b1 meso_b2 meso_g
0.15333 0.03402 0.20881
Resulting formation fractions:
ff
meso_free 1
Estimated disappearance times:
DT50 DT90 DT50back DT50_meso_b1 DT50_meso_b2
meso 14.79 60.81 18.3 4.521 20.37
</code></pre>
<p></p>
<caption>
Hierarchical HS fit with constant variance
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:03 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso/dt = - ifelse(time <= tb, k1, k2) * meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 1.61 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_k1 log_k2 log_tb
92.920 -2.409 -3.295 2.471
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_k1 log_k2 log_tb
meso_0 6.477 0.0000 0.0000 0.00
log_k1 0.000 0.8675 0.0000 0.00
log_k2 0.000 0.0000 0.4035 0.00
log_tb 0.000 0.0000 0.0000 1.16
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
781.9 789.9 -382
Optimised parameters:
est. lower upper
meso_0 93.34242 91.4730 95.2118
log_k1 -2.77312 -3.0826 -2.4637
log_k2 -3.61854 -3.8430 -3.3941
log_tb 2.00266 1.3357 2.6696
a.1 4.47693 3.7059 5.2479
SD.meso_0 0.07963 -63.1661 63.3253
SD.log_k1 0.47817 0.2467 0.7097
SD.log_k2 0.39216 0.2137 0.5706
SD.log_tb 0.94683 0.4208 1.4728
Correlation:
meso_0 log_k1 log_k2
log_k1 0.1627
log_k2 0.0063 -0.0301
log_tb 0.0083 -0.3931 -0.1225
Random effects:
est. lower upper
SD.meso_0 0.07963 -63.1661 63.3253
SD.log_k1 0.47817 0.2467 0.7097
SD.log_k2 0.39216 0.2137 0.5706
SD.log_tb 0.94683 0.4208 1.4728
Variance model:
est. lower upper
a.1 4.477 3.706 5.248
Backtransformed parameters:
est. lower upper
meso_0 93.34242 91.47303 95.21181
k1 0.06247 0.04584 0.08512
k2 0.02682 0.02143 0.03357
tb 7.40872 3.80282 14.43376
Estimated disappearance times:
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 16 76 22.88 11.1 25.84
</code></pre>
<p></p>
</div>
<div class="section level4">
<h4 id="fits-with-covariate-effects">Fits with covariate effects<a class="anchor" aria-label="anchor" href="#fits-with-covariate-effects"></a>
</h4>
<caption>
Hierarchichal SFO fit with pH influence
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:15 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso/dt = - k_meso * meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 0.924 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_k_meso
90.832 -3.192
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_k_meso
meso_0 6.752 0.0000
log_k_meso 0.000 0.9155
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
783.1 787.5 -386.5
Optimised parameters:
est. lower upper
meso_0 91.3481 89.2688 93.4275
log_k_meso -3.2854 -3.4590 -3.1118
beta_pH(log_k_meso) 0.5871 0.3684 0.8059
a.1 5.4750 4.7085 6.2415
SD.log_k_meso 0.3471 0.2258 0.4684
Correlation:
meso_0 lg_k_ms
log_k_meso 0.1797
beta_pH(log_k_meso) -0.0183 -0.2379
Random effects:
est. lower upper
SD.log_k_meso 0.3471 0.2258 0.4684
Variance model:
est. lower upper
a.1 5.475 4.709 6.242
Backtransformed parameters:
est. lower upper
meso_0 91.34814 89.26880 93.42748
k_meso 0.03743 0.03146 0.04452
Covariates used for endpoints below:
pH
50% 5.75
Estimated disappearance times:
DT50 DT90
meso 18.52 61.52
</code></pre>
<p></p>
<caption>
Hierarchichal FOMC fit with pH influence
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:18 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso/dt = - (alpha/beta) * 1/((time/beta) + 1) * meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 1.901 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_alpha log_beta
93.0520 0.6008 3.4176
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_alpha log_beta
meso_0 6.287 0.00 0.000
log_alpha 0.000 1.53 0.000
log_beta 0.000 0.00 1.724
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
770.1 776.3 -378
Optimised parameters:
est. lower upper
meso_0 92.840646 90.750461 94.9308
log_alpha 1.114051 0.475668 1.7524
beta_pH(log_alpha) 0.577505 0.369805 0.7852
log_beta 4.214099 3.438851 4.9893
a.1 5.027768 4.322028 5.7335
SD.log_alpha 0.004034 -23.766993 23.7751
SD.log_beta 0.374640 0.009252 0.7400
Correlation:
meso_0 log_lph bt_H(_)
log_alpha -0.3220
beta_pH(log_alpha) -0.0789 0.1148
log_beta -0.3544 0.9709 0.1628
Random effects:
est. lower upper
SD.log_alpha 0.004034 -23.766993 23.78
SD.log_beta 0.374640 0.009252 0.74
Variance model:
est. lower upper
a.1 5.028 4.322 5.734
Backtransformed parameters:
est. lower upper
meso_0 92.841 90.750 94.931
alpha 3.047 1.609 5.769
beta 67.633 31.151 146.840
Covariates used for endpoints below:
pH
50% 5.75
Estimated disappearance times:
DT50 DT90 DT50back
meso 17.28 76.37 22.99
</code></pre>
<p></p>
<caption>
Refined hierarchichal FOMC fit with pH influence
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:21 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso/dt = - (alpha/beta) * 1/((time/beta) + 1) * meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 3.35 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_alpha log_beta
93.0520 0.6008 3.4176
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_alpha log_beta
meso_0 6.287 0.00 0.000
log_alpha 0.000 1.53 0.000
log_beta 0.000 0.00 1.724
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
766.5 771.8 -377.3
Optimised parameters:
est. lower upper
meso_0 93.1950 91.1047 95.2853
log_alpha 0.8125 0.3292 1.2957
beta_pH(log_alpha) 0.5497 0.3490 0.7504
log_beta 3.8464 3.2179 4.4750
a.1 4.9972 4.2976 5.6968
SD.log_beta 0.3829 0.2499 0.5159
Correlation:
meso_0 log_lph bt_H(_)
log_alpha -0.3094
beta_pH(log_alpha) -0.0779 0.0841
log_beta -0.3493 0.9533 0.1434
Random effects:
est. lower upper
SD.log_beta 0.3829 0.2499 0.5159
Variance model:
est. lower upper
a.1 4.997 4.298 5.697
Backtransformed parameters:
est. lower upper
meso_0 93.195 91.10 95.285
alpha 2.253 1.39 3.654
beta 46.826 24.98 87.793
Covariates used for endpoints below:
pH
50% 5.75
Estimated disappearance times:
DT50 DT90 DT50back
meso 16.86 83.27 25.07
</code></pre>
<p></p>
<caption>
Hierarchichal DFOP fit with pH influence
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:25 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
* meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 2.592 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_k1 log_k2 g_qlogis
93.14689 -2.05241 -3.53079 -0.09522
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_k1 log_k2 g_qlogis
meso_0 6.418 0.000 0.000 0.00
log_k1 0.000 1.018 0.000 0.00
log_k2 0.000 0.000 1.694 0.00
g_qlogis 0.000 0.000 0.000 2.37
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
769.1 777.1 -375.5
Optimised parameters:
est. lower upper
meso_0 92.843344 90.8464 94.84028
log_k1 -2.815685 -3.0888 -2.54261
log_k2 -3.956384 -4.4741 -3.43868
beta_pH(log_k2) 1.308417 0.6948 1.92203
g_qlogis -0.121394 -0.5691 0.32627
beta_pH(g_qlogis) -0.565988 -1.0394 -0.09262
a.1 4.955518 4.2597 5.65135
SD.log_k2 0.758963 0.4685 1.04943
SD.g_qlogis 0.005215 -9.9561 9.96656
Correlation:
meso_0 log_k1 log_k2 b_H(_2) g_qlogs
log_k1 0.2706
log_k2 -0.0457 -0.0667
beta_pH(log_k2) 0.0554 -0.1291 -0.5566
g_qlogis 0.1004 -0.4462 -0.4397 0.5042
beta_pH(g_qlogis) 0.1267 0.4226 0.0123 -0.0438 0.2029
Random effects:
est. lower upper
SD.log_k2 0.758963 0.4685 1.049
SD.g_qlogis 0.005215 -9.9561 9.967
Variance model:
est. lower upper
a.1 4.956 4.26 5.651
Backtransformed parameters:
est. lower upper
meso_0 92.84334 90.84641 94.84028
k1 0.05986 0.04556 0.07866
k2 0.01913 0.01140 0.03211
g 0.46969 0.36145 0.58085
Covariates used for endpoints below:
pH
50% 5.75
Estimated disappearance times:
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 20.23 88.45 26.62 11.58 36.23
</code></pre>
<p></p>
<caption>
Refined hierarchical DFOP fit with pH influence
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:30 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
* meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 4.483 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_k1 log_k2 g_qlogis
93.14689 -2.05241 -3.53079 -0.09522
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_k1 log_k2 g_qlogis
meso_0 6.418 0.000 0.000 0.00
log_k1 0.000 1.018 0.000 0.00
log_k2 0.000 0.000 1.694 0.00
g_qlogis 0.000 0.000 0.000 2.37
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
765.1 772.2 -374.5
Optimised parameters:
est. lower upper
meso_0 93.1612 91.0766 95.24580
log_k1 -1.8857 -2.8975 -0.87395
log_k2 -3.5711 -3.8859 -3.25622
beta_pH(log_k2) 0.8252 0.4952 1.15513
g_qlogis -1.5326 -2.6994 -0.36574
beta_pH(g_qlogis) -0.8365 -1.7163 0.04333
a.1 4.9218 4.2328 5.61070
SD.log_k2 0.4011 0.2604 0.54184
Correlation:
meso_0 log_k1 log_k2 b_H(_2) g_qlogs
log_k1 0.1734
log_k2 0.0553 0.6421
beta_pH(log_k2) -0.0382 -0.5121 -0.5860
g_qlogis 0.0616 -0.8501 -0.7251 0.4941
beta_pH(g_qlogis) 0.1408 -0.3240 -0.2716 -0.0041 0.6264
Random effects:
est. lower upper
SD.log_k2 0.4011 0.2604 0.5418
Variance model:
est. lower upper
a.1 4.922 4.233 5.611
Backtransformed parameters:
est. lower upper
meso_0 93.16118 91.07655 95.24580
k1 0.15172 0.05516 0.41730
k2 0.02813 0.02053 0.03853
g 0.17762 0.06301 0.40957
Covariates used for endpoints below:
pH
50% 5.75
Estimated disappearance times:
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 18.47 74.92 22.55 4.569 24.64
</code></pre>
<p></p>
<caption>
Further refined hierarchical DFOP fit with pH influence
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:37 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
* meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 3.213 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_k1 log_k2 g_qlogis
93.14689 -2.05241 -3.53079 -0.09522
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_k1 log_k2 g_qlogis
meso_0 6.418 0.000 0.000 0.00
log_k1 0.000 1.018 0.000 0.00
log_k2 0.000 0.000 1.694 0.00
g_qlogis 0.000 0.000 0.000 2.37
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
767.4 773.6 -376.7
Optimised parameters:
est. lower upper
meso_0 93.3011 91.1905 95.4118
log_k1 -2.1487 -2.7607 -1.5367
log_k2 -3.6066 -3.9305 -3.2828
beta_pH(log_k2) 0.7821 0.4126 1.1517
g_qlogis -1.0373 -1.9337 -0.1409
a.1 5.0095 4.3082 5.7108
SD.log_k2 0.4622 0.3009 0.6235
Correlation:
meso_0 log_k1 log_k2 b_H(_2)
log_k1 0.2179
log_k2 0.0271 0.5067
beta_pH(log_k2) -0.0326 -0.5546 -0.5485
g_qlogis 0.0237 -0.8479 -0.6866 0.6123
Random effects:
est. lower upper
SD.log_k2 0.4622 0.3009 0.6235
Variance model:
est. lower upper
a.1 5.009 4.308 5.711
Backtransformed parameters:
est. lower upper
meso_0 93.30113 91.19050 95.41175
k1 0.11664 0.06325 0.21508
k2 0.02714 0.01963 0.03752
g 0.26168 0.12635 0.46483
Covariates used for endpoints below:
pH
50% 5.75
Estimated disappearance times:
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 17.09 73.67 22.18 5.943 25.54
</code></pre>
<p></p>
<caption>
Hierarchichal SFORB fit with pH influence
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:41 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso_free/dt = - k_meso_free * meso_free - k_meso_free_bound *
meso_free + k_meso_bound_free * meso_bound
d_meso_bound/dt = + k_meso_free_bound * meso_free - k_meso_bound_free *
meso_bound
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 2.758 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_free_0 log_k_meso_free log_k_meso_free_bound
93.147 -2.305 -4.230
log_k_meso_bound_free
-3.761
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_free_0 log_k_meso_free log_k_meso_free_bound
meso_free_0 6.418 0.0000 0.000
log_k_meso_free 0.000 0.9276 0.000
log_k_meso_free_bound 0.000 0.0000 2.272
log_k_meso_bound_free 0.000 0.0000 0.000
log_k_meso_bound_free
meso_free_0 0.000
log_k_meso_free 0.000
log_k_meso_free_bound 0.000
log_k_meso_bound_free 1.447
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
768.8 776.8 -375.4
Optimised parameters:
est. lower upper
meso_free_0 93.4204 91.3213 95.5195
log_k_meso_free -2.9408 -3.2344 -2.6471
beta_pH(log_k_meso_free) 0.4232 0.1769 0.6695
log_k_meso_free_bound -3.4889 -4.9243 -2.0535
log_k_meso_bound_free -2.9130 -4.3047 -1.5212
beta_pH(log_k_meso_bound_free) 1.2290 -0.2107 2.6687
a.1 4.9031 4.1795 5.6268
SD.log_k_meso_free 0.3454 0.2252 0.4656
SD.log_k_meso_bound_free 0.1277 -1.9459 2.2012
Correlation:
ms_fr_0 lg_k_m_ b_H(___) lg_k_ms_f_ lg_k_ms_b_
log_k_meso_free 0.3460
beta_pH(log_k_meso_free) -0.0930 -0.4206
log_k_meso_free_bound 0.2439 0.7749 -0.3492
log_k_meso_bound_free 0.1350 0.6400 -0.2912 0.9346
beta_pH(log_k_meso_bound_free) -0.2216 -0.4778 -0.0111 -0.6566 -0.6498
Random effects:
est. lower upper
SD.log_k_meso_free 0.3454 0.2252 0.4656
SD.log_k_meso_bound_free 0.1277 -1.9459 2.2012
Variance model:
est. lower upper
a.1 4.903 4.18 5.627
Backtransformed parameters:
est. lower upper
meso_free_0 93.42040 91.321340 95.51946
k_meso_free 0.05282 0.039384 0.07085
k_meso_free_bound 0.03054 0.007268 0.12829
k_meso_bound_free 0.05431 0.013505 0.21845
Covariates used for endpoints below:
pH
50% 5.75
Estimated Eigenvalues of SFORB model(s):
meso_b1 meso_b2 meso_g
0.1121 0.0256 0.3148
Resulting formation fractions:
ff
meso_free 1
Estimated disappearance times:
DT50 DT90 DT50back DT50_meso_b1 DT50_meso_b2
meso 16.42 75.2 22.64 6.185 27.08
</code></pre>
<p> </p>
<caption>
Refined hierarchichal SFORB fit with pH influence
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:44 2025
Date of summary: Fri Nov 28 09:15:24 2025
Equations:
d_meso_free/dt = - k_meso_free * meso_free - k_meso_free_bound *
meso_free + k_meso_bound_free * meso_bound
d_meso_bound/dt = + k_meso_free_bound * meso_free - k_meso_bound_free *
meso_bound
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 2.971 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_free_0 log_k_meso_free log_k_meso_free_bound
93.147 -2.305 -4.230
log_k_meso_bound_free
-3.761
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_free_0 log_k_meso_free log_k_meso_free_bound
meso_free_0 6.418 0.0000 0.000
log_k_meso_free 0.000 0.9276 0.000
log_k_meso_free_bound 0.000 0.0000 2.272
log_k_meso_bound_free 0.000 0.0000 0.000
log_k_meso_bound_free
meso_free_0 0.000
log_k_meso_free 0.000
log_k_meso_free_bound 0.000
log_k_meso_bound_free 1.447
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
770.9 777.2 -378.5
Optimised parameters:
est. lower upper
meso_free_0 93.3196 91.1650 95.4743
log_k_meso_free -3.0207 -3.2655 -2.7759
beta_pH(log_k_meso_free) 0.5435 0.3329 0.7542
log_k_meso_free_bound -3.8001 -5.1809 -2.4193
log_k_meso_bound_free -2.9462 -4.2411 -1.6513
a.1 5.0825 4.3709 5.7940
SD.log_k_meso_free 0.3338 0.2175 0.4502
Correlation:
ms_fr_0 lg_k_m_ b_H(___ lg_k_ms_f_
log_k_meso_free 0.3556
beta_pH(log_k_meso_free) -0.0422 -0.2041
log_k_meso_free_bound 0.2513 0.6866 -0.0400
log_k_meso_bound_free 0.1292 0.5341 -0.0129 0.9214
Random effects:
est. lower upper
SD.log_k_meso_free 0.3338 0.2175 0.4502
Variance model:
est. lower upper
a.1 5.082 4.371 5.794
Backtransformed parameters:
est. lower upper
meso_free_0 93.31965 91.165044 95.47425
k_meso_free 0.04877 0.038177 0.06230
k_meso_free_bound 0.02237 0.005623 0.08898
k_meso_bound_free 0.05254 0.014392 0.19179
Covariates used for endpoints below:
pH
50% 5.75
Estimated Eigenvalues of SFORB model(s):
meso_b1 meso_b2 meso_g
0.09736 0.02632 0.31602
Resulting formation fractions:
ff
meso_free 1
Estimated disappearance times:
DT50 DT90 DT50back DT50_meso_b1 DT50_meso_b2
meso 16.87 73.16 22.02 7.12 26.34
</code></pre>
<p> </p>
<caption>
Hierarchichal HS fit with pH influence
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:48 2025
Date of summary: Fri Nov 28 09:15:25 2025
Equations:
d_meso/dt = - ifelse(time <= tb, k1, k2) * meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 2.254 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_k1 log_k2 log_tb
92.920 -2.409 -3.295 2.471
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_k1 log_k2 log_tb
meso_0 6.477 0.0000 0.0000 0.00
log_k1 0.000 0.8675 0.0000 0.00
log_k2 0.000 0.0000 0.4035 0.00
log_tb 0.000 0.0000 0.0000 1.16
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
769.8 779.6 -373.9
Optimised parameters:
est. lower upper
meso_0 93.32599 91.4658 95.1862
log_k1 -3.08497 -3.2830 -2.8870
beta_pH(log_k1) 0.47472 0.2334 0.7160
log_k2 -3.68267 -3.9077 -3.4577
beta_pH(log_k2) 0.54151 0.2124 0.8706
log_tb 2.67815 2.3846 2.9717
beta_pH(log_tb) -0.09889 -0.4258 0.2280
a.1 4.49487 3.7766 5.2132
SD.log_k1 0.37191 0.2370 0.5068
SD.log_k2 0.29210 0.0994 0.4848
SD.log_tb 0.25353 -0.0664 0.5735
Correlation:
meso_0 log_k1 b_H(_1) log_k2 b_H(_2) log_tb
log_k1 0.2301
beta_pH(log_k1) -0.0452 -0.2842
log_k2 -0.0029 -0.0322 -0.0008
beta_pH(log_k2) -0.0071 0.0000 -0.0391 -0.2655
log_tb -0.1003 -0.2052 0.1180 -0.4197 0.1794
beta_pH(log_tb) 0.0097 0.1125 -0.1265 0.1894 -0.3653 -0.3449
Random effects:
est. lower upper
SD.log_k1 0.3719 0.2370 0.5068
SD.log_k2 0.2921 0.0994 0.4848
SD.log_tb 0.2535 -0.0664 0.5735
Variance model:
est. lower upper
a.1 4.495 3.777 5.213
Backtransformed parameters:
est. lower upper
meso_0 93.32599 91.46575 95.18624
k1 0.04573 0.03752 0.05574
k2 0.02516 0.02009 0.03150
tb 14.55810 10.85502 19.52445
Covariates used for endpoints below:
pH
50% 5.75
Estimated disappearance times:
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 15.65 79.63 23.97 15.16 27.55
</code></pre>
<p> </p>
<caption>
Refined hierarchichal HS fit with pH influence
</caption>
<pre><code>
saemix version used for fitting: 3.4
mkin version used for pre-fitting: 1.2.11
R version used for fitting: 4.5.1
Date of fit: Fri Sep 12 22:15:50 2025
Date of summary: Fri Nov 28 09:15:25 2025
Equations:
d_meso/dt = - ifelse(time <= tb, k1, k2) * meso
Data:
116 observations of 1 variable(s) grouped in 18 datasets
Model predictions using solution type analytical
Fitted in 1.425 s
Using 300, 100 iterations and 3 chains
Variance model: Constant variance
Starting values for degradation parameters:
meso_0 log_k1 log_k2 log_tb
92.920 -2.409 -3.295 2.471
Fixed degradation parameter values:
None
Starting values for random effects (square root of initial entries in omega):
meso_0 log_k1 log_k2 log_tb
meso_0 6.477 0.0000 0.0000 0.00
log_k1 0.000 0.8675 0.0000 0.00
log_k2 0.000 0.0000 0.4035 0.00
log_tb 0.000 0.0000 0.0000 1.16
Starting values for error model parameters:
a.1
1
Results:
Likelihood computed by importance sampling
AIC BIC logLik
766.5 775.4 -373.2
Optimised parameters:
est. lower upper
meso_0 93.3251 91.49823 95.1520
log_k1 -3.0535 -3.24879 -2.8582
beta_pH(log_k1) 0.4567 0.22400 0.6894
log_k2 -3.7439 -3.94307 -3.5447
beta_pH(log_k2) 0.4982 0.20644 0.7899
log_tb 2.7040 2.33033 3.0777
a.1 4.4452 3.73537 5.1551
SD.log_k1 0.3570 0.22104 0.4930
SD.log_k2 0.2252 0.01864 0.4318
SD.log_tb 0.5488 0.24560 0.8521
Correlation:
meso_0 log_k1 b_H(_1) log_k2 b_H(_2)
log_k1 0.2233
beta_pH(log_k1) -0.0453 -0.2955
log_k2 0.0028 -0.0420 0.0126
beta_pH(log_k2) -0.0116 0.0112 -0.0667 -0.2097
log_tb -0.0658 -0.1928 0.0913 -0.2843 0.1210
Random effects:
est. lower upper
SD.log_k1 0.3570 0.22104 0.4930
SD.log_k2 0.2252 0.01864 0.4318
SD.log_tb 0.5488 0.24560 0.8521
Variance model:
est. lower upper
a.1 4.445 3.735 5.155
Backtransformed parameters:
est. lower upper
meso_0 93.32513 91.49823 95.15204
k1 0.04719 0.03882 0.05737
k2 0.02366 0.01939 0.02888
tb 14.93925 10.28132 21.70744
Covariates used for endpoints below:
pH
50% 5.75
Estimated disappearance times:
DT50 DT90 DT50back DT50_k1 DT50_k2
meso 14.69 82.45 24.82 14.69 29.29
</code></pre>
<p></p>
</div>
</div>
<div class="section level3">
<h3 id="session-info">Session info<a class="anchor" aria-label="anchor" href="#session-info"></a>
</h3>
<pre><code>R version 4.5.2 (2025-10-31)
Platform: x86_64-pc-linux-gnu
Running under: Debian GNU/Linux 13 (trixie)
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.12.1
LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.12.1; LAPACK version 3.12.0
locale:
[1] LC_CTYPE=de_DE.UTF-8 LC_NUMERIC=C
[3] LC_TIME=de_DE.UTF-8 LC_COLLATE=de_DE.UTF-8
[5] LC_MONETARY=de_DE.UTF-8 LC_MESSAGES=de_DE.UTF-8
[7] LC_PAPER=de_DE.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C
time zone: Europe/Berlin
tzcode source: system (glibc)
attached base packages:
[1] parallel stats graphics grDevices utils datasets methods
[8] base
other attached packages:
[1] saemix_3.4 npde_3.5 knitr_1.50 mkin_1.2.10
loaded via a namespace (and not attached):
[1] gtable_0.3.6 jsonlite_2.0.0 dplyr_1.1.4 compiler_4.5.2
[5] tidyselect_1.2.1 dichromat_2.0-0.1 gridExtra_2.3 jquerylib_0.1.4
[9] systemfonts_1.2.3 scales_1.4.0 textshaping_1.0.1 yaml_2.3.10
[13] fastmap_1.2.0 lattice_0.22-7 ggplot2_3.5.2 R6_2.6.1
[17] generics_0.1.4 lmtest_0.9-40 MASS_7.3-65 htmlwidgets_1.6.4
[21] tibble_3.3.0 desc_1.4.3 bslib_0.9.0 pillar_1.10.2
[25] RColorBrewer_1.1-3 rlang_1.1.6 cachem_1.1.0 xfun_0.52
[29] fs_1.6.6 sass_0.4.10 cli_3.6.5 pkgdown_2.1.3
[33] magrittr_2.0.3 digest_0.6.37 grid_4.5.2 mclust_6.1.1
[37] lifecycle_1.0.4 nlme_3.1-168 vctrs_0.6.5 evaluate_1.0.3
[41] glue_1.8.0 farver_2.1.2 ragg_1.4.0 zoo_1.8-14
[45] colorspace_2.1-1 rmarkdown_2.29 tools_4.5.2 pkgconfig_2.0.3
[49] htmltools_0.5.8.1 </code></pre>
</div>
<div class="section level3">
<h3 id="hardware-info">Hardware info<a class="anchor" aria-label="anchor" href="#hardware-info"></a>
</h3>
<pre><code>CPU model: AMD Ryzen 9 7950X 16-Core Processor</code></pre>
<pre><code>MemTotal: 64933716 kB</code></pre>
</div>
</div>
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