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<h1>Example evaluation of FOCUS Example Dataset D</h1>
<h4 class="author">Johannes Ranke</h4>
<h4 class="date">2019-09-18</h4>
<div class="hidden name"><code>FOCUS_D.Rmd</code></div>
</div>
<p>This is just a very simple vignette showing how to fit a degradation model for a parent compound with one transformation product using <code>mkin</code>. After loading the library we look at the data. We have observed concentrations in the column named <code>value</code> at the times specified in column <code>time</code> for the two observed variables named <code>parent</code> and <code>m1</code>.</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb1-1" title="1"><span class="kw"><a href="https://rdrr.io/r/base/library.html">library</a></span>(mkin, <span class="dt">quietly =</span> <span class="ot">TRUE</span>)</a>
<a class="sourceLine" id="cb1-2" title="2"><span class="kw"><a href="https://rdrr.io/r/base/print.html">print</a></span>(FOCUS_<span class="dv">2006</span>_D)</a></code></pre></div>
<pre><code>## name time value
## 1 parent 0 99.46
## 2 parent 0 102.04
## 3 parent 1 93.50
## 4 parent 1 92.50
## 5 parent 3 63.23
## 6 parent 3 68.99
## 7 parent 7 52.32
## 8 parent 7 55.13
## 9 parent 14 27.27
## 10 parent 14 26.64
## 11 parent 21 11.50
## 12 parent 21 11.64
## 13 parent 35 2.85
## 14 parent 35 2.91
## 15 parent 50 0.69
## 16 parent 50 0.63
## 17 parent 75 0.05
## 18 parent 75 0.06
## 19 parent 100 NA
## 20 parent 100 NA
## 21 parent 120 NA
## 22 parent 120 NA
## 23 m1 0 0.00
## 24 m1 0 0.00
## 25 m1 1 4.84
## 26 m1 1 5.64
## 27 m1 3 12.91
## 28 m1 3 12.96
## 29 m1 7 22.97
## 30 m1 7 24.47
## 31 m1 14 41.69
## 32 m1 14 33.21
## 33 m1 21 44.37
## 34 m1 21 46.44
## 35 m1 35 41.22
## 36 m1 35 37.95
## 37 m1 50 41.19
## 38 m1 50 40.01
## 39 m1 75 40.09
## 40 m1 75 33.85
## 41 m1 100 31.04
## 42 m1 100 33.13
## 43 m1 120 25.15
## 44 m1 120 33.31</code></pre>
<p>Next we specify the degradation model: The parent compound degrades with simple first-order kinetics (SFO) to one metabolite named m1, which also degrades with SFO kinetics.</p>
<p>The call to mkinmod returns a degradation model. The differential equations represented in R code can be found in the character vector <code>$diffs</code> of the <code>mkinmod</code> object. If a C compiler (gcc) is installed and functional, the differential equation model will be compiled from auto-generated C code.</p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb3-1" title="1">SFO_SFO <-<span class="st"> </span><span class="kw"><a href="../reference/mkinmod.html">mkinmod</a></span>(<span class="dt">parent =</span> <span class="kw"><a href="../reference/mkinsub.html">mkinsub</a></span>(<span class="st">"SFO"</span>, <span class="st">"m1"</span>), <span class="dt">m1 =</span> <span class="kw"><a href="../reference/mkinsub.html">mkinsub</a></span>(<span class="st">"SFO"</span>))</a></code></pre></div>
<pre><code>## Successfully compiled differential equation model from auto-generated C code.</code></pre>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb5-1" title="1"><span class="kw"><a href="https://rdrr.io/r/base/print.html">print</a></span>(SFO_SFO<span class="op">$</span>diffs)</a></code></pre></div>
<pre><code>## parent
## "d_parent = - k_parent_sink * parent - k_parent_m1 * parent"
## m1
## "d_m1 = + k_parent_m1 * parent - k_m1_sink * m1"</code></pre>
<p>We do the fitting without progress report (<code>quiet = TRUE</code>).</p>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb7-1" title="1">fit <-<span class="st"> </span><span class="kw"><a href="../reference/mkinfit.html">mkinfit</a></span>(SFO_SFO, FOCUS_<span class="dv">2006</span>_D, <span class="dt">quiet =</span> <span class="ot">TRUE</span>)</a></code></pre></div>
<pre><code>## Warning in mkinfit(SFO_SFO, FOCUS_2006_D, quiet = TRUE): Observations with
## value of zero were removed from the data</code></pre>
<p>A plot of the fit including a residual plot for both observed variables is obtained using the <code>plot_sep</code> method for <code>mkinfit</code> objects, which shows separate graphs for all compounds and their residuals.</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb9-1" title="1"><span class="kw"><a href="../reference/plot.mkinfit.html">plot_sep</a></span>(fit, <span class="dt">lpos =</span> <span class="kw"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="st">"topright"</span>, <span class="st">"bottomright"</span>))</a></code></pre></div>
<p><img src="FOCUS_D_files/figure-html/plot-1.png" width="768"></p>
<p>Confidence intervals for the parameter estimates are obtained using the <code>mkinparplot</code> function.</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb10-1" title="1"><span class="kw"><a href="../reference/mkinparplot.html">mkinparplot</a></span>(fit)</a></code></pre></div>
<p><img src="FOCUS_D_files/figure-html/plot_2-1.png" width="768"></p>
<p>A comprehensive report of the results is obtained using the <code>summary</code> method for <code>mkinfit</code> objects.</p>
<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb11-1" title="1"><span class="kw"><a href="https://rdrr.io/r/base/summary.html">summary</a></span>(fit)</a></code></pre></div>
<pre><code>## mkin version used for fitting: 0.9.49.6
## R version used for fitting: 3.6.1
## Date of fit: Wed Sep 18 12:57:33 2019
## Date of summary: Wed Sep 18 12:57:33 2019
##
## Equations:
## d_parent/dt = - k_parent_sink * parent - k_parent_m1 * parent
## d_m1/dt = + k_parent_m1 * parent - k_m1_sink * m1
##
## Model predictions using solution type deSolve
##
## Fitted using 389 model solutions performed in 1.01 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 100.750000 state
## k_parent_sink 0.100000 deparm
## k_parent_m1 0.100100 deparm
## k_m1_sink 0.100200 deparm
## sigma 3.125504 error
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 100.750000 -Inf Inf
## log_k_parent_sink -2.302585 -Inf Inf
## log_k_parent_m1 -2.301586 -Inf Inf
## log_k_m1_sink -2.300587 -Inf Inf
## sigma 3.125504 0 Inf
##
## Fixed parameter values:
## value type
## m1_0 0 state
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 99.600 1.57000 96.400 102.800
## log_k_parent_sink -3.038 0.07626 -3.193 -2.883
## log_k_parent_m1 -2.980 0.04033 -3.062 -2.898
## log_k_m1_sink -5.248 0.13320 -5.518 -4.977
## sigma 3.126 0.35850 2.396 3.855
##
## Parameter correlation:
## parent_0 log_k_parent_sink log_k_parent_m1
## parent_0 1.000e+00 6.067e-01 -6.372e-02
## log_k_parent_sink 6.067e-01 1.000e+00 -8.550e-02
## log_k_parent_m1 -6.372e-02 -8.550e-02 1.000e+00
## log_k_m1_sink -1.688e-01 -6.252e-01 4.731e-01
## sigma 1.164e-09 -8.908e-10 1.652e-08
## log_k_m1_sink sigma
## parent_0 -1.688e-01 1.164e-09
## log_k_parent_sink -6.252e-01 -8.908e-10
## log_k_parent_m1 4.731e-01 1.652e-08
## log_k_m1_sink 1.000e+00 -1.340e-10
## sigma -1.340e-10 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 99.600000 63.430 2.298e-36 96.400000 1.028e+02
## k_parent_sink 0.047920 13.110 6.126e-15 0.041030 5.596e-02
## k_parent_m1 0.050780 24.800 3.269e-23 0.046780 5.512e-02
## k_m1_sink 0.005261 7.510 6.165e-09 0.004012 6.898e-03
## sigma 3.126000 8.718 2.235e-10 2.396000 3.855e+00
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 6.398 4 15
## parent 6.827 3 6
## m1 4.490 1 9
##
## Resulting formation fractions:
## ff
## parent_sink 0.4855
## parent_m1 0.5145
## m1_sink 1.0000
##
## Estimated disappearance times:
## DT50 DT90
## parent 7.023 23.33
## m1 131.761 437.70
##
## Data:
## time variable observed predicted residual
## 0 parent 99.46 99.59848 -1.385e-01
## 0 parent 102.04 99.59848 2.442e+00
## 1 parent 93.50 90.23787 3.262e+00
## 1 parent 92.50 90.23787 2.262e+00
## 3 parent 63.23 74.07320 -1.084e+01
## 3 parent 68.99 74.07320 -5.083e+00
## 7 parent 52.32 49.91207 2.408e+00
## 7 parent 55.13 49.91207 5.218e+00
## 14 parent 27.27 25.01257 2.257e+00
## 14 parent 26.64 25.01257 1.627e+00
## 21 parent 11.50 12.53462 -1.035e+00
## 21 parent 11.64 12.53462 -8.946e-01
## 35 parent 2.85 3.14787 -2.979e-01
## 35 parent 2.91 3.14787 -2.379e-01
## 50 parent 0.69 0.71624 -2.624e-02
## 50 parent 0.63 0.71624 -8.624e-02
## 75 parent 0.05 0.06074 -1.074e-02
## 75 parent 0.06 0.06074 -7.382e-04
## 1 m1 4.84 4.80296 3.704e-02
## 1 m1 5.64 4.80296 8.370e-01
## 3 m1 12.91 13.02400 -1.140e-01
## 3 m1 12.96 13.02400 -6.400e-02
## 7 m1 22.97 25.04476 -2.075e+00
## 7 m1 24.47 25.04476 -5.748e-01
## 14 m1 41.69 36.69002 5.000e+00
## 14 m1 33.21 36.69002 -3.480e+00
## 21 m1 44.37 41.65310 2.717e+00
## 21 m1 46.44 41.65310 4.787e+00
## 35 m1 41.22 43.31312 -2.093e+00
## 35 m1 37.95 43.31312 -5.363e+00
## 50 m1 41.19 41.21831 -2.831e-02
## 50 m1 40.01 41.21831 -1.208e+00
## 75 m1 40.09 36.44703 3.643e+00
## 75 m1 33.85 36.44703 -2.597e+00
## 100 m1 31.04 31.98163 -9.416e-01
## 100 m1 33.13 31.98163 1.148e+00
## 120 m1 25.15 28.78984 -3.640e+00
## 120 m1 33.31 28.78984 4.520e+00</code></pre>
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