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<!DOCTYPE html>
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<h1>Two-component error model</h1>
<small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/R/sigma_twocomp.R" class="external-link"><code>R/sigma_twocomp.R</code></a></small>
<div class="d-none name"><code>sigma_twocomp.Rd</code></div>
</div>
<div class="ref-description section level2">
<p>Function describing the standard deviation of the measurement error in
dependence of the measured value \(y\):</p>
</div>
<div class="section level2">
<h2 id="ref-usage">Usage<a class="anchor" aria-label="anchor" href="#ref-usage"></a></h2>
<div class="sourceCode"><pre class="sourceCode r"><code><span><span class="fu">sigma_twocomp</span><span class="op">(</span><span class="va">y</span>, <span class="va">sigma_low</span>, <span class="va">rsd_high</span><span class="op">)</span></span></code></pre></div>
</div>
<div class="section level2">
<h2 id="arguments">Arguments<a class="anchor" aria-label="anchor" href="#arguments"></a></h2>
<dl><dt id="arg-y">y<a class="anchor" aria-label="anchor" href="#arg-y"></a></dt>
<dd><p>The magnitude of the observed value</p></dd>
<dt id="arg-sigma-low">sigma_low<a class="anchor" aria-label="anchor" href="#arg-sigma-low"></a></dt>
<dd><p>The asymptotic minimum of the standard deviation for low
observed values</p></dd>
<dt id="arg-rsd-high">rsd_high<a class="anchor" aria-label="anchor" href="#arg-rsd-high"></a></dt>
<dd><p>The coefficient describing the increase of the standard
deviation with the magnitude of the observed value</p></dd>
</dl></div>
<div class="section level2">
<h2 id="value">Value<a class="anchor" aria-label="anchor" href="#value"></a></h2>
<p>The standard deviation of the response variable.</p>
</div>
<div class="section level2">
<h2 id="details">Details<a class="anchor" aria-label="anchor" href="#details"></a></h2>
<p>$$\sigma = \sqrt{ \sigma_{low}^2 + y^2 * {rsd}_{high}^2}$$ sigma =
sqrt(sigma_low^2 + y^2 * rsd_high^2)</p>
<p>This is the error model used for example by Werner et al. (1978). The model
proposed by Rocke and Lorenzato (1995) can be written in this form as well,
but assumes approximate lognormal distribution of errors for high values of
y.</p>
</div>
<div class="section level2">
<h2 id="references">References<a class="anchor" aria-label="anchor" href="#references"></a></h2>
<p>Werner, Mario, Brooks, Samuel H., and Knott, Lancaster B. (1978)
Additive, Multiplicative, and Mixed Analytical Errors. Clinical Chemistry
24(11), 1895-1898.</p>
<p>Rocke, David M. and Lorenzato, Stefan (1995) A two-component model for
measurement error in analytical chemistry. Technometrics 37(2), 176-184.</p>
<p>Ranke J and Meinecke S (2019) Error Models for the Kinetic Evaluation of Chemical
Degradation Data. <em>Environments</em> 6(12) 124
<a href="https://doi.org/10.3390/environments6120124" class="external-link">doi:10.3390/environments6120124</a>
.</p>
</div>
<div class="section level2">
<h2 id="ref-examples">Examples<a class="anchor" aria-label="anchor" href="#ref-examples"></a></h2>
<div class="sourceCode"><pre class="sourceCode r"><code><span class="r-in"><span><span class="va">times</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">1</span>, <span class="fl">3</span>, <span class="fl">7</span>, <span class="fl">14</span>, <span class="fl">28</span>, <span class="fl">60</span>, <span class="fl">90</span>, <span class="fl">120</span><span class="op">)</span></span></span>
<span class="r-in"><span><span class="va">d_pred</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html" class="external-link">data.frame</a></span><span class="op">(</span>time <span class="op">=</span> <span class="va">times</span>, parent <span class="op">=</span> <span class="fl">100</span> <span class="op">*</span> <span class="fu"><a href="https://rdrr.io/r/base/Log.html" class="external-link">exp</a></span><span class="op">(</span><span class="op">-</span> <span class="fl">0.03</span> <span class="op">*</span> <span class="va">times</span><span class="op">)</span><span class="op">)</span></span></span>
<span class="r-in"><span><span class="fu"><a href="https://rdrr.io/r/base/Random.html" class="external-link">set.seed</a></span><span class="op">(</span><span class="fl">123456</span><span class="op">)</span></span></span>
<span class="r-in"><span><span class="va">d_syn</span> <span class="op"><-</span> <span class="fu"><a href="add_err.html">add_err</a></span><span class="op">(</span><span class="va">d_pred</span>, <span class="kw">function</span><span class="op">(</span><span class="va">y</span><span class="op">)</span> <span class="fu">sigma_twocomp</span><span class="op">(</span><span class="va">y</span>, <span class="fl">1</span>, <span class="fl">0.07</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> reps <span class="op">=</span> <span class="fl">2</span>, n <span class="op">=</span> <span class="fl">1</span><span class="op">)</span><span class="op">[[</span><span class="fl">1</span><span class="op">]</span><span class="op">]</span></span></span>
<span class="r-in"><span><span class="va">f_nls</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/stats/nls.html" class="external-link">nls</a></span><span class="op">(</span><span class="va">value</span> <span class="op">~</span> <span class="fu"><a href="https://rdrr.io/r/stats/SSasymp.html" class="external-link">SSasymp</a></span><span class="op">(</span><span class="va">time</span>, <span class="fl">0</span>, <span class="va">parent_0</span>, <span class="va">lrc</span><span class="op">)</span>, data <span class="op">=</span> <span class="va">d_syn</span>,</span></span>
<span class="r-in"><span> start <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>parent_0 <span class="op">=</span> <span class="fl">100</span>, lrc <span class="op">=</span> <span class="op">-</span><span class="fl">3</span><span class="op">)</span><span class="op">)</span></span></span>
<span class="r-in"><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://svn.r-project.org/R-packages/trunk/nlme/" class="external-link">nlme</a></span><span class="op">)</span></span></span>
<span class="r-in"><span><span class="va">f_gnls</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/pkg/nlme/man/gnls.html" class="external-link">gnls</a></span><span class="op">(</span><span class="va">value</span> <span class="op">~</span> <span class="fu"><a href="https://rdrr.io/r/stats/SSasymp.html" class="external-link">SSasymp</a></span><span class="op">(</span><span class="va">time</span>, <span class="fl">0</span>, <span class="va">parent_0</span>, <span class="va">lrc</span><span class="op">)</span>,</span></span>
<span class="r-in"><span> data <span class="op">=</span> <span class="va">d_syn</span>, na.action <span class="op">=</span> <span class="va">na.omit</span>,</span></span>
<span class="r-in"><span> start <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>parent_0 <span class="op">=</span> <span class="fl">100</span>, lrc <span class="op">=</span> <span class="op">-</span><span class="fl">3</span><span class="op">)</span><span class="op">)</span></span></span>
<span class="r-in"><span><span class="kw">if</span> <span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/length.html" class="external-link">length</a></span><span class="op">(</span><span class="fu">findFunction</span><span class="op">(</span><span class="st">"varConstProp"</span><span class="op">)</span><span class="op">)</span> <span class="op">></span> <span class="fl">0</span><span class="op">)</span> <span class="op">{</span></span></span>
<span class="r-in"><span> <span class="va">f_gnls_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_gnls</span>, weights <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/pkg/nlme/man/varConstProp.html" class="external-link">varConstProp</a></span><span class="op">(</span><span class="op">)</span><span class="op">)</span></span></span>
<span class="r-in"><span> <span class="va">f_gnls_tc_sf</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_gnls_tc</span>, control <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>sigma <span class="op">=</span> <span class="fl">1</span><span class="op">)</span><span class="op">)</span></span></span>
<span class="r-in"><span><span class="op">}</span></span></span>
<span class="r-in"><span><span class="va">f_mkin</span> <span class="op"><-</span> <span class="fu"><a href="mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="va">d_syn</span>, error_model <span class="op">=</span> <span class="st">"const"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></span>
<span class="r-in"><span><span class="va">f_mkin_tc</span> <span class="op"><-</span> <span class="fu"><a href="mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="va">d_syn</span>, error_model <span class="op">=</span> <span class="st">"tc"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></span>
<span class="r-in"><span><span class="fu"><a href="plot.mkinfit.html">plot_res</a></span><span class="op">(</span><span class="va">f_mkin_tc</span>, standardized <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></span>
<span class="r-plt img"><img src="sigma_twocomp-1.png" alt="" width="700" height="433"></span>
<span class="r-in"><span><span class="fu"><a href="https://rdrr.io/r/stats/AIC.html" class="external-link">AIC</a></span><span class="op">(</span><span class="va">f_nls</span>, <span class="va">f_gnls</span>, <span class="va">f_gnls_tc</span>, <span class="va">f_gnls_tc_sf</span>, <span class="va">f_mkin</span>, <span class="va">f_mkin_tc</span><span class="op">)</span></span></span>
<span class="r-out co"><span class="r-pr">#></span> df AIC</span>
<span class="r-out co"><span class="r-pr">#></span> f_nls 3 114.4817</span>
<span class="r-out co"><span class="r-pr">#></span> f_gnls 3 114.4817</span>
<span class="r-out co"><span class="r-pr">#></span> f_gnls_tc 5 103.6447</span>
<span class="r-out co"><span class="r-pr">#></span> f_gnls_tc_sf 4 101.6447</span>
<span class="r-out co"><span class="r-pr">#></span> f_mkin 3 114.4817</span>
<span class="r-out co"><span class="r-pr">#></span> f_mkin_tc 4 101.6446</span>
</code></pre></div>
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