<!DOCTYPE html> <!-- Generated by pkgdown: do not edit by hand --><html lang="en"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><meta charset="utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge"><meta name="viewport" content="width=device-width, initial-scale=1.0"><title>Two-component error model — sigma_twocomp • mkin</title><!-- jquery --><script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min.js" integrity="sha256-CSXorXvZcTkaix6Yvo6HppcZGetbYMGWSFlBw8HfCJo=" crossorigin="anonymous"></script><!-- Bootstrap --><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.4.1/css/bootstrap.min.css" integrity="sha256-bZLfwXAP04zRMK2BjiO8iu9pf4FbLqX6zitd+tIvLhE=" crossorigin="anonymous"><script src="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.4.1/js/bootstrap.min.js" integrity="sha256-nuL8/2cJ5NDSSwnKD8VqreErSWHtnEP9E7AySL+1ev4=" crossorigin="anonymous"></script><!-- bootstrap-toc --><link rel="stylesheet" href="../bootstrap-toc.css"><script src="../bootstrap-toc.js"></script><!-- Font Awesome icons --><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/all.min.css" integrity="sha256-mmgLkCYLUQbXn0B1SRqzHar6dCnv9oZFPEC1g1cwlkk=" crossorigin="anonymous"><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/v4-shims.min.css" integrity="sha256-wZjR52fzng1pJHwx4aV2AO3yyTOXrcDW7jBpJtTwVxw=" crossorigin="anonymous"><!-- clipboard.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/clipboard.js/2.0.6/clipboard.min.js" integrity="sha256-inc5kl9MA1hkeYUt+EC3BhlIgyp/2jDIyBLS6k3UxPI=" crossorigin="anonymous"></script><!-- headroom.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/headroom.min.js" integrity="sha256-AsUX4SJE1+yuDu5+mAVzJbuYNPHj/WroHuZ8Ir/CkE0=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/jQuery.headroom.min.js" integrity="sha256-ZX/yNShbjqsohH1k95liqY9Gd8uOiE1S4vZc+9KQ1K4=" crossorigin="anonymous"></script><!-- pkgdown --><link href="../pkgdown.css" rel="stylesheet"><script src="../pkgdown.js"></script><meta property="og:title" content="Two-component error model — sigma_twocomp"><meta property="og:description" content="Function describing the standard deviation of the measurement error in dependence of the measured value \(y\):"><!-- mathjax --><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js" integrity="sha256-nvJJv9wWKEm88qvoQl9ekL2J+k/RWIsaSScxxlsrv8k=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/config/TeX-AMS-MML_HTMLorMML.js" integrity="sha256-84DKXVJXs0/F8OTMzX4UR909+jtl4G7SPypPavF+GfA=" crossorigin="anonymous"></script><!--[if lt IE 9]> <script src="https://oss.maxcdn.com/html5shiv/3.7.3/html5shiv.min.js"></script> <script src="https://oss.maxcdn.com/respond/1.4.2/respond.min.js"></script> <![endif]--></head><body data-spy="scroll" data-target="#toc"> <div class="container template-reference-topic"> <header><div class="navbar navbar-default navbar-fixed-top" role="navigation"> <div class="container"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <span class="navbar-brand"> <a class="navbar-link" href="../index.html">mkin</a> <span class="version label label-default" data-toggle="tooltip" data-placement="bottom" title="Released version">1.2.0</span> </span> </div> <div id="navbar" class="navbar-collapse collapse"> <ul class="nav navbar-nav"><li> <a href="../reference/index.html">Functions and data</a> </li> <li class="dropdown"> <a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" data-bs-toggle="dropdown" aria-expanded="false"> Articles <span class="caret"></span> </a> <ul class="dropdown-menu" role="menu"><li> <a href="../articles/mkin.html">Introduction to mkin</a> </li> <li> <a href="../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a> </li> <li> <a href="../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a> </li> <li> <a href="../articles/web_only/dimethenamid_2018.html">Example evaluations of dimethenamid data from 2018 with nonlinear mixed-effects models</a> </li> <li> <a href="../articles/web_only/multistart.html">Short demo of the multistart method</a> </li> <li> <a href="../articles/web_only/compiled_models.html">Performance benefit by using compiled model definitions in mkin</a> </li> <li> <a href="../articles/web_only/FOCUS_Z.html">Example evaluation of FOCUS Example Dataset Z</a> </li> <li> <a href="../articles/twa.html">Calculation of time weighted average concentrations with mkin</a> </li> <li> <a href="../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a> </li> <li> <a href="../articles/web_only/benchmarks.html">Benchmark timings for mkin</a> </li> <li> <a href="../articles/web_only/saem_benchmarks.html">Benchmark timings for saem.mmkin</a> </li> </ul></li> <li> <a href="../news/index.html">News</a> </li> </ul><ul class="nav navbar-nav navbar-right"><li> <a href="https://github.com/jranke/mkin/" class="external-link"> <span class="fab fa-github fa-lg"></span> </a> </li> </ul></div><!--/.nav-collapse --> </div><!--/.container --> </div><!--/.navbar --> </header><div class="row"> <div class="col-md-9 contents"> <div class="page-header"> <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="hidden name"><code>sigma_twocomp.Rd</code></div> </div> <div class="ref-description"> <p>Function describing the standard deviation of the measurement error in dependence of the measured value \(y\):</p> </div> <div id="ref-usage"> <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 id="arguments"> <h2>Arguments</h2> <dl><dt>y</dt> <dd><p>The magnitude of the observed value</p></dd> <dt>sigma_low</dt> <dd><p>The asymptotic minimum of the standard deviation for low observed values</p></dd> <dt>rsd_high</dt> <dd><p>The coefficient describing the increase of the standard deviation with the magnitude of the observed value</p></dd> </dl></div> <div id="value"> <h2>Value</h2> <p>The standard deviation of the response variable.</p> </div> <div id="details"> <h2>Details</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 id="references"> <h2>References</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 id="ref-examples"> <h2>Examples</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> </div> </div> <div class="col-md-3 hidden-xs hidden-sm" id="pkgdown-sidebar"> <nav id="toc" data-toggle="toc" class="sticky-top"><h2 data-toc-skip>Contents</h2> </nav></div> </div> <footer><div class="copyright"> <p></p><p>Developed by Johannes Ranke.</p> </div> <div class="pkgdown"> <p></p><p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.0.6.</p> </div> </footer></div> </body></html>