<!-- Generated by pkgdown: do not edit by hand --> <!DOCTYPE html> <html lang="en"> <head> <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 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href="../news/index.html">News</a> </li> </ul> <ul class="nav navbar-nav navbar-right"> <li> <a href="https://github.com/jranke/mkin/"> <span class="fab fa 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/master/R/sigma_twocomp.R'><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> <pre class="usage"><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></pre> <h2 class="hasAnchor" id="arguments"><a class="anchor" href="#arguments"></a>Arguments</h2> <table class="ref-arguments"> <colgroup><col class="name" /><col class="desc" /></colgroup> <tr> <th>y</th> <td><p>The magnitude of the observed value</p></td> </tr> <tr> <th>sigma_low</th> <td><p>The asymptotic minimum of the standard deviation for low observed values</p></td> </tr> <tr> <th>rsd_high</th> <td><p>The coefficient describing the increase of the standard deviation with the magnitude of the observed value</p></td> </tr> </table> <h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</h2> <p>The standard deviation of the response variable.</p> <h2 class="hasAnchor" id="details"><a class="anchor" href="#details"></a>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> <h2 class="hasAnchor" id="references"><a class="anchor" href="#references"></a>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> <h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2> <pre class="examples"><div class='input'><span class='va'>times</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/c.html'>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 class='va'>d_pred</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/data.frame.html'>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'>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 class='fu'><a href='https://rdrr.io/r/base/Random.html'>set.seed</a></span><span class='op'>(</span><span class='fl'>123456</span><span class='op'>)</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>, 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 class='va'>f_nls</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/stats/nls.html'>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'>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>, start <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/list.html'>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 class='kw'><a href='https://rdrr.io/r/base/library.html'>library</a></span><span class='op'>(</span><span class='va'><a href='https://svn.r-project.org/R-packages/trunk/nlme/'>nlme</a></span><span class='op'>)</span> <span class='va'>f_gnls</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/pkg/nlme/man/gnls.html'>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'>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>, na.action <span class='op'>=</span> <span class='va'>na.omit</span>, start <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/list.html'>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 class='kw'>if</span> <span class='op'>(</span><span class='fu'><a href='https://rdrr.io/r/base/length.html'>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 class='va'>f_gnls_tc</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/stats/update.html'>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'>varConstProp</a></span><span class='op'>(</span><span class='op'>)</span><span class='op'>)</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'>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'>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 class='op'>}</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 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 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> </div><div class='img'><img src='sigma_twocomp-1.png' alt='' width='700' height='433' /></div><div class='input'><span class='fu'><a href='https://rdrr.io/r/stats/AIC.html'>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> </div><div class='output co'>#> df AIC #> f_nls 3 114.4817 #> f_gnls 3 114.4817 #> f_gnls_tc 5 103.6447 #> f_gnls_tc_sf 4 101.6447 #> f_mkin 3 114.4817 #> f_mkin_tc 4 101.6446</div></pre> </div> <div class="col-md-3 hidden-xs 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