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<h1>Introduction to chemCal</h1>
<h4 class="author">Johannes Ranke</h4>
<h4 class="date">2019-02-21</h4>
<div class="hidden name"><code>chemCal.Rmd</code></div>
</div>
<div id="basic-calibration-functions" class="section level1">
<h1 class="hasAnchor">
<a href="#basic-calibration-functions" class="anchor"></a>Basic calibration functions</h1>
<p>The <code>chemCal</code> package was first designed in the course of a lecture and lab course on “Analytics of Organic Trace Contaminants” at the University of Bremen from October to December 2004. In the fall 2005, an email exchange with Ron Wehrens led to the belief that it would be desirable to implement the inverse prediction method given in <span class="citation">Massart et al. (1997)</span> since it also covers the case of weighted regression. Studies of the IUPAC orange book and of DIN 32645 (equivalent to ISO 11843), publications by <span class="citation">Currie (1997)</span> and the Analytical Method Committee of the Royal Society of Chemistry <span class="citation">(Analytical Methods Committee 1989)</span> and a nice paper by Castells and Castillo <span class="citation">(Castells and Castillo 2000)</span> provided some further understanding of the matter.</p>
<p>At the moment, the package consists of four functions (<a href="https://pkgdown.jrwb.de/chemCal/reference/calplot.lm.html">calplot</a>, <a href="https://pkgdown.jrwb.de/chemCal/reference/lod.html">lod</a>, <a href="https://pkgdown.jrwb.de/chemCal/reference/loq.html">loq</a> and <a href="https://pkgdown.jrwb.de/chemCal/reference/inverse.predict.html">inverse.predict</a>), working on univariate linear models of class <code>lm</code> or <code>rlm</code>, plus several datasets for validation.</p>
<p>A <a href="https://bugs.r-project.org/bugzilla/show_bug.cgi?id=8877">bug report</a> and the following e-mail exchange on the r-devel mailing list about prediction intervals from weighted regression entailed some further studies on this subject. However, I did not encounter any proof or explanation of the formula cited below yet, so I can’t really confirm that Massart’s method is correct.</p>
<p>In fact, in June 2018 I was made aware of the fact that the inverse prediction method implemented in chemCal version 0.1.37 and before did not take the variance of replicate calibration standards about their means into account, nor the number of replicates when calculating the degrees of freedom. Thanks to PhD student Anna Burniol Figols for reporting this issue!</p>
<p>As a consequence, I rewrote <code>inverse.predict</code> not to automatically work with the mean responses for each calibration standard any more. The example calculations from <span class="citation">Massart et al. (1997)</span> can still be reproduced when the regression model is calculated using the means of the calibration data as shown below.</p>
</div>
<div id="usage" class="section level1">
<h1 class="hasAnchor">
<a href="#usage" class="anchor"></a>Usage</h1>
<p>When calibrating an analytical method, the first task is to generate a suitable model. If we want to use the <code>chemCal</code> functions, we have to restrict ourselves to univariate, possibly weighted, linear regression so far.</p>
<p>Once such a model has been created, the calibration can be graphically shown by using the <code>calplot</code> function:</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb1-1" data-line-number="1"><span class="kw"><a href="https://www.rdocumentation.org/packages/base/topics/library">library</a></span>(chemCal)</a>
<a class="sourceLine" id="cb1-2" data-line-number="2">m0 <-<span class="st"> </span><span class="kw"><a href="https://www.rdocumentation.org/packages/stats/topics/lm">lm</a></span>(y <span class="op">~</span><span class="st"> </span>x, <span class="dt">data =</span> massart97ex3)</a>
<a class="sourceLine" id="cb1-3" data-line-number="3"><span class="kw"><a href="../reference/calplot.lm.html">calplot</a></span>(m0)</a></code></pre></div>
<p><img src="chemCal_files/figure-html/unnamed-chunk-1-1.png" width="700"></p>
<p>As we can see, the scatter increases with increasing x. This is also illustrated by one of the diagnostic plots for linear models provided by R:</p>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb2-1" data-line-number="1"><span class="kw"><a href="https://www.rdocumentation.org/packages/graphics/topics/plot">plot</a></span>(m0, <span class="dt">which=</span><span class="dv">3</span>)</a></code></pre></div>
<p><img src="chemCal_files/figure-html/unnamed-chunk-2-1.png" width="700"></p>
<p>Therefore, in Example 8 in <span class="citation">Massart et al. (1997)</span>, weighted regression is proposed which can be reproduced by the following code. Note that we are building the model on the mean values for each standard in order to be able to reproduce the results given in the book with the current version of chemCal.</p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb3-1" data-line-number="1">weights <-<span class="st"> </span><span class="kw"><a href="https://www.rdocumentation.org/packages/base/topics/with">with</a></span>(massart97ex3, {</a>
<a class="sourceLine" id="cb3-2" data-line-number="2"> yx <-<span class="st"> </span><span class="kw"><a href="https://www.rdocumentation.org/packages/base/topics/split">split</a></span>(y, x)</a>
<a class="sourceLine" id="cb3-3" data-line-number="3"> ybar <-<span class="st"> </span><span class="kw"><a href="https://www.rdocumentation.org/packages/base/topics/lapply">sapply</a></span>(yx, mean)</a>
<a class="sourceLine" id="cb3-4" data-line-number="4"> s <-<span class="st"> </span><span class="kw"><a href="https://www.rdocumentation.org/packages/base/topics/Round">round</a></span>(<span class="kw"><a href="https://www.rdocumentation.org/packages/base/topics/lapply">sapply</a></span>(yx, sd), <span class="dt">digits =</span> <span class="dv">2</span>)</a>
<a class="sourceLine" id="cb3-5" data-line-number="5"> w <-<span class="st"> </span><span class="kw"><a href="https://www.rdocumentation.org/packages/base/topics/Round">round</a></span>(<span class="dv">1</span> <span class="op">/</span><span class="st"> </span>(s<span class="op">^</span><span class="dv">2</span>), <span class="dt">digits =</span> <span class="dv">3</span>)</a>
<a class="sourceLine" id="cb3-6" data-line-number="6">})</a>
<a class="sourceLine" id="cb3-7" data-line-number="7">massart97ex3.means <-<span class="st"> </span><span class="kw"><a href="https://www.rdocumentation.org/packages/stats/topics/aggregate">aggregate</a></span>(y <span class="op">~</span><span class="st"> </span>x, massart97ex3, mean)</a>
<a class="sourceLine" id="cb3-8" data-line-number="8"></a>
<a class="sourceLine" id="cb3-9" data-line-number="9">m <-<span class="st"> </span><span class="kw"><a href="https://www.rdocumentation.org/packages/stats/topics/lm">lm</a></span>(y <span class="op">~</span><span class="st"> </span>x, <span class="dt">w =</span> weights, <span class="dt">data =</span> massart97ex3.means)</a></code></pre></div>
<p>If we now want to predict a new x value from measured y values, we use the <code>inverse.predict</code> function:</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb4-1" data-line-number="1"><span class="kw"><a href="../reference/inverse.predict.html">inverse.predict</a></span>(m, <span class="dv">15</span>, <span class="dt">ws=</span><span class="fl">1.67</span>)</a></code></pre></div>
<pre><code>## $Prediction
## [1] 5.865367
##
## $`Standard Error`
## [1] 0.8926109
##
## $Confidence
## [1] 2.478285
##
## $`Confidence Limits`
## [1] 3.387082 8.343652</code></pre>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb6-1" data-line-number="1"><span class="kw"><a href="../reference/inverse.predict.html">inverse.predict</a></span>(m, <span class="dv">90</span>, <span class="dt">ws =</span> <span class="fl">0.145</span>)</a></code></pre></div>
<pre><code>## $Prediction
## [1] 44.06025
##
## $`Standard Error`
## [1] 2.829162
##
## $Confidence
## [1] 7.855012
##
## $`Confidence Limits`
## [1] 36.20523 51.91526</code></pre>
<p>The weight <code>ws</code> assigned to the measured y value has to be given by the user in the case of weighted regression, or alternatively, the approximate variance <code>var.s</code> at this location.</p>
</div>
<div id="background-for-inverse-predict" class="section level1">
<h1 class="hasAnchor">
<a href="#background-for-inverse-predict" class="anchor"></a>Background for <code>inverse.predict</code>
</h1>
<p>Equation 8.28 in <span class="citation">Massart et al. (1997)</span> gives a general equation for predicting the standard error <span class="math inline">\(s_{\hat{x_s}}\)</span> for an <span class="math inline">\(x\)</span> value predicted from measurements of <span class="math inline">\(y\)</span> according to the linear calibration function <span class="math inline">\(y = b_0 + b_1 \cdot x\)</span>:</p>
<p><span class="math display">\[\begin{equation}
s_{\hat{x_s}} = \frac{s_e}{b_1} \sqrt{\frac{1}{w_s m} + \frac{1}{\sum{w_i}} +
\frac{(\bar{y_s} - \bar{y_w})^2 \sum{w_i}}
{{b_1}^2 \left( \sum{w_i} \sum{w_i {x_i}^2} -
{\left( \sum{ w_i x_i } \right)}^2 \right) }}
\end{equation}\]</span></p>
<p>with</p>
<p><span class="math display">\[\begin{equation}
s_e = \sqrt{ \frac{\sum w_i (y_i - \hat{y_i})^2}{n - 2}}
\end{equation}\]</span></p>
<p>In chemCal version before 0.2, I interpreted <span class="math inline">\(w_i\)</span> to be the weight for calibration standard <span class="math inline">\(i\)</span>, <span class="math inline">\(y_i\)</span> to be the mean value observed for standard <span class="math inline">\(i\)</span>, and <span class="math inline">\(n\)</span> to be the number of calibration standards. With this implementation I was able to reproduce the examples given in the book. However, as noted above, I was made aware of the fact that this way of calculation does not take the variation of the y values about the means into account. Furthermore, I noticed that for the case of unweighted linear calibration with replicate standards, <code>inverse.predict</code> produced different results than <code>calibrate</code> from the <code>investr</code> package when using the Wald method.</p>
<p>Both issues are now addressed in chemCal starting from version 0.2.1. Here, <span class="math inline">\(y_i\)</span> is calibration measurement <span class="math inline">\(i\)</span>, <span class="math inline">\(\hat{y_i}\)</span> is the estimated value for calibration measurement <span class="math inline">\(i\)</span> and <span class="math inline">\(n\)</span> is the total number of calibration measurements.</p>
<p><span class="math inline">\(w_s\)</span> is the weight attributed to the sample <span class="math inline">\(s\)</span>, <span class="math inline">\(m\)</span> is the number of replicate measurements of sample <span class="math inline">\(s\)</span>, <span class="math inline">\(\bar{y_s}\)</span> is the mean response for the sample, <span class="math inline">\(\bar{y_w} = \frac{\sum{w_i y_i}}{\sum{w_i}}\)</span> is the weighted mean of responses <span class="math inline">\(y_i\)</span>, and <span class="math inline">\(x_i\)</span> is the given <span class="math inline">\(x\)</span> value for standard <span class="math inline">\(i\)</span>.</p>
<p>The weight <span class="math inline">\(w_s\)</span> for the sample should be estimated or calculated in accordance to the weights used in the linear regression.</p>
<p>I had also adjusted the above equation in order to be able to take a different precisions in standards and samples into account. In analogy to Equation 8.26 from I am using</p>
<p><span class="math display">\[\begin{equation}
s_{\hat{x_s}} = \frac{1}{b_1} \sqrt{\frac{{s_s}^2}{w_s m} +
{s_e}^2 \left( \frac{1}{\sum{w_i}} +
\frac{(\bar{y_s} - \bar{y_w})^2 \sum{w_i}}
{{b_1}^2 \left( \sum{w_i} \sum{w_i {x_i}^2} - {\left( \sum{ w_i x_i } \right)}^2 \right) } \right) }
\end{equation}\]</span></p>
<p>where I interpret <span class="math inline">\(\frac{{s_s}^2}{w_s}\)</span> as an estimator of the variance at location <span class="math inline">\(\hat{x_s}\)</span>, which can be replaced by a user-specified value using the argument <code>var.s</code> of the function <code>inverse.predict</code>.</p>
<div id="refs" class="references">
<div id="ref-amc89">
<p>Analytical Methods Committee. 1989. “Robust Statistics — How Not to Reject Outliers. Part 1. Basic Concepts.” <em>The Analyst</em> 114: 1693–7.</p>
</div>
<div id="ref-castells00">
<p>Castells, Reynaldo César, and Marcela Alejandra Castillo. 2000. “Systematic Errors: Detection and Correction by Means of Standard Calibration, Youden Calibration and Standard Additions Method in Conjunction with a Method Response Model.” <em>Analytica Chimica Acta</em> 423: 179–85.</p>
</div>
<div id="ref-currie97">
<p>Currie, L. A. 1997. “Nomenclature in Evaluation of Analytical Methods Including Detection and Quantification Capabilities (IUPAC Recommendations 1995).” <em>Analytica Chimica Acta</em> 391: 105–26.</p>
</div>
<div id="ref-massart97">
<p>Massart, D. L, B. G. M. Vandeginste, L. M. C. Buydens, S. De Jong, P. J. Lewi, and J Smeyers-Verbeke. 1997. <em>Handbook of Chemometrics and Qualimetrics: Part A</em>. Amsterdam: Elsevier.</p>
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