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author: Johannes Ranke <jranke@uni-bremen.de> 2019-02-21 16:03:25 +0100
committer: Johannes Ranke <jranke@uni-bremen.de> 2019-02-21 16:03:25 +0100
commit: f55851e21d471a0ff1a4a72893374c30ca4cf819 (patch)
tree: 9e72a99c7985ff6ecba45e209351d6b43bfcf286 /docs/articles/chemCal.html
parent: f8307bd495a0c687e51f600f2bc22f49f9eaccd8 (diff)
1 files changed, 31 insertions, 31 deletions
diff --git a/docs/articles/chemCal.html b/docs/articles/chemCal.html
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@@ -1,18 +1,18 @@
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@@ -22,14 +22,15 @@
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-        <span class="label label-default" data-toggle="tooltip" data-placement="bottom" title="Released package">0.2.1</span>
+        <span class="version label label-default" data-toggle="tooltip" data-placement="bottom" title="Released version">0.2.2</span>
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@@ -66,7 +67,7 @@
       <h1>Introduction to chemCal</h1>
                         <h4 class="author">Johannes Ranke</h4>
             
-            <h4 class="date">2018-07-17</h4>
+            <h4 class="date">2019-02-21</h4>
       
       
       <div class="hidden name"><code>chemCal.Rmd</code></div>
@@ -89,25 +90,25 @@
 <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"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(chemCal)
-m0 &lt;-<span class="st"> </span><span class="kw">lm</span>(y <span class="op">~</span><span class="st"> </span>x, <span class="dt">data =</span> massart97ex3)
-<span class="kw"><a href="../reference/calplot.lm.html">calplot</a></span>(m0)</code></pre></div>
+<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 &lt;-<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"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">plot</span>(m0, <span class="dt">which=</span><span class="dv">3</span>)</code></pre></div>
+<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"><pre class="sourceCode r"><code class="sourceCode r">weights &lt;-<span class="st"> </span><span class="kw">with</span>(massart97ex3, {
-  yx &lt;-<span class="st"> </span><span class="kw">split</span>(y, x)
-  ybar &lt;-<span class="st"> </span><span class="kw">sapply</span>(yx, mean)
-  s &lt;-<span class="st"> </span><span class="kw">round</span>(<span class="kw">sapply</span>(yx, sd), <span class="dt">digits =</span> <span class="dv">2</span>)
-  w &lt;-<span class="st"> </span><span class="kw">round</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>)
-})
-massart97ex3.means &lt;-<span class="st"> </span><span class="kw">aggregate</span>(y <span class="op">~</span><span class="st"> </span>x, massart97ex3, mean)
-
-m &lt;-<span class="st"> </span><span class="kw">lm</span>(y <span class="op">~</span><span class="st"> </span>x, <span class="dt">w =</span> weights, <span class="dt">data =</span> massart97ex3.means)</code></pre></div>
+<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb3-1" data-line-number="1">weights &lt;-<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 &lt;-<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 &lt;-<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 &lt;-<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 &lt;-<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 &lt;-<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 &lt;-<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"><pre class="sourceCode r"><code class="sourceCode r"><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>)</code></pre></div>
+<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
 ## 
@@ -119,7 +120,7 @@ m &lt;-<span class="st"> </span><span class="kw">lm</span>(y <span class="op">~<
 ## 
 ## $`Confidence Limits`
 ## [1] 3.387082 8.343652</code></pre>
-<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><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>)</code></pre></div>
+<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
 ## 
@@ -138,27 +139,27 @@ m &lt;-<span class="st"> </span><span class="kw">lm</span>(y <span class="op">~<
 <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>
-<span class="math display">\[\begin{equation}
+<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>
+\end{equation}\]</span></p>
 <p>with</p>
-<span class="math display">\[\begin{equation}
+<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>
+\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>
-<span class="math display">\[\begin{equation}
+<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>
+\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">
@@ -197,9 +198,8 @@ s_{\hat{x_s}} = \frac{1}{b_1} \sqrt{\frac{{s_s}^2}{w_s m} +
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author	Johannes Ranke <jranke@uni-bremen.de>	2019-02-21 16:03:25 +0100
committer	Johannes Ranke <jranke@uni-bremen.de>	2019-02-21 16:03:25 +0100
commit	f55851e21d471a0ff1a4a72893374c30ca4cf819 (patch)
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