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analyte concentration that is significantly different from the blank signal
with a first order error alpha (one-sided significance test).
The detection limit, or more precise, the minimum detectable value
(German: Erfassungsgrenze), is then defined as the signal or analyte
concentration where the probability that the signal is not detected although
the analyte is present (type II or false negative error), is beta (also a
one-sided significance test)."><!-- 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]>
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<h1>Estimate a limit of detection (LOD)</h1>
<div class="hidden name"><code>lod.Rd</code></div>
</div>
<div class="ref-description">
<p>The decision limit (German: Nachweisgrenze) is defined as the signal or
analyte concentration that is significantly different from the blank signal
with a first order error alpha (one-sided significance test).
The detection limit, or more precise, the minimum detectable value
(German: Erfassungsgrenze), is then defined as the signal or analyte
concentration where the probability that the signal is not detected although
the analyte is present (type II or false negative error), is beta (also a
one-sided significance test).</p>
</div>
<div id="ref-usage">
<div class="sourceCode"><pre class="sourceCode r"><code><span class="fu">lod</span><span class="op">(</span><span class="va">object</span>, <span class="va">...</span>, alpha <span class="op">=</span> <span class="fl">0.05</span>, beta <span class="op">=</span> <span class="fl">0.05</span>, method <span class="op">=</span> <span class="st">"default"</span>, tol <span class="op">=</span> <span class="st">"default"</span><span class="op">)</span></code></pre></div>
</div>
<div id="arguments">
<h2>Arguments</h2>
<dl><dt>object</dt>
<dd><p>A univariate model object of class <code><a href="https://rdrr.io/r/stats/lm.html" class="external-link">lm</a></code> or
<code><a href="https://rdrr.io/pkg/MASS/man/rlm.html" class="external-link">rlm</a></code>
with model formula <code>y ~ x</code> or <code>y ~ x - 1</code>,
optionally from a weighted regression.</p></dd>
<dt>...</dt>
<dd><p>Placeholder for further arguments that might be needed by
future implementations.</p></dd>
<dt>alpha</dt>
<dd><p>The error tolerance for the decision limit (critical value).</p></dd>
<dt>beta</dt>
<dd><p>The error tolerance beta for the detection limit.</p></dd>
<dt>method</dt>
<dd><p>The “default” method uses a prediction interval at the LOD
for the estimation of the LOD, which obviously requires
iteration. This is described for example in Massart, p. 432 ff.
The “din” method uses the prediction interval at
x = 0 as an approximation.</p></dd>
<dt>tol</dt>
<dd><p>When the “default” method is used, the default tolerance
for the LOD on the x scale is the value of the smallest non-zero standard
divided by 1000. Can be set to a numeric value to override this.</p></dd>
</dl></div>
<div id="value">
<h2>Value</h2>
<p>A list containig the corresponding x and y values of the estimated limit of
detection of a model used for calibration.</p>
</div>
<div id="note">
<h2>Note</h2>
<p>- The default values for alpha and beta are the ones recommended by IUPAC.
- The estimation of the LOD in terms of the analyte amount/concentration
xD from the LOD in the signal domain SD is done by simply inverting the
calibration function (i.e. assuming a known calibration function).
- The calculation of a LOD from weighted calibration models requires
a weights argument for the internally used <code><a href="https://rdrr.io/r/stats/predict.lm.html" class="external-link">predict.lm</a></code>
function, which is currently not supported in R.</p>
</div>
<div id="references">
<h2>References</h2>
<p>Massart, L.M, Vandenginste, B.G.M., Buydens, L.M.C., De Jong, S., Lewi, P.J.,
Smeyers-Verbeke, J. (1997) Handbook of Chemometrics and Qualimetrics: Part A,
Chapter 13.7.8</p>
<p>J. Inczedy, T. Lengyel, and A.M. Ure (2002) International Union of Pure and
Applied Chemistry Compendium of Analytical Nomenclature: Definitive Rules.
Web edition.</p>
<p>Currie, L. A. (1997) Nomenclature in evaluation of analytical methods including
detection and quantification capabilities (IUPAC Recommendations 1995).
Analytica Chimica Acta 391, 105 - 126.</p>
</div>
<div id="see-also">
<h2>See also</h2>
<div class="dont-index"><p>Examples for <code><a href="din32645.html">din32645</a></code></p></div>
</div>
<div id="ref-examples">
<h2>Examples</h2>
<div class="sourceCode"><pre class="sourceCode r"><code><span class="r-in"><span class="va">m</span> <span class="op"><-</span> <span class="fu"><a href="https://rdrr.io/r/stats/lm.html" class="external-link">lm</a></span><span class="op">(</span><span class="va">y</span> <span class="op">~</span> <span class="va">x</span>, data <span class="op">=</span> <span class="va">din32645</span><span class="op">)</span></span>
<span class="r-in"><span class="fu">lod</span><span class="op">(</span><span class="va">m</span><span class="op">)</span> </span>
<span class="r-out co"><span class="r-pr">#></span> $x</span>
<span class="r-out co"><span class="r-pr">#></span> [1] 0.08655484</span>
<span class="r-out co"><span class="r-pr">#></span> </span>
<span class="r-out co"><span class="r-pr">#></span> $y</span>
<span class="r-out co"><span class="r-pr">#></span> [1] 3317.154</span>
<span class="r-out co"><span class="r-pr">#></span> </span>
<span class="r-in"></span>
<span class="r-in"><span class="co"># The critical value (decision limit, German Nachweisgrenze) can be obtained</span></span>
<span class="r-in"><span class="co"># by using beta = 0.5:</span></span>
<span class="r-in"><span class="fu">lod</span><span class="op">(</span><span class="va">m</span>, alpha <span class="op">=</span> <span class="fl">0.01</span>, beta <span class="op">=</span> <span class="fl">0.5</span><span class="op">)</span></span>
<span class="r-out co"><span class="r-pr">#></span> $x</span>
<span class="r-out co"><span class="r-pr">#></span> [1] 0.0698127</span>
<span class="r-out co"><span class="r-pr">#></span> </span>
<span class="r-out co"><span class="r-pr">#></span> $y</span>
<span class="r-out co"><span class="r-pr">#></span> [1] 3155.393</span>
<span class="r-out co"><span class="r-pr">#></span> </span>
</code></pre></div>
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