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of the quantification given the calibration model reaches a prespecified
value 1/k. Thus, it is the solution of the equation
$$L = k c(L)$$
where c(L) is half of the length of the confidence interval at the limit L
(DIN 32645, equivalent to ISO 11843). c(L) is internally estimated by
inverse.predict, and L is obtained by iteration."><!-- 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 quantification (LOQ)</h1>
<div class="hidden name"><code>loq.Rd</code></div>
</div>
<div class="ref-description">
<p>The limit of quantification is the x value, where the relative error
of the quantification given the calibration model reaches a prespecified
value 1/k. Thus, it is the solution of the equation
$$L = k c(L)$$
where c(L) is half of the length of the confidence interval at the limit L
(DIN 32645, equivalent to ISO 11843). c(L) is internally estimated by
<code><a href="inverse.predict.html">inverse.predict</a></code>, and L is obtained by iteration.</p>
</div>
<div id="ref-usage">
<div class="sourceCode"><pre class="sourceCode r"><code><span class="fu">loq</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>, k <span class="op">=</span> <span class="fl">3</span>, n <span class="op">=</span> <span class="fl">1</span>, w.loq <span class="op">=</span> <span class="st">"auto"</span>,
var.loq <span class="op">=</span> <span class="st">"auto"</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. If weights are specified
in the model, either <code>w.loq</code> or <code>var.loq</code> have to
be specified.</p></dd>
<dt>alpha</dt>
<dd><p>The error tolerance for the prediction of x values in the calculation.</p></dd>
<dt>...</dt>
<dd><p>Placeholder for further arguments that might be needed by
future implementations.</p></dd>
<dt>k</dt>
<dd><p>The inverse of the maximum relative error tolerated at the
desired LOQ.</p></dd>
<dt>n</dt>
<dd><p>The number of replicate measurements for which the LOQ should be
specified.</p></dd>
<dt>w.loq</dt>
<dd><p>The weight that should be attributed to the LOQ. Defaults
to one for unweighted regression, and to the mean of the weights
for weighted regression. See <code><a href="massart97ex3.html">massart97ex3</a></code> for
an example how to take advantage of knowledge about the
variance function.</p></dd>
<dt>var.loq</dt>
<dd><p>The approximate variance at the LOQ. The default value is
calculated from the model.</p></dd>
<dt>tol</dt>
<dd><p>The default tolerance for the LOQ 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>The estimated limit of quantification for a model used for calibration.</p>
</div>
<div id="note">
<h2>Note</h2>
<p>- IUPAC recommends to base the LOQ on the standard deviation of the signal
where x = 0.
- The calculation of a LOQ based on weighted regression is non-standard
and therefore not tested. Feedback is welcome.</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">massart97ex1</span><span class="op">)</span></span>
<span class="r-in"><span class="fu">loq</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] 13.97764</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] 30.6235</span>
<span class="r-out co"><span class="r-pr">#></span> </span>
<span class="r-in"></span>
<span class="r-in"><span class="co"># We can get better by using replicate measurements</span></span>
<span class="r-in"><span class="fu">loq</span><span class="op">(</span><span class="va">m</span>, n <span class="op">=</span> <span class="fl">3</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] 9.971963</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] 22.68539</span>
<span class="r-out co"><span class="r-pr">#></span> </span>
</code></pre></div>
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