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<title>Coefficient of Quartile Variation: cqv_versatile</title>



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<h1 class="title toc-ignore">Coefficient of Quartile Variation: cqv_versatile</h1>
<h4 class="author">Maani Beigy</h4>
<h4 class="date">February 25, 2019</h4>



<div id="coefficient-of-quartile-variation" class="section level1">
<h1>Coefficient of Quartile Variation</h1>
<p>Coefficient of quartile variation <em>(<span class="math inline">\(CQV\)</span>)</em> is a measure of relative dispersion that is based on interquartile range (<code>IQR</code>). Since <em>cqv</em> is unitless, it is useful for comparison of variables with different units. It is also a measure of homogeneity <span class="citation">(Altunkaynak &amp; Gamgam, 2018; Bonett, 2006)</span>.<br />
The <em>population</em> coefficient of quartile variation is:<br />
<span class="math display">\[
CQV = \biggl(\frac{Q_3-Q_1}{Q_3+Q_1}\biggr)\times100
\]</span> where <span class="math inline">\(q_3\)</span> and <span class="math inline">\(q_1\)</span> are the population third quartile (<em>i.e.,</em> <span class="math inline">\(75^{th}\)</span> percentile) and first quartile (<em>i.e.,</em> <span class="math inline">\(25^{th}\)</span> percentile), respectively. Almost always, we analyze data from samples but want to generalize it as the population’s parameter <span class="citation">(Albatineh, Kibria, Wilcox, &amp; Zogheib, 2014)</span>. Its sample’s estimate is given as:<br />
<span class="math display">\[
cqv = \biggl(\frac{q_3-q_1}{q_3+q_1}\biggr)\times100
\]</span> There are different methods for the calculation of <strong>confidence intervals (CI)</strong> for <em>CQV</em>. All of them are fruitful and have particular use cases. For sake of versatility, we cover almost all of these methods in <code>cvcqv</code> package. Here, we explain them along with some examples:</p>
<div id="bonett-confidence-interval" class="section level3">
<h3>Bonett Confidence Interval</h3>
<p>Bonett <span class="citation">(Bonett, 2006)</span> introduced the following confidence interval for <em>CQV</em>:<br />
<span class="math display">\[
\exp\{\ln{(D/S)c\  \pm\ Z_{1-\alpha/2}\sqrt{v} }\}
\]</span> where <span class="math inline">\(c = n/(n-1)\)</span> is a centering adjustment which helps to equalize the tail error probabilities <span class="citation">(Altunkaynak &amp; Gamgam, 2018; Bonett, 2006)</span>. <span class="math inline">\(D = \hat{Q3}-\hat{Q1}\)</span> and <span class="math inline">\(S = \hat{Q3}+\hat{Q1}\)</span> are the sample <span class="math inline">\(25^{th}\)</span> and <span class="math inline">\(75^{th}\)</span> percentiles, respectively; <span class="math inline">\(Z_{1-\alpha/2}\)</span> is the <span class="math inline">\(1-\alpha/2\)</span> quantile of the standard normal distribution. Computation of <span class="math inline">\(v\)</span> which is <span class="math inline">\(Var\{\ln{(D/S)}\}\)</span> is long and a bit complicated, but has been implemented for <code>cqv</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">x &lt;-<span class="st"> </span><span class="kw">c</span>(</a>
<a class="sourceLine" id="cb1-2" data-line-number="2">    <span class="fl">0.2</span>, <span class="fl">0.5</span>, <span class="fl">1.1</span>, <span class="fl">1.4</span>, <span class="fl">1.8</span>, <span class="fl">2.3</span>, <span class="fl">2.5</span>, <span class="fl">2.7</span>, <span class="fl">3.5</span>, <span class="fl">4.4</span>,</a>
<a class="sourceLine" id="cb1-3" data-line-number="3">    <span class="fl">4.6</span>, <span class="fl">5.4</span>, <span class="fl">5.4</span>, <span class="fl">5.7</span>, <span class="fl">5.8</span>, <span class="fl">5.9</span>, <span class="fl">6.0</span>, <span class="fl">6.6</span>, <span class="fl">7.1</span>, <span class="fl">7.9</span></a>
<a class="sourceLine" id="cb1-4" data-line-number="4">)</a>
<a class="sourceLine" id="cb1-5" data-line-number="5"><span class="kw">cqv_versatile</span>(</a>
<a class="sourceLine" id="cb1-6" data-line-number="6">    x, </a>
<a class="sourceLine" id="cb1-7" data-line-number="7">    <span class="dt">na.rm =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb1-8" data-line-number="8">    <span class="dt">digits =</span> <span class="dv">3</span>, </a>
<a class="sourceLine" id="cb1-9" data-line-number="9">    <span class="dt">method =</span> <span class="st">&quot;bonett&quot;</span></a>
<a class="sourceLine" id="cb1-10" data-line-number="10">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;cqv with Bonett 95% CI&quot;
## 
## $statistics
##      est  lower  upper
##   45.625 24.785 77.329</code></pre>
</div>
<div id="bootstrap-confidence-intervals" class="section level3">
<h3>Bootstrap Confidence Intervals</h3>
<p>Thanks to package <code>boot</code> <span class="citation">(Canty &amp; Ripley, 2017)</span> we can obtain bootstrap CI around <span class="math inline">\(cqv\)</span>:</p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb3-1" data-line-number="1"><span class="kw">cqv_versatile</span>(</a>
<a class="sourceLine" id="cb3-2" data-line-number="2">    x, </a>
<a class="sourceLine" id="cb3-3" data-line-number="3">    <span class="dt">na.rm =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb3-4" data-line-number="4">    <span class="dt">digits =</span> <span class="dv">3</span>, </a>
<a class="sourceLine" id="cb3-5" data-line-number="5">    <span class="dt">method =</span> <span class="st">&quot;bca&quot;</span></a>
<a class="sourceLine" id="cb3-6" data-line-number="6">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;cqv with adjusted bootstrap percentile (BCa) 95% CI&quot;
## 
## $statistics
##      est  lower  upper
##   45.625 23.168 83.956</code></pre>
</div>
<div id="all-available-methods" class="section level3">
<h3>All Available Methods</h3>
<p>In conclusion, we can observe CIs calculated by all available methods:</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb5-1" data-line-number="1"><span class="kw">cqv_versatile</span>(</a>
<a class="sourceLine" id="cb5-2" data-line-number="2">    x, </a>
<a class="sourceLine" id="cb5-3" data-line-number="3">    <span class="dt">na.rm =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb5-4" data-line-number="4">    <span class="dt">digits =</span> <span class="dv">3</span>, </a>
<a class="sourceLine" id="cb5-5" data-line-number="5">    <span class="dt">method =</span> <span class="st">&quot;all&quot;</span></a>
<a class="sourceLine" id="cb5-6" data-line-number="6">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;All methods&quot;
## 
## $statistics
##            est  lower  upper
## bonett  45.625 24.785 77.329
## norm    45.625 19.403 69.740
## basic   45.625 20.710 73.815
## percent 45.625 17.435 70.540
## bca     45.625 20.735 80.710
##                                                 description
## bonett                               cqv with Bonett 95% CI
## norm                   cqv with normal approximation 95% CI
## basic                       cqv with basic bootstrap 95% CI
## percent                cqv with bootstrap percentile 95% CI
## bca     cqv with adjusted bootstrap percentile (BCa) 95% CI</code></pre>
</div>
</div>
<div id="references" class="section level1 unnumbered">
<h1>References</h1>
<div id="refs" class="references">
<div id="ref-Albatineh2014">
<p>Albatineh, A. N., Kibria, B. M., Wilcox, M. L., &amp; Zogheib, B. (2014). Confidence interval estimation for the population coefficient of variation using ranked set sampling: A simulation study. <em>Journal of Applied Statistics</em>, <em>41</em>(4), 733–751. <a href="https://doi.org/10.1080/02664763.2013.847405">https://doi.org/10.1080/02664763.2013.847405</a></p>
</div>
<div id="ref-Altunkaynak2018">
<p>Altunkaynak, B., &amp; Gamgam, H. (2018). Bootstrap confidence intervals for the coefficient of quartile variation. <em>Communications in Statistics: Simulation and Computation</em>, <em>0</em>(0), 1–9. <a href="https://doi.org/10.1080/03610918.2018.1435800">https://doi.org/10.1080/03610918.2018.1435800</a></p>
</div>
<div id="ref-Bonett2006">
<p>Bonett, D. G. (2006). Confidence interval for a coefficient of quartile variation. <em>Computational Statistics and Data Analysis</em>, <em>50</em>(11), 2953–2957. <a href="https://doi.org/10.1016/j.csda.2005.05.007">https://doi.org/10.1016/j.csda.2005.05.007</a></p>
</div>
<div id="ref-Canty2017">
<p>Canty, A., &amp; Ripley, B. (2017). boot: Bootstrap R (S-Plus) Functions. R package version 1.3-20.</p>
</div>
</div>
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