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<title>Coefficient of Variation: cv_versatile</title>



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<h1 class="title toc-ignore">Coefficient of Variation: cv_versatile</h1>
<h4 class="author">Maani Beigy</h4>
<h4 class="date">February 18, 2019</h4>



<div id="coefficient-of-variation" class="section level1">
<h1>Coefficient of Variation</h1>
<p>Coefficient of variation <em>(<span class="math inline">\(CV\)</span>)</em> is a measure of relative dispersion representing the degree of variability relative to the mean <span class="citation">(Albatineh, Kibria, Wilcox, &amp; Zogheib, 2014)</span>. Since cv is unitless, it is useful for comparison of variables with different units <span class="citation">(Albatineh et al., 2014)</span>. It is also a measure of homogeneity. The <em>population</em> coefficient of variation is:<br />
<span class="math display">\[CV = \frac{\sigma}{\mu},\]</span> where <span class="math inline">\(\sigma\)</span> is the population standard deviation and <span class="math inline">\(\mu\)</span> is the population mean. Almost always, we analyze data from samples but want to generalize it as the population’s parameter <span class="citation">(Albatineh et al., 2014)</span>. Its sample’s estimate is given as:<br />
<span class="math display">\[cv = \frac{sd}{\bar{X}}\]</span><br />
where <span class="math inline">\(sd\)</span> is the sample standard deviation, the square root of the unbiased estimator of population variance, and <span class="math inline">\(\bar{X}\)</span> is the sample mean. The corrected <em>cv</em> to account for the sample size is: <span class="math display">\[
cv_{corr} = cv * \biggl(1 - \frac{1}{4(n-1)} 
+ \frac{1}{n}cv^2 
+ \frac{1}{2 (n-1)^2} \biggr)
\]</span> There are various methods for the calculation of <strong>confidence intervals (CI)</strong> for <em>cv</em>. All of them are fruitful and have particular use cases. Some of them are model-based hence their usage depends the assumptions regarding the distribution of data. 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="kelley-confidence-interval" class="section level3">
<h3>Kelley Confidence Interval</h3>
<p>Let us assume that <em>CV</em> follows a noncentral <em>t</em> distribution, when the parent population of the scores is <em>normally-distributed</em>, with noncentrality (<span class="math inline">\(\lambda\)</span>) parameter:<br />
<span class="math display">\[
\lambda = \frac{\sqrt{n}}{cv}
\]</span> with <em>v</em> degrees of freedom, where <span class="math inline">\(v = n - 1\)</span>. Let <span class="math inline">\(1 - \alpha\)</span> be the CI coverage with <span class="math inline">\(\alpha_L + \alpha_U = \alpha\)</span> in which <span class="math inline">\(\alpha_L\)</span> is the the proportion of times that <em>cv</em> will be less than the lower confidence bound and <span class="math inline">\(\alpha_U\)</span> the proportion of times that <em>cv</em> will be greater than the upper confidence bound in the CI procedure <span class="citation">(Kelley, 2007)</span>. The lower confidence tile for <span class="math inline">\(\lambda\)</span> is is the noncentrality parameter that results in <span class="math inline">\(t_{(1-\alpha_L,v,\lambda_L)}=\hat{\lambda}\)</span> and the upper confidence tile for <span class="math inline">\(\lambda\)</span> is is the noncentrality parameter that results in <span class="math inline">\(t_{(\alpha_U,v,\lambda_U)}=\hat{\lambda}\)</span>, where <span class="math inline">\(t_{(1-\alpha_L,v,\lambda_L)}=\hat{\lambda}\)</span> is the value of noncentral <em>t</em> distribution at the <span class="math inline">\(1-\alpha_L\)</span> <strong>quantile</strong> with noncentrality parameter <span class="math inline">\(\lambda_L\)</span> and <span class="math inline">\(t_{(\alpha_U,v,\lambda_U)}=\hat{\lambda}\)</span> is the value of noncentral <em>t</em> distribution at the <span class="math inline">\(\alpha_U\)</span> <strong>quantile</strong> with noncentrality parameter <span class="math inline">\(\lambda_U\)</span>, respectively <span class="citation">(Kelley, 2007)</span>.<br />
Afterwards, we transform the tiles of the confidence interval for <span class="math inline">\(\lambda\)</span>, by dividing the tiles by <span class="math inline">\(\sqrt{n}\)</span> and thereafter inverting them; the CI limits of <span class="math inline">\(cv\)</span> will be obtained:<br />
<span class="math display">\[
p\left[\biggl(\frac{\lambda_U}{\sqrt{n}}\biggr)^{-1}
\le CV \le \biggl(\frac{\lambda_L}{\sqrt{n}}\biggr)^{-1}\right] = 1-\alpha
\]</span> where <span class="math inline">\(p\)</span> stands for <em>probability</em>. Thanks to package <code>MBESS</code> <span class="citation">(Kelley, 2018)</span> for the computation of confidence limits for the noncentrality parameter from a <em>t</em> distribution (<code>conf.limits.nct</code>), <span class="math inline">\(cv\)</span> will be obtained as:</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">cv_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;kelley&quot;</span>, </a>
<a class="sourceLine" id="cb1-10" data-line-number="10">    <span class="dt">correction =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb1-11" data-line-number="11">    <span class="dt">alpha =</span> <span class="fl">0.05</span></a>
<a class="sourceLine" id="cb1-12" data-line-number="12">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;Corrected cv with Kelley 95% CI&quot;
## 
## $statistics
##      est  lower upper
##   58.058 41.467 98.51</code></pre>
</div>
<div id="mckay-confidence-interval" class="section level3">
<h3>McKay Confidence Interval</h3>
<p>McKay <span class="citation">(McKay, 1932)</span> introduced the following CI for <span class="math inline">\(cv\)</span>; considering <span class="math inline">\(u_1 = \chi_{v,1-\alpha/2}^2\)</span> and <span class="math inline">\(u_1 = \chi_{v,\alpha/2}^2\)</span> being the <span class="math inline">\(100(1-\alpha/2)\%\)</span> and <span class="math inline">\(100(\alpha/2)\%\)</span> percentile of the <span class="math inline">\(\chi^2\)</span> distribution with <span class="math inline">\(v = n-1\)</span> degrees of freedom, respectively <span class="citation">(Albatineh et al., 2014)</span>:<br />
<span class="math display">\[
\biggl(cv\left[\biggl(\frac{u_1}{v}-1\biggr)(cv)^{2}+\frac{u_1}{v}\right]^{-1/2}
\le CV \le cv
\left[\biggl(\frac{u_2}{v}-1\biggr)(cv)^{2}+\frac{u_2}{v}\right]^{-1/2}\biggr)
\]</span> Let us calculate the 95% CI for our variable <span class="math inline">\(x\)</span> according to McKay’s method <span class="citation">(McKay, 1932)</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">cv_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;mckay&quot;</span>, </a>
<a class="sourceLine" id="cb3-6" data-line-number="6">    <span class="dt">correction =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb3-7" data-line-number="7">    <span class="dt">alpha =</span> <span class="fl">0.05</span></a>
<a class="sourceLine" id="cb3-8" data-line-number="8">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;Corrected cv with McKay 95% CI&quot;
## 
## $statistics
##      est  lower   upper
##   58.058 41.622 109.367</code></pre>
</div>
<div id="miller-confidence-interval" class="section level3">
<h3>Miller Confidence Interval</h3>
<p>Miller <span class="citation">(Edward Miller, 1991)</span> introduced the following CI for <span class="math inline">\(cv\)</span>; considering <span class="math inline">\(Z_{\alpha/2}\)</span> being the <span class="math inline">\((1-\alpha/2)\)</span> percentile of the standard normal distribution <span class="citation">(Albatineh et al., 2014)</span>: <span class="math display">\[
\biggl(cv - Z_{\alpha/2}\sqrt{
\biggl(\frac{cv^2}{v}\biggr)\biggl(\frac{1}{2}+cv^2\biggr)} \le
CV \le cv + Z_{\alpha/2}\sqrt{
\biggl(\frac{cv^2}{v}\biggr)\biggl(\frac{1}{2}+cv^2\biggr)}
\biggr)
\]</span> where <span class="math inline">\(v = n-1\)</span> is the degree of freedom.<br />
Let us calculate the 95% CI for <span class="math inline">\(x\)</span> according to Miller’s method <span class="citation">(Edward Miller, 1991)</span>:</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">cv_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;miller&quot;</span>, </a>
<a class="sourceLine" id="cb5-6" data-line-number="6">    <span class="dt">correction =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb5-7" data-line-number="7">    <span class="dt">alpha =</span> <span class="fl">0.05</span></a>
<a class="sourceLine" id="cb5-8" data-line-number="8">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;Corrected cv with Miller 95% CI&quot;
## 
## $statistics
##      est  lower  upper
##   58.058 34.173 81.942</code></pre>
</div>
<div id="vangel-confidence-interval" class="section level3">
<h3>Vangel Confidence Interval</h3>
<p>Vangel <span class="citation">(Vangel, 1996)</span> proposed the following CI for <span class="math inline">\(cv\)</span>; which is a modification on McKay’s CI: <span class="math display">\[
\biggl(cv\left[\biggl(\frac{u_1+1}{v}-1\biggr)(cv)^{2}+\frac{u_1}{v}\right]^{-1/2}
\le CV \le cv
\left[\biggl(\frac{u_2+1}{v}-1\biggr)(cv)^{2}+\frac{u_2}{v}\right]^{-1/2}\biggr)
\]</span> Let us calculate the 95% CI for <span class="math inline">\(x\)</span> according to Vangel’s method <span class="citation">(Vangel, 1996)</span>:</p>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb7-1" data-line-number="1"><span class="kw">cv_versatile</span>(</a>
<a class="sourceLine" id="cb7-2" data-line-number="2">    x, </a>
<a class="sourceLine" id="cb7-3" data-line-number="3">    <span class="dt">na.rm =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb7-4" data-line-number="4">    <span class="dt">digits =</span> <span class="dv">3</span>, </a>
<a class="sourceLine" id="cb7-5" data-line-number="5">    <span class="dt">method =</span> <span class="st">&quot;vangel&quot;</span>, </a>
<a class="sourceLine" id="cb7-6" data-line-number="6">    <span class="dt">correction =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb7-7" data-line-number="7">    <span class="dt">alpha =</span> <span class="fl">0.05</span></a>
<a class="sourceLine" id="cb7-8" data-line-number="8">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;Corrected cv with Vangel 95% CI&quot;
## 
## $statistics
##      est  lower   upper
##   58.058 41.443 106.237</code></pre>
</div>
<div id="mahmoudvand-hassani-confidence-interval" class="section level3">
<h3>Mahmoudvand-Hassani Confidence Interval</h3>
<p>Mahmoudvand and Hassani <span class="citation">(Mahmoudvand &amp; Hassani, 2009)</span> proposed the following CI for <span class="math inline">\(cv\)</span>; which is obtained using ranked set sampling <em>(RSS)</em>:<br />
<span class="math display">\[
\biggl(\frac{cv}{2-C_n+Z_{1-\alpha/2}\sqrt{1-C_n^2}}
\le CV \le
\frac{cv}{2-C_n-Z_{1-\alpha/2}\sqrt{1-C_n^2}}
\biggr)
\]</span> where <span class="math display">\[
C_n=\sqrt{\frac{2}{n-1}}\frac{\Gamma{(n/2)}}{\Gamma{((n-1)/2)}},
\Gamma(n)=(n-1)!
\]</span> Let us now calculate the 95% CI for <span class="math inline">\(x\)</span> according to Mahmoudvand-Hassani’s method <span class="citation">(Mahmoudvand &amp; Hassani, 2009)</span>:</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb9-1" data-line-number="1"><span class="kw">cv_versatile</span>(</a>
<a class="sourceLine" id="cb9-2" data-line-number="2">    x, </a>
<a class="sourceLine" id="cb9-3" data-line-number="3">    <span class="dt">na.rm =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb9-4" data-line-number="4">    <span class="dt">digits =</span> <span class="dv">3</span>, </a>
<a class="sourceLine" id="cb9-5" data-line-number="5">    <span class="dt">method =</span> <span class="st">&quot;mahmoudvand_hassani&quot;</span>, </a>
<a class="sourceLine" id="cb9-6" data-line-number="6">    <span class="dt">correction =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb9-7" data-line-number="7">    <span class="dt">alpha =</span> <span class="fl">0.05</span></a>
<a class="sourceLine" id="cb9-8" data-line-number="8">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;Corrected cv with Mahmoudvand-Hassani 95% CI&quot;
## 
## $statistics
##      est lower  upper
##   58.058 43.69 83.264</code></pre>
</div>
<div id="normal-approximation-confidence-interval" class="section level3">
<h3>Normal Approximation Confidence Interval</h3>
<p>Wararit Panichkitkosolkul <span class="citation">(Panichkitkosolkul, 2013)</span> proposed the following CI for <span class="math inline">\(cv\)</span>; which is a <em>normal approximation</em>: <span class="math display">\[
\biggl(\frac{cv}{C_{n+1}+Z_{1-\alpha/2}\sqrt{1-C_{n+1}^2}}
\le CV \le
\frac{cv}{C_{n+1}-Z_{1-\alpha/2}\sqrt{1-C_{n+1}^2}}
\biggr)
\]</span> where <span class="math inline">\(C_{n+1}=\sqrt{1-(1/2n)}\)</span><br />
Now we calculate the normal approximation 95% CI for <span class="math inline">\(x\)</span> according to Panichkitkosolkul <span class="citation">(Panichkitkosolkul, 2013)</span>:</p>
<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb11-1" data-line-number="1"><span class="kw">cv_versatile</span>(</a>
<a class="sourceLine" id="cb11-2" data-line-number="2">    x, </a>
<a class="sourceLine" id="cb11-3" data-line-number="3">    <span class="dt">na.rm =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb11-4" data-line-number="4">    <span class="dt">digits =</span> <span class="dv">3</span>, </a>
<a class="sourceLine" id="cb11-5" data-line-number="5">    <span class="dt">method =</span> <span class="st">&quot;normal_approximation&quot;</span>, </a>
<a class="sourceLine" id="cb11-6" data-line-number="6">    <span class="dt">correction =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb11-7" data-line-number="7">    <span class="dt">alpha =</span> <span class="fl">0.05</span></a>
<a class="sourceLine" id="cb11-8" data-line-number="8">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;Corrected cv with Normal Approximation 95% CI&quot;
## 
## $statistics
##      est  lower  upper
##   58.058 44.752 85.691</code></pre>
</div>
<div id="shortest-length-confidence-interval" class="section level3">
<h3>Shortest-Length Confidence Interval</h3>
<p>Panichkitkosolkul <span class="citation">(Panichkitkosolkul, 2013)</span> has also introduced the following CI for <span class="math inline">\(cv\)</span>:<br />
<span class="math display">\[
\biggl(\frac{cv\sqrt{v}}{\sqrt{b}}
\le CV \le
\frac{cv\sqrt{v}}{\sqrt{a}}
\biggr)
\]</span> with <span class="math inline">\(v = n-1\)</span> degrees of freedom. Then, shortest-length 95% CI for <span class="math inline">\(x\)</span> is:</p>
<div class="sourceCode" id="cb13"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb13-1" data-line-number="1"><span class="kw">cv_versatile</span>(</a>
<a class="sourceLine" id="cb13-2" data-line-number="2">    x, </a>
<a class="sourceLine" id="cb13-3" data-line-number="3">    <span class="dt">na.rm =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb13-4" data-line-number="4">    <span class="dt">digits =</span> <span class="dv">3</span>, </a>
<a class="sourceLine" id="cb13-5" data-line-number="5">    <span class="dt">method =</span> <span class="st">&quot;shortest_length&quot;</span>, </a>
<a class="sourceLine" id="cb13-6" data-line-number="6">    <span class="dt">correction =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb13-7" data-line-number="7">    <span class="dt">alpha =</span> <span class="fl">0.05</span></a>
<a class="sourceLine" id="cb13-8" data-line-number="8">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;Corrected cv with Shortest-Length 95% CI&quot;
## 
## $statistics
##      est  lower  upper
##   58.058 42.221 81.411</code></pre>
</div>
<div id="equal-tailed-confidence-interval" class="section level3">
<h3>Equal-Tailed Confidence Interval</h3>
<p>The <span class="math inline">\(100(1-\alpha)\%\)</span> equal-tailed CI for <span class="math inline">\(cv\)</span> can be obtained as: <span class="math display">\[
\biggl(\frac{cv\sqrt{v}}{\sqrt{\chi_{v,1-\alpha/2}^2}}
\le CV \le
\frac{cv\sqrt{v}}{\sqrt{\chi_{v,\alpha/2}^2}}
\biggr)
\]</span> where <span class="math inline">\(\chi_{v,\alpha/2}^2\)</span> and <span class="math inline">\(\chi_{v,1-\alpha/2}^2\)</span> are the <span class="math inline">\(100(\alpha/2)\)</span> and <span class="math inline">\(100(1-\alpha/2)\)</span> percentiles of the central <span class="math inline">\(\chi^2\)</span> distribution with <span class="math inline">\(v\)</span> degrees of freedom, respectively <span class="citation">(Panichkitkosolkul, 2013)</span>.<br />
Then, equal-tailed 95% CI for <span class="math inline">\(x\)</span> is:</p>
<div class="sourceCode" id="cb15"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb15-1" data-line-number="1"><span class="kw">cv_versatile</span>(</a>
<a class="sourceLine" id="cb15-2" data-line-number="2">    x, </a>
<a class="sourceLine" id="cb15-3" data-line-number="3">    <span class="dt">na.rm =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb15-4" data-line-number="4">    <span class="dt">digits =</span> <span class="dv">3</span>, </a>
<a class="sourceLine" id="cb15-5" data-line-number="5">    <span class="dt">method =</span> <span class="st">&quot;equal_tailed&quot;</span>, </a>
<a class="sourceLine" id="cb15-6" data-line-number="6">    <span class="dt">correction =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb15-7" data-line-number="7">    <span class="dt">alpha =</span> <span class="fl">0.05</span></a>
<a class="sourceLine" id="cb15-8" data-line-number="8">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;Corrected cv with Equal-Tailed 95% CI&quot;
## 
## $statistics
##      est  lower  upper
##   58.058 44.152 84.797</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">\(cv\)</span>:</p>
<div class="sourceCode" id="cb17"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb17-1" data-line-number="1"><span class="kw">cv_versatile</span>(</a>
<a class="sourceLine" id="cb17-2" data-line-number="2">    x, </a>
<a class="sourceLine" id="cb17-3" data-line-number="3">    <span class="dt">na.rm =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb17-4" data-line-number="4">    <span class="dt">digits =</span> <span class="dv">3</span>, </a>
<a class="sourceLine" id="cb17-5" data-line-number="5">    <span class="dt">method =</span> <span class="st">&quot;basic&quot;</span>, </a>
<a class="sourceLine" id="cb17-6" data-line-number="6">    <span class="dt">correction =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb17-7" data-line-number="7">    <span class="dt">alpha =</span> <span class="fl">0.05</span></a>
<a class="sourceLine" id="cb17-8" data-line-number="8">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;Corrected cv with Basic Bootstrap 95% CI&quot;
## 
## $statistics
##      est  lower upper
##   58.058 37.768 77.95</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="cb19"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb19-1" data-line-number="1"><span class="kw">cv_versatile</span>(</a>
<a class="sourceLine" id="cb19-2" data-line-number="2">    x, </a>
<a class="sourceLine" id="cb19-3" data-line-number="3">    <span class="dt">na.rm =</span> <span class="ot">TRUE</span>, </a>
<a class="sourceLine" id="cb19-4" data-line-number="4">    <span class="dt">digits =</span> <span class="dv">3</span>, </a>
<a class="sourceLine" id="cb19-5" data-line-number="5">    <span class="dt">method =</span> <span class="st">&quot;all&quot;</span>, </a>
<a class="sourceLine" id="cb19-6" data-line-number="6">    <span class="dt">correction =</span> <span class="ot">FALSE</span>, </a>
<a class="sourceLine" id="cb19-7" data-line-number="7">    <span class="dt">alpha =</span> <span class="fl">0.05</span></a>
<a class="sourceLine" id="cb19-8" data-line-number="8">)</a></code></pre></div>
<pre><code>## $method
## [1] &quot;All methods&quot;
## 
## $statistics
##                         est  lower   upper
## kelley               57.774 41.467  98.510
## mckay                57.774 41.441 108.482
## miller               57.774 34.053  81.494
## vangel               57.774 41.264 105.424
## mahmoudvand_hassani  57.774 43.476  82.857
## equal_tailed         57.774 43.936  84.382
## shortest_length      57.774 42.014  81.012
## normal_approximation 57.774 44.533  85.272
## norm                 57.774 38.497  78.012
## basic                57.774 36.354  78.490
##                                                        description
## kelley                                       cv with Kelley 95% CI
## mckay                                         cv with McKay 95% CI
## miller                                       cv with Miller 95% CI
## vangel                                       cv with Vangel 95% CI
## mahmoudvand_hassani             cv with Mahmoudvand-Hassani 95% CI
## equal_tailed                           cv with Equal-Tailed 95% CI
## shortest_length                     cv with Shortest-Length 95% CI
## normal_approximation           cv with Normal Approximation 95% CI
## norm                 cv with Normal Approximation Bootstrap 95% CI
## basic                               cv with Basic Bootstrap 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-Canty2017">
<p>Canty, A., &amp; Ripley, B. (2017). boot: Bootstrap R (S-Plus) Functions. R package version 1.3-20.</p>
</div>
<div id="ref-EdwardMiller1991">
<p>Edward Miller, G. (1991). Asymptotic test statistics for coefficients of variation. <em>Communications in Statistics-Theory and Methods</em>, <em>20</em>(10), 3351–3363.</p>
</div>
<div id="ref-Kelley2007">
<p>Kelley, K. (2007). Sample size planning for the coefficient of variation from the accuracy in parameter estimation approach. <em>Behavior Research Methods</em>, <em>39</em>(4), 755–766. <a href="https://doi.org/10.3758/BF03192966">https://doi.org/10.3758/BF03192966</a></p>
</div>
<div id="ref-Kelley2018">
<p>Kelley, K. (2018). MBESS: The MBESS R Package. R package version 4.4. 3. Retrieved from <a href="https://cran.r-project.org/package=MBESS">https://cran.r-project.org/package=MBESS</a></p>
</div>
<div id="ref-Mahmoudvand2009">
<p>Mahmoudvand, R., &amp; Hassani, H. (2009). Two new confidence intervals for the coefficient of variation in a normal distribution. <em>Journal of Applied Statistics</em>, <em>36</em>(4), 429–442.</p>
</div>
<div id="ref-McKay1932">
<p>McKay, A. T. (1932). Distribution of the Coefficient of Variation and the Extended&quot; t&quot; Distribution. <em>Journal of the Royal Statistical Society</em>, <em>95</em>(4), 695–698.</p>
</div>
<div id="ref-Panichkitkosolkul2013">
<p>Panichkitkosolkul, W. (2013). Confidence Intervals for the Coefficient of Variation in a Normal Distribution with a Known Population Mean. <em>Journal of Probability and Statistics</em>, <em>2013</em>, 1–11. <a href="https://doi.org/10.1155/2013/324940">https://doi.org/10.1155/2013/324940</a></p>
</div>
<div id="ref-Vangel1996">
<p>Vangel, M. G. (1996). Confidence intervals for a normal coefficient of variation. <em>The American Statistician</em>, <em>50</em>(1), 21–26.</p>
</div>
</div>
</div>



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