{"id":11699,"date":"2019-12-26T11:01:04","date_gmt":"2019-12-26T09:01:04","guid":{"rendered":"https:\/\/www.datanovia.com\/en\/?post_type=dt_lessons&#038;p=11699"},"modified":"2019-12-26T11:01:04","modified_gmt":"2019-12-26T09:01:04","slug":"t-test-effect-size-using-cohens-d-measure","status":"publish","type":"dt_lessons","link":"https:\/\/www.datanovia.com\/en\/lessons\/t-test-effect-size-using-cohens-d-measure\/","title":{"rendered":"T-test Effect Size using Cohen&#8217;s d Measure"},"content":{"rendered":"<div id=\"rdoc\">\n<p>This article describe the <strong>t-test effect size<\/strong>. The most commonly used measure of effect size for a t-test is the <strong>Cohen\u2019s d<\/strong> <span class=\"citation\">(Cohen 1998)<\/span>.<\/p>\n<p>The <code>d<\/code> statistic redefines the difference in means as the number of standard deviations that separates those means. The formula looks like this <span class=\"citation\">(Navarro 2015)<\/span>:<\/p>\n<p><span class=\"math display\">\\[<br \/>\nd = \\frac{\\mbox{(mean 1)} - \\mbox{(mean 2)}}{\\mbox{std dev}}<br \/>\n\\]<\/span><\/p>\n<p>In this article, you will learn:<\/p>\n<ul>\n<li><strong>Cohen\u2019s d formula<\/strong> to calculate the effect size for one-sample t-test, for independent t-test (with pooled standard deviation or not) and for paired samples t-test (also known as repeated measures t-test).<\/li>\n<li><strong>Effect size interpretation<\/strong> describing the critical value corresponding to <strong>small, medium and large<\/strong> effect sizes.<\/li>\n<li><strong>Calculation of the Cohen\u2019s d in R<\/strong><\/li>\n<\/ul>\n<p>Contents:<\/p>\n<div id=\"TOC\">\n<ul>\n<li><a href=\"#prerequisites\">Prerequisites<\/a><\/li>\n<li><a href=\"#effect-size-interpretation\">Effect size interpretation<\/a><\/li>\n<li><a href=\"#cohens-d-for-one-sample-t-test\">Cohen\u2019s d for one-sample t-test<\/a><\/li>\n<li><a href=\"#cohens-d-for-independent-t-test\">Cohen\u2019s d for independent t-test<\/a>\n<ul>\n<li><a href=\"#cohens-d-for-student-t-test\">Cohen\u2019s d for Student t-test<\/a><\/li>\n<li><a href=\"#cohens-d-for-welch-test\">Cohen\u2019s d for Welch test<\/a><\/li>\n<\/ul>\n<\/li>\n<li><a href=\"#cohens-d-for-paired-samples-t-test\">Cohen\u2019s d for paired samples t-test<\/a><\/li>\n<li><a href=\"#summary\">Summary<\/a><\/li>\n<li><a href=\"#related-article\">Related article<\/a><\/li>\n<li><a href=\"#references\">References<\/a><\/li>\n<\/ul>\n<\/div>\n<div class='dt-sc-hr-invisible-medium  '><\/div>\n<div class='dt-sc-ico-content type1'><div class='custom-icon' ><a href='https:\/\/www.datanovia.com\/en\/product\/practical-statistics-in-r-for-comparing-groups-numerical-variables\/' target='_blank'><span class='fa fa-book'><\/span><\/a><\/div><h4><a href='https:\/\/www.datanovia.com\/en\/product\/practical-statistics-in-r-for-comparing-groups-numerical-variables\/' target='_blank'> Related Book <\/a><\/h4>Practical Statistics in R II - Comparing Groups: Numerical Variables<\/div>\n<div class='dt-sc-hr-invisible-medium  '><\/div>\n<div id=\"prerequisites\" class=\"section level2\">\n<h2>Prerequisites<\/h2>\n<p>Load the required R package for computing the Cohen\u2019s d:<\/p>\n<pre class=\"r\"><code>library(rstatix)<\/code><\/pre>\n<p>Demo dataset:<\/p>\n<pre class=\"r\"><code>head(ToothGrowth, 3)<\/code><\/pre>\n<pre><code>##    len supp dose\r\n## 1  4.2   VC  0.5\r\n## 2 11.5   VC  0.5\r\n## 3  7.3   VC  0.5<\/code><\/pre>\n<\/div>\n<div id=\"effect-size-interpretation\" class=\"section level2\">\n<h2>Effect size interpretation<\/h2>\n<p>T-test conventional effect sizes, poposed by Cohen, are: 0.2 (small efect), 0.5 (moderate effect) and 0.8 (large effect) <span class=\"citation\">(Cohen 1998, <span class=\"citation\">Navarro (2015)<\/span>)<\/span>. This means that if two groups\u2019 means don\u2019t differ by 0.2 standard deviations or more, the difference is trivial, even if it is statistically significant.<\/p>\n<table>\n<thead>\n<tr class=\"header\">\n<th>d-value<\/th>\n<th>rough interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"odd\">\n<td>0.2<\/td>\n<td>Small effect<\/td>\n<\/tr>\n<tr class=\"even\">\n<td>0.5<\/td>\n<td>Moderate effect<\/td>\n<\/tr>\n<tr class=\"odd\">\n<td>0.8<\/td>\n<td>Large effect<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"cohens-d-for-one-sample-t-test\" class=\"section level2\">\n<h2>Cohen\u2019s d for one-sample t-test<\/h2>\n<p>To calculate an effect size, called <code>Cohen's d<\/code>, for the one-sample t-test you need to divide the mean difference by the standard deviation of the difference, as shown below. Note that, here: <code>sd(x-mu) = sd(x)<\/code>.<\/p>\n<p><strong>Cohen\u2019s d formula<\/strong>:<\/p>\n<p><span class=\"math display\">\\[<br \/>\nd = \\frac{m-\\mu}{s}<br \/>\n\\]<\/span><\/p>\n<ul>\n<li><span class=\"math inline\">\\(m\\)<\/span> is the sample mean<\/li>\n<li><span class=\"math inline\">\\(s\\)<\/span> is the sample standard deviation with <span class=\"math inline\">\\(n-1\\)<\/span> degrees of freedom<\/li>\n<li><span class=\"math inline\">\\(\\mu\\)<\/span> is the theoretical mean against which the mean of our sample is compared (default value is mu = 0).<\/li>\n<\/ul>\n<p><strong>Calculation<\/strong>:<\/p>\n<pre class=\"r\"><code>ToothGrowth %&gt;% cohens_d(len ~ 1, mu = 0)<\/code><\/pre>\n<pre><code>## # A tibble: 1 x 6\r\n##   .y.   group1 group2     effsize     n magnitude\r\n## * &lt;chr&gt; &lt;chr&gt;  &lt;chr&gt;        &lt;dbl&gt; &lt;int&gt; &lt;ord&gt;    \r\n## 1 len   1      null model    2.46    60 large<\/code><\/pre>\n<\/div>\n<div id=\"cohens-d-for-independent-t-test\" class=\"section level2\">\n<h2>Cohen\u2019s d for independent t-test<\/h2>\n<p>The independent samples t-test comes in two different forms:<\/p>\n<ul>\n<li>the standard <em>Student\u2019s t-test<\/em>, which assumes that the variance of the two groups are equal.<\/li>\n<li>the <em>Welch\u2019s t-test<\/em>, which is less restrictive compared to the original Student\u2019s test. This is the test where you do not assume that the variance is the same in the two groups, which results in the fractional degrees of freedom.<\/li>\n<\/ul>\n<div id=\"cohens-d-for-student-t-test\" class=\"section level3\">\n<h3>Cohen\u2019s d for Student t-test<\/h3>\n<p>There are multiple version of Cohen\u2019s d for Student t-test. The most commonly used version of the Student t-test effect size, comparing two groups (<span class=\"math inline\">\\(A\\)<\/span> and <span class=\"math inline\">\\(B\\)<\/span>), is calculated by dividing the mean difference between the groups by the pooled standard deviation.<\/p>\n<p><strong>Cohen\u2019s d formula<\/strong>:<\/p>\n<p><span class=\"math display\">\\[<br \/>\nd = \\frac{m_A - m_B}{SD_{pooled}}<br \/>\n\\]<\/span><\/p>\n<p>where,<\/p>\n<ul>\n<li><span class=\"math inline\">\\(m_A\\)<\/span> and <span class=\"math inline\">\\(m_B\\)<\/span> represent the mean value of the group A and B, respectively.<\/li>\n<li><span class=\"math inline\">\\(n_A\\)<\/span> and <span class=\"math inline\">\\(n_B\\)<\/span> represent the sizes of the group A and B, respectively.<\/li>\n<li><span class=\"math inline\">\\(SD_{pooled}\\)<\/span> is an estimator of the pooled standard deviation of the two groups. It can be calculated as follow :<br \/>\n<span class=\"math display\">\\[<br \/>\nSD_{pooled} = \\sqrt{\\frac{\\sum{(x-m_A)^2}+\\sum{(x-m_B)^2}}{n_A+n_B-2}}<br \/>\n\\]<\/span><\/li>\n<\/ul>\n<p><strong>Calculation<\/strong>. If the option <code>var.equal = TRUE<\/code>, then the pooled SD is used when compting the Cohen\u2019s d.<\/p>\n<pre class=\"r\"><code>ToothGrowth %&gt;% cohens_d(len ~ supp, var.equal = TRUE)<\/code><\/pre>\n<pre><code>## # A tibble: 1 x 7\r\n##   .y.   group1 group2 effsize    n1    n2 magnitude\r\n## * &lt;chr&gt; &lt;chr&gt;  &lt;chr&gt;    &lt;dbl&gt; &lt;int&gt; &lt;int&gt; &lt;ord&gt;    \r\n## 1 len   OJ     VC       0.495    30    30 small<\/code><\/pre>\n<p>Note that, for small sample size (&lt; 50), the Cohen\u2019s d tends to over-inflate results. There exists a <strong>Hedge\u2019s Corrected version of the Cohen\u2019s d<\/strong> <span class=\"citation\">(Hedges and Olkin 1985)<\/span>, which reduces effect sizes for small samples by a few percentage points. The correction is introduced by multiplying the usual value of d by <code>(N-3)\/(N-2.25)<\/code> (for unpaired t-test) and by <code>(n1-2)\/(n1-1.25)<\/code> for paired t-test; where N is the total size of the two groups being compared <code>(N = n1 + n2)<\/code>.<\/p>\n<pre class=\"r\"><code>ToothGrowth %&gt;% cohens_d(\r\n  len ~ supp, var.equal = TRUE, \r\n  hedges.correction = TRUE\r\n  )<\/code><\/pre>\n<pre><code>## # A tibble: 1 x 7\r\n##   .y.   group1 group2 effsize    n1    n2 magnitude\r\n## * &lt;chr&gt; &lt;chr&gt;  &lt;chr&gt;    &lt;dbl&gt; &lt;int&gt; &lt;int&gt; &lt;ord&gt;    \r\n## 1 len   OJ     VC       0.488    30    30 small<\/code><\/pre>\n<\/div>\n<div id=\"cohens-d-for-welch-test\" class=\"section level3\">\n<h3>Cohen\u2019s d for Welch test<\/h3>\n<p>The Welch test is a variant of t-test used when the equality of variance can\u2019t be assumed. The effect size can be computed by dividing the mean difference between the groups by the \u201caveraged\u201d standard deviation.<\/p>\n<p><strong>Cohen\u2019s d formula<\/strong>:<\/p>\n<p><span class=\"math display\">\\[<br \/>\nd = \\frac{m_A - m_B}{\\sqrt{(Var_1 + Var_2)\/2}}<br \/>\n\\]<\/span><\/p>\n<p>where,<\/p>\n<ul>\n<li><span class=\"math inline\">\\(m_A\\)<\/span> and <span class=\"math inline\">\\(m_B\\)<\/span> represent the mean value of the group A and B, respectively.<\/li>\n<li><span class=\"math inline\">\\(Var_1\\)<\/span> and <span class=\"math inline\">\\(Var_2\\)<\/span> are the variance of the two groups.<\/li>\n<\/ul>\n<p><strong>Calculation<\/strong>:<\/p>\n<pre class=\"r\"><code>ToothGrowth %&gt;% cohens_d(len ~ supp, var.equal = FALSE)<\/code><\/pre>\n<pre><code>## # A tibble: 1 x 7\r\n##   .y.   group1 group2 effsize    n1    n2 magnitude\r\n## * &lt;chr&gt; &lt;chr&gt;  &lt;chr&gt;    &lt;dbl&gt; &lt;int&gt; &lt;int&gt; &lt;ord&gt;    \r\n## 1 len   OJ     VC       0.495    30    30 small<\/code><\/pre>\n<\/div>\n<\/div>\n<div id=\"cohens-d-for-paired-samples-t-test\" class=\"section level2\">\n<h2>Cohen\u2019s d for paired samples t-test<\/h2>\n<p>The effect size for a paired-samples t-test can be calculated by dividing the mean difference by the standard deviation of the difference, as shown below.<\/p>\n<p><strong>Cohen\u2019s d formula<\/strong>:<\/p>\n<p><span class=\"math display\">\\[<br \/>\nd = \\frac{mean_D}{SD_D}<br \/>\n\\]<\/span><\/p>\n<p>Where <code>D<\/code> is the differences of the paired samples values.<\/p>\n<p><strong>Calculation<\/strong>:<\/p>\n<pre class=\"r\"><code>ToothGrowth %&gt;% cohens_d(len ~ supp, paired = TRUE)<\/code><\/pre>\n<pre><code>## # A tibble: 1 x 7\r\n##   .y.   group1 group2 effsize    n1    n2 magnitude\r\n## * &lt;chr&gt; &lt;chr&gt;  &lt;chr&gt;    &lt;dbl&gt; &lt;int&gt; &lt;int&gt; &lt;ord&gt;    \r\n## 1 len   OJ     VC       0.603    30    30 moderate<\/code><\/pre>\n<\/div>\n<div id=\"summary\" class=\"section level2\">\n<h2>Summary<\/h2>\n<p>This article shows how to compute and interpret the t-test effect using the Cohen\u2019s d statistic. We describe the formula of the Cohen\u2019s d for one-sample, two-samples and paired samples t-test. Examples of R codes are provided for the calculations.<\/p>\n<\/div>\n<div id=\"related-article\" class=\"section level2\">\n<h2>Related article<\/h2>\n<p><a href=\"\/?p=10861\">T-test in R<\/a><\/p>\n<\/div>\n<div id=\"references\" class=\"section level2 unnumbered\">\n<h2>References<\/h2>\n<div id=\"refs\" class=\"references\">\n<div id=\"ref-cohen1998\">\n<p>Cohen, J. 1998. <em>Statistical Power Analysis for the Behavioral Sciences<\/em>. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates.<\/p>\n<\/div>\n<div id=\"ref-hedges1985\">\n<p>Hedges, Larry, and Ingram Olkin. 1985. \u201cStatistical Methods in Meta-Analysis.\u201d In <em>Stat Med<\/em>. Vol. 20. doi:<a href=\"https:\/\/doi.org\/10.2307\/1164953\">10.2307\/1164953<\/a>.<\/p>\n<\/div>\n<div id=\"ref-Navarro_learningstatistics\">\n<p>Navarro, Daniel. 2015. <em>Learning Statistics with R: A Tutorial for Psychology Students and Other Beginners (Version 0.5)<\/em>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><!--end rdoc--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Describes the t-test effect size using the Cohen&#8217;s d. 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