{"id":11654,"date":"2019-12-25T12:51:28","date_gmt":"2019-12-25T10:51:28","guid":{"rendered":"https:\/\/www.datanovia.com\/en\/?post_type=dt_lessons&#038;p=11654"},"modified":"2019-12-25T12:51:28","modified_gmt":"2019-12-25T10:51:28","slug":"paired-t-test","status":"publish","type":"dt_lessons","link":"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/","title":{"rendered":"Paired T-Test"},"content":{"rendered":"<div id=\"rdoc\">\n<p>The <strong>paired t-test<\/strong> is used to compare the means of two related groups of samples. Put into another words, it\u2019s used in a situation where you have two values (i.e., pair of values) for the same samples.<\/p>\n<p>It is also referred as:<\/p>\n<ul>\n<li><em>dependent t-test<\/em>,<\/li>\n<li><em>dependent samples t-test<\/em>,<\/li>\n<li><em>repeated measures t-test<\/em>,<\/li>\n<li><em>related samples t-test<\/em>,<\/li>\n<li><em>paired two sample t-test<\/em>,<\/li>\n<li><em>paired sample t-test<\/em> and<\/li>\n<li><em>t-test for dependent means<\/em>.<\/li>\n<\/ul>\n<p>For example, you might want to compare the average weight of 20 mice before and after treatment. The data contain 20 sets of values before treatment and 20 sets of values after treatment from measuring twice the weight of the same mice. In such situations, paired t-test can be used to compare the mean weights before and after treatment.<\/p>\n<p>The procedure of the paired t-test analysis is as follow:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Calculate the difference (<span class=\"math inline\">\\(d\\)<\/span>) between each pair of value<\/li>\n<li>Compute the mean (<span class=\"math inline\">\\(m\\)<\/span>) and the standard deviation (<span class=\"math inline\">\\(s\\)<\/span>) of <span class=\"math inline\">\\(d\\)<\/span><\/li>\n<li>Compare the average difference to 0. If there is any significant difference between the two pairs of samples, then the mean of d (<span class=\"math inline\">\\(m\\)<\/span>) is expected to be far from 0.<\/li>\n<\/ol>\n<div class=\"error\">\n<p>Paired t-test can be used only when the difference <span class=\"math inline\"><span class=\"math inline\">\\(d\\)<\/span><\/span> is normally distributed. This can be checked using the Shapiro-Wilk test.<\/p>\n<\/div>\n<p>In this chapter, you will learn the <em>paired t-test formula<\/em>, as well as, how to:<\/p>\n<ul>\n<li><em>Compute the paired t-test in R<\/em>. The pipe-friendly function <code>t_test()<\/code> [rstatix package] will be used.<\/li>\n<li><em>Check the paired t-test assumptions<\/em><\/li>\n<li><em>Calculate and report the paired t-test effect size<\/em> using the <em>Cohen\u2019s d<\/em>. The <code>d<\/code> statistic redefines the difference in means as the number of standard deviations that separates those means. T-test conventional effect sizes, proposed by Cohen, are: 0.2 (small effect), 0.5 (moderate effect) and 0.8 (large effect) <span class=\"citation\">(Cohen 1998)<\/span>.<\/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=\"#research-questions\">Research questions<\/a><\/li>\n<li><a href=\"#statistical-hypotheses\">statistical hypotheses<\/a><\/li>\n<li><a href=\"#formula\">Formula<\/a><\/li>\n<li><a href=\"#demo-data\">Demo data<\/a><\/li>\n<li><a href=\"#summary-statistics\">Summary statistics<\/a><\/li>\n<li><a href=\"#visualization\">Visualization<\/a><\/li>\n<li><a href=\"#assumptions-and-preleminary-tests\">Assumptions and preleminary tests<\/a>\n<ul>\n<li><a href=\"#identify-outliers\">Identify outliers<\/a><\/li>\n<li><a href=\"#check-normality-assumption\">Check normality assumption<\/a><\/li>\n<\/ul>\n<\/li>\n<li><a href=\"#computation\">Computation<\/a><\/li>\n<li><a href=\"#effect-size\">Effect size<\/a><\/li>\n<li><a href=\"#report\">Report<\/a><\/li>\n<li><a href=\"#summary\">Summary<\/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>Make sure you have installed the following R packages:<\/p>\n<ul>\n<li><code>tidyverse<\/code> for data manipulation and visualization<\/li>\n<li><code>ggpubr<\/code> for creating easily publication ready plots<\/li>\n<li><code>rstatix<\/code> provides pipe-friendly R functions for easy statistical analyses.<\/li>\n<li><code>datarium<\/code>: contains required data sets for this chapter.<\/li>\n<\/ul>\n<p>Start by loading the following required packages:<\/p>\n<pre class=\"r\"><code>library(tidyverse)\r\nlibrary(ggpubr)\r\nlibrary(rstatix)<\/code><\/pre>\n<\/div>\n<div id=\"research-questions\" class=\"section level2\">\n<h2>Research questions<\/h2>\n<p>Typical research questions are:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>whether the mean difference (<span class=\"math inline\">\\(m\\)<\/span>) <em>is equal<\/em> to 0?<\/li>\n<li>whether the mean difference (<span class=\"math inline\">\\(m\\)<\/span>) <em>is less than<\/em> 0?<\/li>\n<li>whether the mean difference (<span class=\"math inline\">\\(m\\)<\/span>) <em>is greather than<\/em> 0?<\/li>\n<\/ol>\n<\/div>\n<div id=\"statistical-hypotheses\" class=\"section level2\">\n<h2>statistical hypotheses<\/h2>\n<p>In statistics, we can define the corresponding <em>null hypothesis<\/em> (<span class=\"math inline\">\\(H_0\\)<\/span>) as follow:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li><span class=\"math inline\">\\(H_0: m = 0\\)<\/span><\/li>\n<li><span class=\"math inline\">\\(H_0: m \\leq 0\\)<\/span><\/li>\n<li><span class=\"math inline\">\\(H_0: m \\geq 0\\)<\/span><\/li>\n<\/ol>\n<p>The corresponding <em>alternative hypotheses<\/em> (<span class=\"math inline\">\\(H_a\\)<\/span>) are as follow:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li><span class=\"math inline\">\\(H_a: m \\ne 0\\)<\/span> (different)<\/li>\n<li><span class=\"math inline\">\\(H_a: m &gt; 0\\)<\/span> (greater)<\/li>\n<li><span class=\"math inline\">\\(H_a: m &lt; 0\\)<\/span> (less)<\/li>\n<\/ol>\n<p>Note that:<\/p>\n<ul>\n<li>Hypotheses 1) are called <em>two-tailed tests<\/em><\/li>\n<li>Hypotheses 2) and 3) are called <em>one-tailed tests<\/em><\/li>\n<\/ul>\n<\/div>\n<div id=\"formula\" class=\"section level2\">\n<h2>Formula<\/h2>\n<p>The paired t-test statistics value can be calculated using the following formula:<\/p>\n<p><span class=\"math display\">\\[<br \/>\nt = \\frac{m}{s\/\\sqrt{n}}<br \/>\n\\]<\/span><\/p>\n<p>where,<\/p>\n<ul>\n<li><code>m<\/code> is the mean differences<\/li>\n<li><code>n<\/code> is the sample size (i.e., size of d).<\/li>\n<li><code>s<\/code> is the standard deviation of d<\/li>\n<\/ul>\n<p>We can compute the p-value corresponding to the absolute value of the t-test statistics (|t|) for the degrees of freedom (df): <span class=\"math inline\">\\(df = n - 1\\)<\/span>.<\/p>\n<div class=\"success\">\n<p>If the p-value is inferior or equal to 0.05, we can conclude that the difference between the two paired samples are significantly different.<\/p>\n<\/div>\n<\/div>\n<div id=\"demo-data\" class=\"section level2\">\n<h2>Demo data<\/h2>\n<p>Here, we\u2019ll use a demo dataset <code>mice2<\/code> [datarium package], which contains the weight of 10 mice before and after the treatment.<\/p>\n<pre class=\"r\"><code># Wide format\r\ndata(\"mice2\", package = \"datarium\")\r\nhead(mice2, 3)<\/code><\/pre>\n<pre><code>##   id before after\r\n## 1  1    187   430\r\n## 2  2    194   404\r\n## 3  3    232   406<\/code><\/pre>\n<pre class=\"r\"><code># Transform into long data: \r\n# gather the before and after values in the same column\r\nmice2.long &lt;- mice2 %&gt;%\r\n  gather(key = \"group\", value = \"weight\", before, after)\r\nhead(mice2.long, 3)<\/code><\/pre>\n<pre><code>##   id  group weight\r\n## 1  1 before    187\r\n## 2  2 before    194\r\n## 3  3 before    232<\/code><\/pre>\n<\/div>\n<div id=\"summary-statistics\" class=\"section level2\">\n<h2>Summary statistics<\/h2>\n<p>Compute some summary statistics (mean and sd) by groups:<\/p>\n<pre class=\"r\"><code>mice2.long %&gt;%\r\n  group_by(group) %&gt;%\r\n  get_summary_stats(weight, type = \"mean_sd\")<\/code><\/pre>\n<pre><code>## # A tibble: 2 x 5\r\n##   group  variable     n  mean    sd\r\n##   &lt;chr&gt;  &lt;chr&gt;    &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt;\r\n## 1 after  weight      10  400.  30.1\r\n## 2 before weight      10  201.  20.0<\/code><\/pre>\n<\/div>\n<div id=\"visualization\" class=\"section level2\">\n<h2>Visualization<\/h2>\n<pre class=\"r\"><code>bxp &lt;- ggpaired(mice2.long, x = \"group\", y = \"weight\", \r\n         order = c(\"before\", \"after\"),\r\n         ylab = \"Weight\", xlab = \"Groups\")\r\nbxp<\/code><\/pre>\n<p><img decoding=\"async\" src=\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/dn-tutorials\/r-statistics-2-comparing-groups-means\/figures\/072-paired-t-test-box-plot-1.png\" width=\"364.8\" \/><\/p>\n<\/div>\n<div id=\"assumptions-and-preleminary-tests\" class=\"section level2\">\n<h2>Assumptions and preleminary tests<\/h2>\n<p>The paired samples t-test assume the following characteristics about the data:<\/p>\n<ul>\n<li><strong>the two groups are paired<\/strong>. In our example, this is the case since the data have been collected from measuring twice the weight of the same mice.<\/li>\n<li><strong>No significant outliers<\/strong> in the difference between the two related groups<\/li>\n<li><strong>Normality<\/strong>. the difference of pairs follow a normal distribution.<\/li>\n<\/ul>\n<p>In this section, we\u2019ll perform some preliminary tests to check whether these assumptions are met.<\/p>\n<p>First, start by computing the difference between groups:<\/p>\n<pre class=\"r\"><code>mice2 &lt;- mice2 %&gt;% mutate(differences = before - after)\r\nhead(mice2, 3)<\/code><\/pre>\n<pre><code>##   id before after differences\r\n## 1  1    187   430        -242\r\n## 2  2    194   404        -210\r\n## 3  3    232   406        -174<\/code><\/pre>\n<div id=\"identify-outliers\" class=\"section level3\">\n<h3>Identify outliers<\/h3>\n<p>Outliers can be easily identified using boxplot methods, implemented in the R function <code>identify_outliers()<\/code> [rstatix package].<\/p>\n<pre class=\"r\"><code>mice2 %&gt;% identify_outliers(differences)<\/code><\/pre>\n<pre><code>## [1] id          before      after       differences is.outlier  is.extreme \r\n## &lt;0 rows&gt; (or 0-length row.names)<\/code><\/pre>\n<div class=\"success\">\n<p>There were no extreme outliers.<\/p>\n<\/div>\n<div class=\"warning\">\n<p>Note that, in the situation where you have extreme outliers, this can be due to: 1) data entry errors, measurement errors or unusual values.<\/p>\n<p>You can include the outlier in the analysis anyway if you do not believe the result will be substantially affected. This can be evaluated by comparing the result of the t-test with and without the outlier.<\/p>\n<p>It\u2019s also possible to keep the outliers in the data and perform Wilcoxon test or robust t-test using the WRS2 package.<\/p>\n<\/div>\n<\/div>\n<div id=\"check-normality-assumption\" class=\"section level3\">\n<h3>Check normality assumption<\/h3>\n<p>The normality assumption can be checked by computing the Shapiro-Wilk test for each group. If the data is normally distributed, the p-value should be greater than 0.05.<\/p>\n<pre class=\"r\"><code>mice2 %&gt;% shapiro_test(differences) <\/code><\/pre>\n<pre><code>## # A tibble: 1 x 3\r\n##   variable    statistic     p\r\n##   &lt;chr&gt;           &lt;dbl&gt; &lt;dbl&gt;\r\n## 1 differences     0.968 0.867<\/code><\/pre>\n<div class=\"success\">\n<p>From the output, the two p-values are greater than the significance level 0.05 indicating that the distribution of the data are not significantly different from the normal distribution. In other words, we can assume the normality.<\/p>\n<\/div>\n<p>You can also create QQ plots for each group. QQ plot draws the correlation between a given data and the normal distribution.<\/p>\n<pre class=\"r\"><code>ggqqplot(mice2, \"differences\")<\/code><\/pre>\n<p><img decoding=\"async\" src=\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/dn-tutorials\/r-statistics-2-comparing-groups-means\/figures\/072-paired-t-test-qqplot-1.png\" width=\"288\" \/><\/p>\n<div class=\"success\">\n<p>All the points fall approximately along the (45-degree) reference line, for each group. So we can assume normality of the data.<\/p>\n<\/div>\n<div class=\"warning\">\n<p>Note that, if your sample size is greater than 50, the normal QQ plot is preferred because at larger sample sizes the Shapiro-Wilk test becomes very sensitive even to a minor deviation from normality.<\/p>\n<p>In the situation where the data are not normally distributed, it\u2019s recommended to use the non parametric Wilcoxon test.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"computation\" class=\"section level2\">\n<h2>Computation<\/h2>\n<p>We want to know, if there is any significant difference in the mean weights after treatment?<\/p>\n<p>We\u2019ll use the pipe-friendly <code>t_test()<\/code> function [rstatix package], a wrapper around the R base function <code>t.test()<\/code>.<\/p>\n<pre class=\"r\"><code>stat.test &lt;- mice2.long  %&gt;% \r\n  t_test(weight ~ group, paired = TRUE) %&gt;%\r\n  add_significance()\r\nstat.test<\/code><\/pre>\n<pre><code>## # A tibble: 1 x 9\r\n##   .y.    group1 group2    n1    n2 statistic    df             p p.signif\r\n##   &lt;chr&gt;  &lt;chr&gt;  &lt;chr&gt;  &lt;int&gt; &lt;int&gt;     &lt;dbl&gt; &lt;dbl&gt;         &lt;dbl&gt; &lt;chr&gt;   \r\n## 1 weight after  before    10    10      25.5     9 0.00000000104 ****<\/code><\/pre>\n<p>The results above show the following components:<\/p>\n<ul>\n<li><code>.y.<\/code>: the y variable used in the test.<\/li>\n<li><code>group1,group2<\/code>: the compared groups in the pairwise tests.<\/li>\n<li><code>statistic<\/code>: Test statistic used to compute the p-value.<\/li>\n<li><code>df<\/code>: degrees of freedom.<\/li>\n<li><code>p<\/code>: p-value.<\/li>\n<\/ul>\n<div class=\"warning\">\n<p>Note that, you can obtain a detailed result by specifying the option <code>detailed = TRUE<\/code>.<\/p>\n<\/div>\n<p>To compute one tailed paired t-test, you can specify the option <code>alternative<\/code> as follow.<\/p>\n<ul>\n<li>if you want to test whether the average weight before treatment is less than the average weight after treatment, type this:<\/li>\n<\/ul>\n<pre class=\"r\"><code>mice2.long  %&gt;%\r\n  t_test(weight ~ group, paired = TRUE, alternative = \"less\")<\/code><\/pre>\n<ul>\n<li>Or, if you want to test whether the average weight before treatment is greater than the average weight after treatment, type this<\/li>\n<\/ul>\n<pre class=\"r\"><code>mice2.long  %&gt;%\r\n  t_test(weight ~ group, paired = TRUE, alternative = \"greater\")<\/code><\/pre>\n<\/div>\n<div id=\"effect-size\" class=\"section level2\">\n<h2>Effect size<\/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>mice2.long  %&gt;% cohens_d(weight ~ group, 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 weight after  before    8.08    10    10 large<\/code><\/pre>\n<div class=\"success\">\n<p>There is a large effect size, Cohen\u2019s d = 8.07.<\/p>\n<\/div>\n<\/div>\n<div id=\"report\" class=\"section level2\">\n<h2>Report<\/h2>\n<p>We could report the result as follow: The average weight of mice was significantly increased after treatment, t(9) = 25.5, p &lt; 0.0001, d = 8.07.<\/p>\n<pre class=\"r\"><code>stat.test &lt;- stat.test %&gt;% add_xy_position(x = \"group\")\r\nbxp + \r\n  stat_pvalue_manual(stat.test, tip.length = 0) +\r\n  labs(subtitle = get_test_label(stat.test, detailed= TRUE))<\/code><\/pre>\n<p><img decoding=\"async\" src=\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/dn-tutorials\/r-statistics-2-comparing-groups-means\/figures\/072-paired-t-test-paired-t-test-box-plot-with-p-values-1.png\" width=\"364.8\" \/><\/p>\n<\/div>\n<div id=\"summary\" class=\"section level2\">\n<h2>Summary<\/h2>\n<p>This article describes the formula and the basics of the paired t-test or dependent t-test. Examples of R codes are provided to check the assumptions, computing the test and the effect size, interpreting and reporting the results.<\/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>\n<\/div>\n<\/div>\n<p><!--end rdoc--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Describes the paired t-test, which is used to compare the mean of two related groups of samples. You will learn the formula, assumptions, calculation, visualization, effect size measure using the Cohen&#8217;s d, interpretation and reporting in R.<\/p>\n","protected":false},"author":1,"featured_media":9026,"parent":11648,"menu_order":72,"comment_status":"open","ping_status":"closed","template":"","class_list":["post-11654","dt_lessons","type-dt_lessons","status-publish","has-post-thumbnail","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Paired T-Test : Excellent Reference You Will Love - Datanovia<\/title>\n<meta name=\"description\" content=\"Describes the paired t-test, which is used to compare the mean of two related groups. You will learn the formula, assumptions, calculation, visualization, effect size measure using the Cohen&#039;s d, interpretation and reporting in R.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Paired T-Test : Excellent Reference You Will Love - Datanovia\" \/>\n<meta property=\"og:description\" content=\"Describes the paired t-test, which is used to compare the mean of two related groups. You will learn the formula, assumptions, calculation, visualization, effect size measure using the Cohen&#039;s d, interpretation and reporting in R.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/\" \/>\n<meta property=\"og:site_name\" content=\"Datanovia\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2019\/05\/X37388380_672019819804632_7612320531290783744_n.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1024\" \/>\n\t<meta property=\"og:image:height\" content=\"512\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"8 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/\",\"url\":\"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/\",\"name\":\"Paired T-Test : Excellent Reference You Will Love - Datanovia\",\"isPartOf\":{\"@id\":\"https:\/\/www.datanovia.com\/en\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2019\/05\/X37388380_672019819804632_7612320531290783744_n.jpg\",\"datePublished\":\"2019-12-25T10:51:28+00:00\",\"description\":\"Describes the paired t-test, which is used to compare the mean of two related groups. You will learn the formula, assumptions, calculation, visualization, effect size measure using the Cohen's d, interpretation and reporting in R.\",\"breadcrumb\":{\"@id\":\"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/#primaryimage\",\"url\":\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2019\/05\/X37388380_672019819804632_7612320531290783744_n.jpg\",\"contentUrl\":\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2019\/05\/X37388380_672019819804632_7612320531290783744_n.jpg\",\"width\":1024,\"height\":512},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/www.datanovia.com\/en\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Lessons\",\"item\":\"https:\/\/www.datanovia.com\/en\/lessons\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Types of T-Test\",\"item\":\"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/\"},{\"@type\":\"ListItem\",\"position\":4,\"name\":\"Paired T-Test\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.datanovia.com\/en\/#website\",\"url\":\"https:\/\/www.datanovia.com\/en\/\",\"name\":\"Datanovia\",\"description\":\"Data Mining and Statistics for Decision Support\",\"publisher\":{\"@id\":\"https:\/\/www.datanovia.com\/en\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.datanovia.com\/en\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/www.datanovia.com\/en\/#organization\",\"name\":\"Datanovia\",\"url\":\"https:\/\/www.datanovia.com\/en\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/www.datanovia.com\/en\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2018\/09\/datanovia-logo.png\",\"contentUrl\":\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2018\/09\/datanovia-logo.png\",\"width\":98,\"height\":99,\"caption\":\"Datanovia\"},\"image\":{\"@id\":\"https:\/\/www.datanovia.com\/en\/#\/schema\/logo\/image\/\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Paired T-Test : Excellent Reference You Will Love - Datanovia","description":"Describes the paired t-test, which is used to compare the mean of two related groups. You will learn the formula, assumptions, calculation, visualization, effect size measure using the Cohen's d, interpretation and reporting in R.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/","og_locale":"en_US","og_type":"article","og_title":"Paired T-Test : Excellent Reference You Will Love - Datanovia","og_description":"Describes the paired t-test, which is used to compare the mean of two related groups. You will learn the formula, assumptions, calculation, visualization, effect size measure using the Cohen's d, interpretation and reporting in R.","og_url":"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/","og_site_name":"Datanovia","og_image":[{"width":1024,"height":512,"url":"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2019\/05\/X37388380_672019819804632_7612320531290783744_n.jpg","type":"image\/jpeg"}],"twitter_card":"summary_large_image","twitter_misc":{"Est. reading time":"8 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/","url":"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/","name":"Paired T-Test : Excellent Reference You Will Love - Datanovia","isPartOf":{"@id":"https:\/\/www.datanovia.com\/en\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/#primaryimage"},"image":{"@id":"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/#primaryimage"},"thumbnailUrl":"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2019\/05\/X37388380_672019819804632_7612320531290783744_n.jpg","datePublished":"2019-12-25T10:51:28+00:00","description":"Describes the paired t-test, which is used to compare the mean of two related groups. You will learn the formula, assumptions, calculation, visualization, effect size measure using the Cohen's d, interpretation and reporting in R.","breadcrumb":{"@id":"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/#primaryimage","url":"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2019\/05\/X37388380_672019819804632_7612320531290783744_n.jpg","contentUrl":"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2019\/05\/X37388380_672019819804632_7612320531290783744_n.jpg","width":1024,"height":512},{"@type":"BreadcrumbList","@id":"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/paired-t-test\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.datanovia.com\/en\/"},{"@type":"ListItem","position":2,"name":"Lessons","item":"https:\/\/www.datanovia.com\/en\/lessons\/"},{"@type":"ListItem","position":3,"name":"Types of T-Test","item":"https:\/\/www.datanovia.com\/en\/lessons\/types-of-t-test\/"},{"@type":"ListItem","position":4,"name":"Paired T-Test"}]},{"@type":"WebSite","@id":"https:\/\/www.datanovia.com\/en\/#website","url":"https:\/\/www.datanovia.com\/en\/","name":"Datanovia","description":"Data Mining and Statistics for Decision Support","publisher":{"@id":"https:\/\/www.datanovia.com\/en\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.datanovia.com\/en\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/www.datanovia.com\/en\/#organization","name":"Datanovia","url":"https:\/\/www.datanovia.com\/en\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.datanovia.com\/en\/#\/schema\/logo\/image\/","url":"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2018\/09\/datanovia-logo.png","contentUrl":"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/2018\/09\/datanovia-logo.png","width":98,"height":99,"caption":"Datanovia"},"image":{"@id":"https:\/\/www.datanovia.com\/en\/#\/schema\/logo\/image\/"}}]}},"multi-rating":{"mr_rating_results":[]},"_links":{"self":[{"href":"https:\/\/www.datanovia.com\/en\/wp-json\/wp\/v2\/dt_lessons\/11654","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.datanovia.com\/en\/wp-json\/wp\/v2\/dt_lessons"}],"about":[{"href":"https:\/\/www.datanovia.com\/en\/wp-json\/wp\/v2\/types\/dt_lessons"}],"author":[{"embeddable":true,"href":"https:\/\/www.datanovia.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.datanovia.com\/en\/wp-json\/wp\/v2\/comments?post=11654"}],"version-history":[{"count":0,"href":"https:\/\/www.datanovia.com\/en\/wp-json\/wp\/v2\/dt_lessons\/11654\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/www.datanovia.com\/en\/wp-json\/wp\/v2\/dt_lessons\/11648"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.datanovia.com\/en\/wp-json\/wp\/v2\/media\/9026"}],"wp:attachment":[{"href":"https:\/\/www.datanovia.com\/en\/wp-json\/wp\/v2\/media?parent=11654"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}