{"id":1007,"date":"2025-02-06T21:53:25","date_gmt":"2025-02-06T16:08:25","guid":{"rendered":"https:\/\/pokharelsugam.com.np\/bioinformatics\/?p=1007"},"modified":"2025-02-06T22:36:24","modified_gmt":"2025-02-06T16:51:24","slug":"mann-whitney-u-test","status":"publish","type":"post","link":"https:\/\/pokharelsugam.com.np\/bioinformatics\/mann-whitney-u-test\/","title":{"rendered":"Mann-Whitney U Test (Wilcoxon Rank-Sum Test)"},"content":{"rendered":"\n<script type=\"text\/javascript\" async \n  src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\">\n<\/script>\n\n\n\n<p class=\"wp-block-paragraph\">The <strong>Mann-Whitney U Test<\/strong> (also known as the <strong>Wilcoxon Rank-Sum Test<\/strong>) is a <strong>non-parametric<\/strong> statistical test used to compare <strong>two independent groups<\/strong> to determine whether there is a significant difference in their distributions. It is often used as an alternative to the <strong>independent samples t-test<\/strong> when the assumption of normality is violated.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>1. Introduction<\/strong><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It is used to compare two independent samples.<\/li>\n\n\n\n<li>It tests whether one population tends to have <strong>higher or lower<\/strong> values than the other.<\/li>\n\n\n\n<li>It does <strong>not<\/strong> assume normal distribution, making it useful for <strong>skewed<\/strong> or <strong>ordinal<\/strong> data.<\/li>\n\n\n\n<li>It ranks all observations and then analyzes differences based on ranks rather than raw scores.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>2. Assumptions<\/strong><\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li>The <strong>dependent variable<\/strong> is either <strong>ordinal<\/strong> or <strong>continuous<\/strong>.<\/li>\n\n\n\n<li>The two groups are <strong>independent<\/strong> (no repeated measures).<\/li>\n\n\n\n<li>The observations should be <strong>randomly sampled<\/strong> from the populations.<\/li>\n\n\n\n<li>The shape of the distributions should be similar if comparing medians.<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">3.<strong>Hypotheses<\/strong><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Null Hypothesis (H\u2080):<\/strong> The two groups have the same distribution (no difference).<\/li>\n\n\n\n<li><strong>Alternative Hypothesis (H\u2081):<\/strong> The distributions of the two groups are different.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Depending on the research question, the alternative hypothesis may be:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Two-tailed:<\/strong> The distributions are different in some way.<\/li>\n\n\n\n<li><strong>One-tailed:<\/strong> One distribution tends to have larger values than the other.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">4.Test statistics<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>For small sample size (n1 and n2 &lt;=20)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(U_1 = n_1n_2 + \\frac{n_1(n_1+1)}{2}-R_1 \\) &#8230;&#8230;&#8230;&#8230;.. i<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(U_2 = n_1n_2 + \\frac{n_2(n_2+1)}{2}-R_2 \\) &#8230;&#8230;&#8230;&#8230;&#8230;&#8230; ii<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>n1 = Number of observation in first sample.<\/li>\n\n\n\n<li>n2 = Number of observation in second sample.<\/li>\n\n\n\n<li>R1 = Sum of ranks of first sample.<\/li>\n\n\n\n<li>R2 = Sum of ranks of second sample.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Decision<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If \\(U_{min} > U_{tab}, H_0\\) is accepted<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>For large sample size  (n1 and n2  >20)<\/strong><\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">\\(Z =\\frac{U_{min}-\\mu_u}{\\sigma_u}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where <\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">\\(\\mu_u = \\frac{n_1n_2}{2}\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">\\(\\sigma_u = \\sqrt{\\frac{n_1n_2(n_1+n_2+1)}{12}}\\)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Mann-Whitney U Test (also known as the Wilcoxon Rank-Sum Test) is a non-parametric statistical test used to compare two [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":null,"ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":null,"ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":null,"ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":null,"stick-header-meta":null,"header-above-stick-meta":null,"header-main-stick-meta":null,"header-below-stick-meta":null,"astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center 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