{"id":79,"date":"2025-01-15T13:40:54","date_gmt":"2025-01-15T13:40:54","guid":{"rendered":"https:\/\/pokharelsugam.com.np\/bioinformatics\/?p=79"},"modified":"2025-01-22T16:03:40","modified_gmt":"2025-01-22T10:18:40","slug":"chi-squared-test","status":"publish","type":"post","link":"https:\/\/pokharelsugam.com.np\/bioinformatics\/chi-squared-test\/","title":{"rendered":"Chi Squared 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\">It is the most popular non parametric test of statistical significance for bivariate tabular analysis with a contingency table. This test can be applied to nominal or categorical data, but cannot be used in ranking technique.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Conditions to apply \\(\\chi^2 \\) test<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The frequency most be absolute not relative<\/li>\n\n\n\n<li>The sample observation should be independent<\/li>\n\n\n\n<li>The total frequency should be reasonably large that is n >50<\/li>\n\n\n\n<li>The expected frequency  of each cell should not be less than 5<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Conditions to apply \\(\\chi^2 \\) test<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Test of independence<\/li>\n\n\n\n<li>Test of Goodness of Fit<\/li>\n\n\n\n<li>Test of Homogeneity<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Formulae<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\chi^2 = \\sum \\frac{(O &#8211; E)^2}{E} \\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where,<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">O : Observed Frequency, E: Expected Frequency<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(E = \\frac{RT \\times CT}{GT} \\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where, RT: Row Total for individual frequency, CT: Column Total for individual Frequency, GT: Grand Total of all frequency<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">1. Test of Independence<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">It is used to test the relationships between two categorical variable where df = (r-1)(c-1). It use contingency table.<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">a. By Direct Method<\/h5>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td><\/td><td>Category A<\/td><td><\/td><td>RT<\/td><\/tr><tr><td>Category B<\/td><td>a<\/td><td>b<\/td><td>a+b<\/td><\/tr><tr><td><\/td><td>c<\/td><td>d<\/td><td>c+d<\/td><\/tr><tr><td>CT<\/td><td>a+c<\/td><td>b+d<\/td><td>GT: a+b+c+d<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>\\(O\\)<\/td><td>\\(E = \\frac{RT \\times CT}{GT} \\)<\/td><td>\\(O-E\\)<\/td><td>\\((O-E)^2 \\)<\/td><td>\\(\\frac{(O-E)^2}{E} \\)<\/td><\/tr><tr><td>a<\/td><td>\\(E_a = \\frac{(a+b) \\times (a+c)}{GT} \\)<\/td><td>\\(a-E_a\\)<\/td><td><br>\\((a-E_a)^2\\)<\/td><td>\\(\\frac{(a-E_a)^2}{E_a}\\)<\/td><\/tr><tr><td>b<\/td><td><br>\\(E_b = \\frac{(a+b) \\times (b+d)}{GT} \\)<\/td><td>\\(b-E_b\\)<\/td><td>\\((b-E_b)^2\\)<\/td><td>\\(\\frac{(b-E_b)^2}{E_b}\\)<\/td><\/tr><tr><td>c<\/td><td>\\(E_c = \\frac{(c+d) \\times (a+c)}{GT} \\)<\/td><td><\/td><td><\/td><td><\/td><\/tr><tr><td>d<\/td><td>\\(E_d = \\frac{(c+d) \\times (b+d)}{GT} \\)<\/td><td><\/td><td><\/td><td><\/td><\/tr><tr><td><\/td><td>\\(\\sum O\\)<\/td><td>\\(\\sum E\\)<\/td><td><\/td><td>\\(\\sum \\frac{(O &#8211; E)^2}{E} \\)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h5 class=\"wp-block-heading\">Test Statistics<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\chi^2 = \\sum \\frac{(O &#8211; E)^2}{E} \\)<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">Degree of Freedom<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">\\(df = (r-1)(c-1) = (2-1)(2-1) = 1  \\) <\/p>\n\n\n\n<h5 class=\"wp-block-heading\">Level of Significance<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">Use generally 5% level of significance \\(\\alpha = 0.05\\) <\/p>\n\n\n\n<h5 class=\"wp-block-heading\">Critical Value<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">Use chi squared test critical value table and find critical value by using level of significance (0.05) and degree of freedom (1).<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">Decision<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">If \\(\\chi^2_{cal} \\leq \\chi^2_{tab} \\) ; Accept \\(H_0\\) and Reject \\(H_1\\) and vice versa<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">b. By Contingency Table<\/h5>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td>a<\/td><td>b<\/td><td>a+b<\/td><\/tr><tr><td>c<\/td><td>d<\/td><td>c+d<\/td><\/tr><tr><td>a+c<\/td><td>b+d<\/td><td>N = a+b+c+d<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\chi^2 = \\frac{N(ad-bc)^2}{(a+b)(c+d)(a+c)(b+d)} \\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If value of one of the cell(that is either a or b or c or d or both or all) is less than 5 then use <strong>Yates Correction<\/strong><\/p>\n\n\n\n<h5 class=\"wp-block-heading\">Yates Correction<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\chi^2 = \\frac{N[|ad-bc| &#8211; \\frac{N}{2}]^2}{(a+b)(c+d)(a+c)(b+d)}\\)<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">2. Test of Goodness of Fit<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">It is used to test if observed data matches a specific distribution use one categorical variable where df = k-1.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td>x<\/td><td>x1<\/td><td>x2<\/td><td>x3<\/td><td>x4<\/td><td>x5<\/td><\/tr><tr><td>f<\/td><td>f1<\/td><td>f2<\/td><td>f3<\/td><td>f4<\/td><td>f5<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Observed Frequency (O) = f1, f2, f3, f4, f5<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Expected Frequency (E) = \\(\\frac{\\sum O}{n}\\)<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td>x<\/td><td>f(O)<\/td><td>f(E)<\/td><td>O-E<\/td><td>\\((O-E)^2 \\)<\/td><td>\\(\\frac{(O-E)^2}{E} \\)<\/td><\/tr><tr><td>x1<\/td><td>f1<\/td><td>\\(\\frac{\\sum O}{n}\\)<\/td><td><\/td><td><\/td><td><\/td><\/tr><tr><td>x2<\/td><td>f2<\/td><td>\\(\\frac{\\sum O}{n}\\)<\/td><td><\/td><td><\/td><td><\/td><\/tr><tr><td>x3<\/td><td>f3<\/td><td>\\(\\frac{\\sum O}{n}\\)<\/td><td><\/td><td><\/td><td><\/td><\/tr><tr><td>x4<\/td><td>f4<\/td><td>\\(\\frac{\\sum O}{n}\\)<\/td><td><\/td><td><\/td><td><\/td><\/tr><tr><td>x5<\/td><td>f5<\/td><td>\\(\\frac{\\sum O}{n}\\)<\/td><td><\/td><td><\/td><td><\/td><\/tr><tr><td>n=5<\/td><td>\\(\\sum O\\)<\/td><td>\\(\\sum E\\)<\/td><td><\/td><td><\/td><td><br>\\(\\sum \\frac{(O &#8211; E)^2}{E} \\)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h5 class=\"wp-block-heading\">Test Statistics<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\chi^2 = \\sum \\frac{(O &#8211; E)^2}{E} \\)<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">Degree of Freedom<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">\\(df = k-1 = 5-1 = 4 \\)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>It is the most popular non parametric test of statistical significance for bivariate tabular analysis with a contingency table. This [&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 center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[7],"tags":[],"class_list":["post-79","post","type-post","status-publish","format-standard","hentry","category-biostatistics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.7 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Chi Squared Test - Bioinformatics Notes<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/pokharelsugam.com.np\/bioinformatics\/chi-squared-test\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Chi Squared Test - Bioinformatics Notes\" \/>\n<meta property=\"og:description\" content=\"It is the most popular non parametric test of statistical significance for bivariate tabular analysis with a contingency table. 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