CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Family-wise error rate - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"27fd9dc1-c2f8-4b68-a523-4ae7052e8016","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Family-wise_error_rate","wgTitle":"Family-wise error rate","wgCurRevisionId":1258017002,"wgRevisionId":1258017002,"wgArticleId":4621448,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 maint: DOI inactive as of November 2024","Articles with short description","Short description is different from Wikidata","Articles needing additional references from June 2016","All articles needing additional references","All articles with unsourced statements","Articles with unsourced statements from June 2016","Articles to be expanded from February 2013","All articles to be expanded","Articles with empty sections from February 2013", "All articles with empty sections","Statistical hypothesis testing","Multiple comparisons","Rates"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Family-wise_error_rate","wgRelevantArticleId":4621448,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage", "wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q5433418","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements", "site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Family-wise error rate - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Family-wise_error_rate"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Family-wise_error_rate&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Family-wise_error_rate"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Family-wise_error_rate rootpage-Family-wise_error_rate skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Family-wise+error+rate" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Family-wise+error+rate" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Family-wise+error+rate" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Family-wise+error+rate" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Familywise_and_Experimentwise_Error_Rates" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Familywise_and_Experimentwise_Error_Rates"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Familywise and Experimentwise Error Rates</span> </div> </a> <ul id="toc-Familywise_and_Experimentwise_Error_Rates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Background" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Background"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Background</span> </div> </a> <button aria-controls="toc-Background-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Background subsection</span> </button> <ul id="toc-Background-sublist" class="vector-toc-list"> <li id="toc-Classification_of_multiple_hypothesis_tests" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classification_of_multiple_hypothesis_tests"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Classification of multiple hypothesis tests</span> </div> </a> <ul id="toc-Classification_of_multiple_hypothesis_tests-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Definition</span> </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Controlling_procedures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Controlling_procedures"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Controlling procedures</span> </div> </a> <button aria-controls="toc-Controlling_procedures-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Controlling procedures subsection</span> </button> <ul id="toc-Controlling_procedures-sublist" class="vector-toc-list"> <li id="toc-The_Bonferroni_procedure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Bonferroni_procedure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>The Bonferroni procedure</span> </div> </a> <ul id="toc-The_Bonferroni_procedure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Šidák_procedure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Šidák_procedure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>The Šidák procedure</span> </div> </a> <ul id="toc-The_Šidák_procedure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tukey's_procedure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tukey's_procedure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Tukey's procedure</span> </div> </a> <ul id="toc-Tukey's_procedure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Holm's_step-down_procedure_(1979)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Holm's_step-down_procedure_(1979)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Holm's step-down procedure (1979)</span> </div> </a> <ul id="toc-Holm's_step-down_procedure_(1979)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hochberg's_step-up_procedure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hochberg's_step-up_procedure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Hochberg's step-up procedure</span> </div> </a> <ul id="toc-Hochberg's_step-up_procedure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dunnett's_correction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dunnett's_correction"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Dunnett's correction</span> </div> </a> <ul id="toc-Dunnett's_correction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scheffé's_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scheffé's_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Scheffé's method</span> </div> </a> <ul id="toc-Scheffé's_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Resampling_procedures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Resampling_procedures"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Resampling procedures</span> </div> </a> <ul id="toc-Resampling_procedures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Harmonic_mean_p-value_procedure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harmonic_mean_p-value_procedure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.9</span> <span>Harmonic mean <i>p</i>-value procedure</span> </div> </a> <ul id="toc-Harmonic_mean_p-value_procedure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Alternative_approaches" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Alternative_approaches"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Alternative approaches</span> </div> </a> <ul id="toc-Alternative_approaches-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Family-wise error rate</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 3 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-3" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">3 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/FWER" title="FWER – Hungarian" lang="hu" hreflang="hu" data-title="FWER" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A1%E3%83%9F%E3%83%AA%E3%83%BC%E3%83%AF%E3%82%A4%E3%82%BA%E3%82%A8%E3%83%A9%E3%83%BC%E7%8E%87" title="ファミリーワイズエラー率 – Japanese" lang="ja" hreflang="ja" data-title="ファミリーワイズエラー率" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/FWER" title="FWER – Polish" lang="pl" hreflang="pl" data-title="FWER" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q5433418#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Family-wise_error_rate" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Family-wise_error_rate" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Family-wise_error_rate"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Family-wise_error_rate&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Family-wise_error_rate&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Family-wise_error_rate"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Family-wise_error_rate&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Family-wise_error_rate&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Family-wise_error_rate" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Family-wise_error_rate" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Family-wise_error_rate&oldid=1258017002" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Family-wise_error_rate&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Family-wise_error_rate&id=1258017002&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FFamily-wise_error_rate"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FFamily-wise_error_rate"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Family-wise_error_rate&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Family-wise_error_rate&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q5433418" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Probability of making type I errors when performing multiple hypotheses tests</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Family-wise_error_rate" title="Special:EditPage/Family-wise error rate">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Family-wise+error+rate%22">"Family-wise error rate"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Family-wise+error+rate%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Family-wise+error+rate%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Family-wise+error+rate%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Family-wise+error+rate%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Family-wise+error+rate%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">June 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, <b>family-wise error rate</b> (<b>FWER</b>) is the <a href="/wiki/Probability" title="Probability">probability</a> of making one or more false discoveries, or <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">type I errors</a> when performing <a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">multiple hypotheses tests</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Familywise_and_Experimentwise_Error_Rates">Familywise and Experimentwise Error Rates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=1" title="Edit section: Familywise and Experimentwise Error Rates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/John_Tukey" title="John Tukey">John Tukey</a> developed in 1953 the concept of a familywise error rate as the probability of making a Type I error among a specified group, or "family," of tests.<sup id="cite_ref-Tukey_(1953)_1-0" class="reference"><a href="#cite_note-Tukey_(1953)-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Ryan (1959) proposed the related concept of an <i>experimentwise error rate</i>, which is the probability of making a Type I error in a given experiment.<sup id="cite_ref-Ryan_(1959)_2-0" class="reference"><a href="#cite_note-Ryan_(1959)-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Hence, an experimentwise error rate is a familywise error rate where the family includes all the tests that are conducted within an experiment. </p><p>As Ryan (1959, Footnote 3) explained, an experiment may contain two or more families of multiple comparisons, each of which relates to a particular statistical inference and each of which has its own separate familywise error rate.<sup id="cite_ref-Ryan_(1959)_2-1" class="reference"><a href="#cite_note-Ryan_(1959)-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Hence, familywise error rates are usually based on theoretically informative collections of multiple comparisons. In contrast, an experimentwise error rate may be based on a collection of simultaneous comparisons that refer to a diverse range of separate inferences. Some have argued that it may not be useful to control the experimentwise error rate in such cases.<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Indeed, Tukey suggested that familywise control was preferable in such cases (Tukey, 1956, personal communication, in Ryan, 1962, p. 302).<sup id="cite_ref-Ryan_(1962)_4-0" class="reference"><a href="#cite_note-Ryan_(1962)-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=2" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Within the statistical framework, there are several definitions for the term "family": </p> <ul><li>Hochberg & Tamhane (1987) defined "family" as "any collection of inferences for which it is meaningful to take into account some combined measure of error".<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li>According to Cox (1982), a set of inferences should be regarded a family:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2016)">citation needed</span></a></i>]</sup></li></ul> <ol><li>To take into account the selection effect due to <a href="/wiki/Data_dredging" title="Data dredging">data dredging</a></li> <li>To ensure simultaneous correctness of a set of inferences as to guarantee a correct overall decision</li></ol> <p>To summarize, a family could best be defined by the potential <a href="/wiki/Selective_inference" class="mw-redirect" title="Selective inference">selective inference</a> that is being faced: A family is the smallest set of items of inference in an analysis, interchangeable about their meaning for the goal of research, from which selection of results for action, presentation or highlighting could be made (<a href="/wiki/Yoav_Benjamini" title="Yoav Benjamini">Yoav Benjamini</a>).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2016)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Classification_of_multiple_hypothesis_tests">Classification of multiple hypothesis tests</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=3" title="Edit section: Classification of multiple hypothesis tests"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Classification_of_multiple_hypothesis_tests" class="mw-redirect" title="Classification of multiple hypothesis tests">Classification of multiple hypothesis tests</a></div> <p>The following table defines the possible outcomes when testing multiple null hypotheses. Suppose we have a number <i>m</i> of null hypotheses, denoted by: <span class="texhtml"><i>H</i><sub>1</sub>, <i>H</i><sub>2</sub>, ..., <i>H</i><sub><i>m</i></sub>.</span> Using a <a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">statistical test</a>, we reject the null hypothesis if the test is declared significant. We do not reject the null hypothesis if the test is non-significant. Summing each type of outcome over all <i>H<sub>i</sub></i>  yields the following random variables: </p> <table class="wikitable"> <tbody><tr> <td> </td> <th>Null hypothesis is true (H<sub>0</sub>) </th> <th>Alternative hypothesis is true (H<sub>A</sub>) </th> <th>Total </th></tr> <tr align="center"> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: left;" class="table-rh">Test is declared significant </th> <td><span class="texhtml mvar" style="font-style:italic;">V</span> </td> <td><span class="texhtml mvar" style="font-style:italic;">S</span> </td> <td><span class="texhtml mvar" style="font-style:italic;">R</span> </td></tr> <tr align="center"> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: left;" class="table-rh">Test is declared non-significant </th> <td><span class="texhtml mvar" style="font-style:italic;">U</span> </td> <td><span class="texhtml mvar" style="font-style:italic;">T</span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m-R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>−</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m-R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c9bf03e2b488956ca5af783661fc8c734e6440a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.645ex; height:2.343ex;" alt="{\displaystyle m-R}"></span> </td></tr> <tr align="center"> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: left;" class="table-rh">Total </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6ff51ee949104fe6fae553cfbdfba29d5fac1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{0}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m-m_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m-m_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5aa8c55f7307feb806e7d5f8df5f40c0843497c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.975ex; height:2.343ex;" alt="{\displaystyle m-m_{0}}"></span> </td> <td><span class="texhtml mvar" style="font-style:italic;">m</span> </td></tr></tbody></table> <ul><li><span class="texhtml mvar" style="font-style:italic;">m</span> is the total number hypotheses tested</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6ff51ee949104fe6fae553cfbdfba29d5fac1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{0}}"></span> is the number of true <a href="/wiki/Null_hypothesis" title="Null hypothesis">null hypotheses</a>, an unknown parameter</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m-m_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m-m_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5aa8c55f7307feb806e7d5f8df5f40c0843497c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.975ex; height:2.343ex;" alt="{\displaystyle m-m_{0}}"></span> is the number of true <a href="/wiki/Alternative_hypothesis" title="Alternative hypothesis">alternative hypotheses</a></li> <li><span class="texhtml mvar" style="font-style:italic;">V</span> is the number of <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">false positives (Type I error)</a> (also called "false discoveries")</li> <li><span class="texhtml mvar" style="font-style:italic;">S</span> is the number of <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">true positives</a> (also called "true discoveries")</li> <li><span class="texhtml mvar" style="font-style:italic;">T</span> is the number of <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">false negatives (Type II error)</a></li> <li><span class="texhtml mvar" style="font-style:italic;">U</span> is the number of <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">true negatives</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=V+S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>V</mi> <mo>+</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=V+S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0c6c98e2eeaa462ce03b0f2c46e83ad2d35960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.989ex; height:2.343ex;" alt="{\displaystyle R=V+S}"></span> is the number of rejected null hypotheses (also called "discoveries", either true or false)</li></ul> <p>In <span class="texhtml mvar" style="font-style:italic;">m</span> hypothesis tests of which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6ff51ee949104fe6fae553cfbdfba29d5fac1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{0}}"></span> are true null hypotheses, <span class="texhtml mvar" style="font-style:italic;">R</span> is an observable random variable, and <span class="texhtml mvar" style="font-style:italic;">S</span>, <span class="texhtml mvar" style="font-style:italic;">T</span>, <span class="texhtml mvar" style="font-style:italic;">U</span>, and <span class="texhtml mvar" style="font-style:italic;">V</span> are unobservable <a href="/wiki/Random_variable" title="Random variable">random variables</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=4" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The FWER is the probability of making at least one <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">type I error</a> in the family, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {FWER} =\Pr(V\geq 1),\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">W</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">R</mi> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>≥</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {FWER} =\Pr(V\geq 1),\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f66915efd7c5d64aa69089c78d333f81acb981a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.684ex; height:2.843ex;" alt="{\displaystyle \mathrm {FWER} =\Pr(V\geq 1),\,}"></span></dd></dl> <p>or equivalently, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {FWER} =1-\Pr(V=0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">W</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">R</mi> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {FWER} =1-\Pr(V=0).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a6f35d7de7f3d41aebd6c6191e3da8c867f3775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.3ex; height:2.843ex;" alt="{\displaystyle \mathrm {FWER} =1-\Pr(V=0).}"></span></dd></dl> <p>Thus, by assuring <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {FWER} \leq \alpha \,\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">W</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">R</mi> </mrow> <mo>≤</mo> <mi>α</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {FWER} \leq \alpha \,\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d0501102ff64635e1a341a2b42e3d54817bd23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.173ex; height:2.343ex;" alt="{\displaystyle \mathrm {FWER} \leq \alpha \,\!\,}"></span>, the probability of making one or more <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">type I errors</a> in the family is controlled at level <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e383db6745c5569c37ab2290d41e06a27510337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle \alpha \,\!}"></span>. </p><p>A procedure controls the FWER <i>in the weak sense</i> if the FWER control at level <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e383db6745c5569c37ab2290d41e06a27510337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle \alpha \,\!}"></span> is guaranteed <i>only</i> when all null hypotheses are true (i.e. when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{0}=m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}=m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78b853711e8bea0d5acbff7712cd94970e2c243d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.233ex; height:2.009ex;" alt="{\displaystyle m_{0}=m}"></span>, meaning the "global null hypothesis" is true).<sup id="cite_ref-CRC_Press_5-0" class="reference"><a href="#cite_note-CRC_Press-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>A procedure controls the FWER <i>in the strong sense</i> if the FWER control at level <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e383db6745c5569c37ab2290d41e06a27510337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle \alpha \,\!}"></span> is guaranteed for <i>any</i> configuration of true and non-true null hypotheses (whether the global null hypothesis is true or not).<sup id="cite_ref-CRC_Press_5-1" class="reference"><a href="#cite_note-CRC_Press-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Controlling_procedures">Controlling procedures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=5" title="Edit section: Controlling procedures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="/wiki/Multiple_testing_correction" class="mw-redirect" title="Multiple testing correction">Multiple testing correction</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/False_coverage_rate#Controlling_procedures" title="False coverage rate">False coverage rate § Controlling procedures</a>, and <a href="/wiki/False_discovery_rate#Controlling_procedures" title="False discovery rate">False discovery rate § Controlling procedures</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Post_hoc_analysis#Tests" title="Post hoc analysis">List of <i>post hoc</i> tests</a></div> <p>Some classical solutions that ensure strong level <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> FWER control, and some newer solutions exist. </p> <div class="mw-heading mw-heading3"><h3 id="The_Bonferroni_procedure">The Bonferroni procedure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=6" title="Edit section: The Bonferroni procedure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bonferroni_correction" title="Bonferroni correction">Bonferroni correction</a></div> <ul><li>Denote by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> the <i>p</i>-value for testing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd0312f590cc5a400008938f3cc304d42ad3986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.731ex; height:2.509ex;" alt="{\displaystyle H_{i}}"></span></li> <li>reject <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd0312f590cc5a400008938f3cc304d42ad3986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.731ex; height:2.509ex;" alt="{\displaystyle H_{i}}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}\leq {\frac {\alpha }{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>m</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}\leq {\frac {\alpha }{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16358edfbf546100fa79428a8264d2a9e4788554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:8.034ex; height:4.676ex;" alt="{\displaystyle p_{i}\leq {\frac {\alpha }{m}}}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="The_Šidák_procedure"><span id="The_.C5.A0id.C3.A1k_procedure"></span>The Šidák procedure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=7" title="Edit section: The Šidák procedure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/%C5%A0id%C3%A1k_correction" title="Šidák correction">Šidák correction</a></div> <ul><li>Testing each hypothesis at level <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{SID}=1-(1-\alpha )^{\frac {1}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>I</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{SID}=1-(1-\alpha )^{\frac {1}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/808614e23ca37209a294e38f20c1df21d6a554f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.61ex; height:3.843ex;" alt="{\displaystyle \alpha _{SID}=1-(1-\alpha )^{\frac {1}{m}}}"></span> is Sidak's multiple testing procedure.</li> <li>This procedure is more powerful than Bonferroni but the gain is small.</li> <li>This procedure can fail to control the FWER when the tests are negatively dependent.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Tukey's_procedure"><span id="Tukey.27s_procedure"></span>Tukey's procedure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=8" title="Edit section: Tukey's procedure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Tukey%27s_range_test" title="Tukey's range test">Tukey's range test</a></div> <ul><li>Tukey's procedure is only applicable for pairwise comparisons.</li> <li>It assumes independence of the observations being tested, as well as equal variation across observations (<a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">homoscedasticity</a>).</li> <li>The procedure calculates for each pair the <a href="/wiki/Studentized_range" title="Studentized range">studentized range</a> statistic: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {Y_{A}-Y_{B}}{SE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mrow> <mi>S</mi> <mi>E</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {Y_{A}-Y_{B}}{SE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d3ffc6815b5e0baf9ae173b8631eb7893b1699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.322ex; height:5.509ex;" alt="{\displaystyle {\frac {Y_{A}-Y_{B}}{SE}}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc361b3764d071879684a21d6ab84a043df7923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.815ex; height:2.509ex;" alt="{\displaystyle Y_{A}}"></span> is the larger of the two means being compared, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b19091196b5ba697a6374f3eaa3bd9d6e115eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle Y_{B}}"></span> is the smaller, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a7d50ca655033f9f4ad0fac6dde1eac170e4c08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.275ex; height:2.176ex;" alt="{\displaystyle SE}"></span> is the standard error of the data in question.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2016)">citation needed</span></a></i>]</sup></li> <li>Tukey's test is essentially a <a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's t-test</a>, except that it corrects for <b>family-wise error-rate</b>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2016)">citation needed</span></a></i>]</sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Holm's_step-down_procedure_(1979)"><span id="Holm.27s_step-down_procedure_.281979.29"></span>Holm's step-down procedure (1979)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=9" title="Edit section: Holm's step-down procedure (1979)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Holm%E2%80%93Bonferroni_method" title="Holm–Bonferroni method">Holm–Bonferroni method</a></div> <ul><li>Start by ordering the <i>p</i>-values (from lowest to highest) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{(1)}\ldots P_{(m)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>…</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{(1)}\ldots P_{(m)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45dc5d65cead7a60a722bff3133f835d263f3274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.77ex; height:3.009ex;" alt="{\displaystyle P_{(1)}\ldots P_{(m)}}"></span> and let the associated hypotheses be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{(1)}\ldots H_{(m)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>…</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{(1)}\ldots H_{(m)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d535eef24378fd505aa83922ce7a021c595dfbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.648ex; height:3.009ex;" alt="{\displaystyle H_{(1)}\ldots H_{(m)}}"></span></li> <li>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> be the minimal index such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{(k)}>{\frac {\alpha }{m+1-k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{(k)}>{\frac {\alpha }{m+1-k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd323db7c2870302aec6cf4bd7f1643052b70c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.89ex; height:5.009ex;" alt="{\displaystyle P_{(k)}>{\frac {\alpha }{m+1-k}}}"></span></li> <li>Reject the null hypotheses <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{(1)}\ldots H_{(k-1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>…</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{(1)}\ldots H_{(k-1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e715d7d88c534faf6a0d98828461426c977e69d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.162ex; height:3.009ex;" alt="{\displaystyle H_{(1)}\ldots H_{(k-1)}}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"></span> then none of the hypotheses are rejected.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2016)">citation needed</span></a></i>]</sup></li></ul> <p>This procedure is uniformly more powerful than the Bonferroni procedure.<sup id="cite_ref-Aickin1996_6-0" class="reference"><a href="#cite_note-Aickin1996-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The reason why this procedure controls the family-wise error rate for all the m hypotheses at level α in the strong sense is, because it is a <a href="/wiki/Closed_testing_procedure" title="Closed testing procedure">closed testing procedure</a>. As such, each intersection is tested using the simple Bonferroni test.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2016)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Hochberg's_step-up_procedure"><span id="Hochberg.27s_step-up_procedure"></span>Hochberg's step-up procedure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=10" title="Edit section: Hochberg's step-up procedure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hochberg's step-up procedure (1988) is performed using the following steps:<sup id="cite_ref-Hochberg1988_7-0" class="reference"><a href="#cite_note-Hochberg1988-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Start by ordering the <i>p</i>-values (from lowest to highest) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{(1)}\ldots P_{(m)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>…</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{(1)}\ldots P_{(m)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45dc5d65cead7a60a722bff3133f835d263f3274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.77ex; height:3.009ex;" alt="{\displaystyle P_{(1)}\ldots P_{(m)}}"></span> and let the associated hypotheses be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{(1)}\ldots H_{(m)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>…</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{(1)}\ldots H_{(m)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d535eef24378fd505aa83922ce7a021c595dfbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.648ex; height:3.009ex;" alt="{\displaystyle H_{(1)}\ldots H_{(m)}}"></span></li> <li>For a given <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> be the largest <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{(k)}\leq {\frac {\alpha }{m-k+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>m</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{(k)}\leq {\frac {\alpha }{m-k+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b85c65af33288ea217272958bb5056fd7a8e27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.89ex; height:5.009ex;" alt="{\displaystyle P_{(k)}\leq {\frac {\alpha }{m-k+1}}}"></span></li> <li>Reject the null hypotheses <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{(1)}\ldots H_{(R)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>…</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{(1)}\ldots H_{(R)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c63798593d497edfe0ae3a0f2ac34e83016ac0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.452ex; height:3.009ex;" alt="{\displaystyle H_{(1)}\ldots H_{(R)}}"></span></li></ul> <p>Hochberg's procedure is more powerful than Holm's. Nevertheless, while Holm’s is a closed testing procedure (and thus, like Bonferroni, has no restriction on the joint distribution of the test statistics), Hochberg’s is based on the Simes test, so it holds only under non-negative dependence.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2016)">citation needed</span></a></i>]</sup> The Simes test is derived under assumption of independent tests;<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> it is conservative for tests that are positively dependent in a certain sense<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and is anti-conservative for certain cases of negative dependence.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> However, it has been suggested that a modified version of the Hochberg procedure remains valid under general negative dependence.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Dunnett's_correction"><span id="Dunnett.27s_correction"></span>Dunnett's correction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=11" title="Edit section: Dunnett's correction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Dunnett%27s_test" title="Dunnett's test">Dunnett's test</a></div> <p><a href="/wiki/Charles_Dunnett" title="Charles Dunnett">Charles Dunnett</a> (1955, 1966) described an alternative alpha error adjustment when <i>k</i> groups are compared to the same control group. Now known as Dunnett's test, this method is less conservative than the Bonferroni adjustment.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2016)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Scheffé's_method"><span id="Scheff.C3.A9.27s_method"></span>Scheffé's method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=12" title="Edit section: Scheffé's method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Scheff%C3%A9%27s_method" title="Scheffé's method">Scheffé's method</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Empty_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span"><b>This section is empty.</b> You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Family-wise_error_rate&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">February 2013</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Resampling_procedures">Resampling procedures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=13" title="Edit section: Resampling procedures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The procedures of Bonferroni and Holm control the FWER under any dependence structure of the <i>p</i>-values (or equivalently the individual test statistics). Essentially, this is achieved by accommodating a `worst-case' dependence structure (which is close to independence for most practical purposes). But such an approach is conservative if dependence is actually positive. To give an extreme example, under perfect positive dependence, there is effectively only one test and thus, the FWER is uninflated. </p><p>Accounting for the dependence structure of the <i>p</i>-values (or of the individual test statistics) produces more powerful procedures. This can be achieved by applying resampling methods, such as bootstrapping and permutations methods. The procedure of Westfall and Young (1993) requires a certain condition that does not always hold in practice (namely, subset pivotality).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The procedures of Romano and Wolf (2005a,b) dispense with this condition and are thus more generally valid.<sup id="cite_ref-Romano_and_Wolf_2005a_15-0" class="reference"><a href="#cite_note-Romano_and_Wolf_2005a-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Romano_and_Wolf_2005b_16-0" class="reference"><a href="#cite_note-Romano_and_Wolf_2005b-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Harmonic_mean_p-value_procedure">Harmonic mean <i>p</i>-value procedure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=14" title="Edit section: Harmonic mean p-value procedure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Harmonic_mean_p-value" title="Harmonic mean p-value">Harmonic mean p-value</a></div> <p>The harmonic mean <i>p</i>-value (HMP) procedure<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> provides a multilevel test that improves on the power of Bonferroni correction by assessing the significance of <i>groups</i> of hypotheses while controlling the strong-sense family-wise error rate. The significance of any subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca7ac47ed296b8a36c32c5df2627fa49fbddb50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\textstyle {\mathcal {R}}}"></span> of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3296968aacae7d971a6d649f6b32b5969ed6d4d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\textstyle m}"></span> tests is assessed by calculating the HMP for the subset,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overset {\circ }{p}}_{\mathcal {R}}={\frac {\sum _{i\in {\mathcal {R}}}w_{i}}{\sum _{i\in {\mathcal {R}}}w_{i}/p_{i}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>∘</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\circ }{p}}_{\mathcal {R}}={\frac {\sum _{i\in {\mathcal {R}}}w_{i}}{\sum _{i\in {\mathcal {R}}}w_{i}/p_{i}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad2afa533157350b5625b87b0817cfb240f1e85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-left: -0.089ex; width:19.192ex; height:6.843ex;" alt="{\displaystyle {\overset {\circ }{p}}_{\mathcal {R}}={\frac {\sum _{i\in {\mathcal {R}}}w_{i}}{\sum _{i\in {\mathcal {R}}}w_{i}/p_{i}}},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle w_{1},\dots ,w_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle w_{1},\dots ,w_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5239f303dc3d33196036b6d5be3011a4998ed7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.236ex; height:2.009ex;" alt="{\textstyle w_{1},\dots ,w_{m}}"></span> are weights that sum to one (i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=1}^{m}w_{i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{m}w_{i}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e024007932f34fb5f1498b85e83d180c4b57ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.466ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{m}w_{i}=1}"></span>). An approximate procedure that controls the strong-sense family-wise error rate at level approximately <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d86dbd6183264b2f8569da1751380b173c7b185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\textstyle \alpha }"></span> rejects the null hypothesis that none of the <i>p</i>-values in subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca7ac47ed296b8a36c32c5df2627fa49fbddb50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\textstyle {\mathcal {R}}}"></span> are significant when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\overset {\circ }{p}}_{\mathcal {R}}\leq \alpha \,w_{\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>∘</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </msub> <mo>≤</mo> <mi>α</mi> <mspace width="thinmathspace" /> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\overset {\circ }{p}}_{\mathcal {R}}\leq \alpha \,w_{\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f532a24bdfa19a0945c66cce84b8e77c6c858c4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.148ex; height:3.509ex;" alt="{\textstyle {\overset {\circ }{p}}_{\mathcal {R}}\leq \alpha \,w_{\mathcal {R}}}"></span><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle w_{\mathcal {R}}=\sum _{i\in {\mathcal {R}}}w_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle w_{\mathcal {R}}=\sum _{i\in {\mathcal {R}}}w_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e64cc0d1bb1bb7d57bd9a24b61a303046109e4d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.983ex; height:3.009ex;" alt="{\textstyle w_{\mathcal {R}}=\sum _{i\in {\mathcal {R}}}w_{i}}"></span>). This approximation is reasonable for small <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d86dbd6183264b2f8569da1751380b173c7b185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\textstyle \alpha }"></span> (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha <0.05}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>α</mi> <mo><</mo> <mn>0.05</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha <0.05}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4d48487aae3280e88321a88a10ac8a3cf96892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.72ex; height:2.176ex;" alt="{\textstyle \alpha <0.05}"></span>) and becomes arbitrarily good as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d86dbd6183264b2f8569da1751380b173c7b185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\textstyle \alpha }"></span> approaches zero. An asymptotically exact test is also available (see <a href="/wiki/Harmonic_mean_p-value" title="Harmonic mean p-value">main article</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Alternative_approaches">Alternative approaches</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=15" title="Edit section: Alternative approaches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/False_discovery_rate" title="False discovery rate">False discovery rate</a></div> <p>FWER control exerts a more stringent control over false discovery compared to false discovery rate (FDR) procedures. FWER control limits the probability of <i>at least one</i> false discovery, whereas FDR control limits (in a loose sense) the expected proportion of false discoveries. Thus, FDR procedures have greater <a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">power</a> at the cost of increased rates of <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">type I</a> errors, i.e., rejecting null hypotheses that are actually true.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>On the other hand, FWER control is less stringent than per-family error rate control, which limits the expected number of errors per family. Because FWER control is concerned with <i>at least one</i> false discovery, unlike per-family error rate control it does not treat multiple simultaneous false discoveries as any worse than one false discovery. The <a href="/wiki/Bonferroni_correction" title="Bonferroni correction">Bonferroni correction</a> is often considered as merely controlling the FWER, but in fact also controls the per-family error rate.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Tukey_(1953)-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tukey_(1953)_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFTukey1953" class="citation book cs1">Tukey, J. W. (1953). <i>The problem of multiple comparisons</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+problem+of+multiple+comparisons&rft.date=1953&rft.aulast=Tukey&rft.aufirst=J.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span> Based on Tukey (1953),</span> </li> <li id="cite_note-Ryan_(1959)-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ryan_(1959)_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ryan_(1959)_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRyan1959" class="citation journal cs1">Ryan, Thomas A. (1959). "Multiple comparison in psychological research". <i>Psychological Bulletin</i>. <b>56</b> (1). American Psychological Association (APA): 26–47. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2Fh0042478">10.1037/h0042478</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1939-1455">1939-1455</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/13623958">13623958</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Bulletin&rft.atitle=Multiple+comparison+in+psychological+research.&rft.volume=56&rft.issue=1&rft.pages=26-47&rft.date=1959&rft.issn=1939-1455&rft_id=info%3Apmid%2F13623958&rft_id=info%3Adoi%2F10.1037%2Fh0042478&rft.aulast=Ryan&rft.aufirst=Thomas+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHochbergTamhane1987" class="citation book cs1">Hochberg, Y.; Tamhane, A. C. (1987). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/multiplecomparis00hoch_295"><i>Multiple Comparison Procedures</i></a></span>. New York: Wiley. p. <a rel="nofollow" class="external text" href="https://archive.org/details/multiplecomparis00hoch_295/page/n25">5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-82222-6" title="Special:BookSources/978-0-471-82222-6"><bdi>978-0-471-82222-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multiple+Comparison+Procedures&rft.place=New+York&rft.pages=5&rft.pub=Wiley&rft.date=1987&rft.isbn=978-0-471-82222-6&rft.aulast=Hochberg&rft.aufirst=Y.&rft.au=Tamhane%2C+A.+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmultiplecomparis00hoch_295&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-Ryan_(1962)-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ryan_(1962)_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRyan1962" class="citation journal cs1">Ryan, T. A. (1962). "The experiment as the unit for computing rates of error". <i>Psychological Bulletin</i>. <b>59</b> (4): 301–305. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2Fh0040562">10.1037/h0040562</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/14495585">14495585</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Bulletin&rft.atitle=The+experiment+as+the+unit+for+computing+rates+of+error&rft.volume=59&rft.issue=4&rft.pages=301-305&rft.date=1962&rft_id=info%3Adoi%2F10.1037%2Fh0040562&rft_id=info%3Apmid%2F14495585&rft.aulast=Ryan&rft.aufirst=T.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-CRC_Press-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-CRC_Press_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-CRC_Press_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDmitrienkoTamhaneBretz2009" class="citation book cs1">Dmitrienko, Alex; Tamhane, Ajit; Bretz, Frank (2009). <i>Multiple Testing Problems in Pharmaceutical Statistics</i> (1 ed.). CRC Press. p. 37. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781584889847" title="Special:BookSources/9781584889847"><bdi>9781584889847</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multiple+Testing+Problems+in+Pharmaceutical+Statistics&rft.pages=37&rft.edition=1&rft.pub=CRC+Press&rft.date=2009&rft.isbn=9781584889847&rft.aulast=Dmitrienko&rft.aufirst=Alex&rft.au=Tamhane%2C+Ajit&rft.au=Bretz%2C+Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-Aickin1996-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Aickin1996_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAickinGensler1996" class="citation journal cs1">Aickin, M; Gensler, H (1996). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1380484">"Adjusting for multiple testing when reporting research results: the Bonferroni vs Holm methods"</a>. <i><a href="/wiki/American_Journal_of_Public_Health" title="American Journal of Public Health">American Journal of Public Health</a></i>. <b>86</b> (5): 726–728. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2105%2Fajph.86.5.726">10.2105/ajph.86.5.726</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1380484">1380484</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/8629727">8629727</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Public+Health&rft.atitle=Adjusting+for+multiple+testing+when+reporting+research+results%3A+the+Bonferroni+vs+Holm+methods&rft.volume=86&rft.issue=5&rft.pages=726-728&rft.date=1996&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1380484%23id-name%3DPMC&rft_id=info%3Apmid%2F8629727&rft_id=info%3Adoi%2F10.2105%2Fajph.86.5.726&rft.aulast=Aickin&rft.aufirst=M&rft.au=Gensler%2C+H&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1380484&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-Hochberg1988-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hochberg1988_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHochberg1988" class="citation journal cs1">Hochberg, Yosef (1988). <a rel="nofollow" class="external text" href="http://www-stat.wharton.upenn.edu/~steele/Courses/956/Resource/MultipleComparision/Hochberg88.pdf">"A Sharper Bonferroni Procedure for Multiple Tests of Significance"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Biometrika" title="Biometrika">Biometrika</a></i>. <b>75</b> (4): 800–802. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F75.4.800">10.1093/biomet/75.4.800</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=A+Sharper+Bonferroni+Procedure+for+Multiple+Tests+of+Significance&rft.volume=75&rft.issue=4&rft.pages=800-802&rft.date=1988&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F75.4.800&rft.aulast=Hochberg&rft.aufirst=Yosef&rft_id=http%3A%2F%2Fwww-stat.wharton.upenn.edu%2F~steele%2FCourses%2F956%2FResource%2FMultipleComparision%2FHochberg88.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimes1986" class="citation journal cs1">Simes, R. J. (1986). "An improved Bonferroni procedure for multiple tests of significance". <i>Biometrika</i>. <b>73</b> (3): 751–754. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F73.3.751">10.1093/biomet/73.3.751</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=An+improved+Bonferroni+procedure+for+multiple+tests+of+significance&rft.volume=73&rft.issue=3&rft.pages=751-754&rft.date=1986&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F73.3.751&rft.aulast=Simes&rft.aufirst=R.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSarkarChang1997" class="citation journal cs1">Sarkar, Sanat K.; Chang, Chung-Kuei (1997). "The Simes method for multiple hypothesis testing with positively dependent test statistics". <i>Journal of the American Statistical Association</i>. <b>92</b> (440): 1601–1608. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1997.10473682">10.1080/01621459.1997.10473682</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Statistical+Association&rft.atitle=The+Simes+method+for+multiple+hypothesis+testing+with+positively+dependent+test+statistics&rft.volume=92&rft.issue=440&rft.pages=1601-1608&rft.date=1997&rft_id=info%3Adoi%2F10.1080%2F01621459.1997.10473682&rft.aulast=Sarkar&rft.aufirst=Sanat+K.&rft.au=Chang%2C+Chung-Kuei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSarkar1998" class="citation journal cs1">Sarkar, Sanat K. (1998). "Some probability inequalities for ordered MTP2 random variables: a proof of the Simes conjecture". <i>The Annals of Statistics</i>. <b>26</b> (2): 494–504. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1028144846">10.1214/aos/1028144846</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=Some+probability+inequalities+for+ordered+MTP2+random+variables%3A+a+proof+of+the+Simes+conjecture&rft.volume=26&rft.issue=2&rft.pages=494-504&rft.date=1998&rft_id=info%3Adoi%2F10.1214%2Faos%2F1028144846&rft.aulast=Sarkar&rft.aufirst=Sanat+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSamuel-Cahn1996" class="citation journal cs1">Samuel-Cahn, Ester (1996). "Is the Simes improved Bonferroni procedure conservative?". <i>Biometrika</i>. <b>83</b> (4): 928–933. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F83.4.928">10.1093/biomet/83.4.928</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=Is+the+Simes+improved+Bonferroni+procedure+conservative%3F&rft.volume=83&rft.issue=4&rft.pages=928-933&rft.date=1996&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F83.4.928&rft.aulast=Samuel-Cahn&rft.aufirst=Ester&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlockSavitsWang2008" class="citation journal cs1">Block, Henry W.; Savits, Thomas H.; Wang, Jie (2008). "Negative dependence and the Simes inequality". <i>Journal of Statistical Planning and Inference</i>. <b>138</b> (12): 4107–4110. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jspi.2008.03.026">10.1016/j.jspi.2008.03.026</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Planning+and+Inference&rft.atitle=Negative+dependence+and+the+Simes+inequality&rft.volume=138&rft.issue=12&rft.pages=4107-4110&rft.date=2008&rft_id=info%3Adoi%2F10.1016%2Fj.jspi.2008.03.026&rft.aulast=Block&rft.aufirst=Henry+W.&rft.au=Savits%2C+Thomas+H.&rft.au=Wang%2C+Jie&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGouTamhane2018" class="citation journal cs1">Gou, Jiangtao; Tamhane, Ajit C. (2018). <a rel="nofollow" class="external text" href="https://www3.stat.sinica.edu.tw/statistica/oldpdf/A28n116.pdf">"Hochberg procedure under negative dependence"</a> <span class="cs1-format">(PDF)</span>. <i>Statistica Sinica</i>. <b>28</b>: 339–362. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5705%2Fss.202016.0306">10.5705/ss.202016.0306</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistica+Sinica&rft.atitle=Hochberg+procedure+under+negative+dependence&rft.volume=28&rft.pages=339-362&rft.date=2018&rft_id=info%3Adoi%2F10.5705%2Fss.202016.0306&rft.aulast=Gou&rft.aufirst=Jiangtao&rft.au=Tamhane%2C+Ajit+C.&rft_id=https%3A%2F%2Fwww3.stat.sinica.edu.tw%2Fstatistica%2Foldpdf%2FA28n116.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWestfallYoung1993" class="citation book cs1">Westfall, P. H.; Young, S. S. (1993). <i>Resampling-Based Multiple Testing: Examples and Methods for p-Value Adjustment</i>. New York: John Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-55761-6" title="Special:BookSources/978-0-471-55761-6"><bdi>978-0-471-55761-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Resampling-Based+Multiple+Testing%3A+Examples+and+Methods+for+p-Value+Adjustment&rft.place=New+York&rft.pub=John+Wiley&rft.date=1993&rft.isbn=978-0-471-55761-6&rft.aulast=Westfall&rft.aufirst=P.+H.&rft.au=Young%2C+S.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-Romano_and_Wolf_2005a-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Romano_and_Wolf_2005a_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRomanoWolf2005a" class="citation journal cs1">Romano, J.P.; Wolf, M. (2005a). "Exact and approximate stepdown methods for multiple hypothesis testing". <i><a href="/wiki/Journal_of_the_American_Statistical_Association" title="Journal of the American Statistical Association">Journal of the American Statistical Association</a></i>. <b>100</b> (469): 94–108. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1198%2F016214504000000539">10.1198/016214504000000539</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10230%2F576">10230/576</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:219594470">219594470</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Statistical+Association&rft.atitle=Exact+and+approximate+stepdown+methods+for+multiple+hypothesis+testing&rft.volume=100&rft.issue=469&rft.pages=94-108&rft.date=2005&rft_id=info%3Ahdl%2F10230%2F576&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A219594470%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1198%2F016214504000000539&rft.aulast=Romano&rft.aufirst=J.P.&rft.au=Wolf%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-Romano_and_Wolf_2005b-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Romano_and_Wolf_2005b_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRomanoWolf2005b" class="citation journal cs1">Romano, J.P.; Wolf, M. (2005b). "Stepwise multiple testing as formalized data snooping". <i><a href="/wiki/Econometrica" title="Econometrica">Econometrica</a></i>. <b>73</b> (4): 1237–1282. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.198.2473">10.1.1.198.2473</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1468-0262.2005.00615.x">10.1111/j.1468-0262.2005.00615.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Econometrica&rft.atitle=Stepwise+multiple+testing+as+formalized+data+snooping&rft.volume=73&rft.issue=4&rft.pages=1237-1282&rft.date=2005&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.198.2473%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1111%2Fj.1468-0262.2005.00615.x&rft.aulast=Romano&rft.aufirst=J.P.&rft.au=Wolf%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodI1958" class="citation journal cs1">Good, I J (1958). "Significance tests in parallel and in series". <i>Journal of the American Statistical Association</i>. <b>53</b> (284): 799–813. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1958.10501480">10.1080/01621459.1958.10501480</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2281953">2281953</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Statistical+Association&rft.atitle=Significance+tests+in+parallel+and+in+series&rft.volume=53&rft.issue=284&rft.pages=799-813&rft.date=1958&rft_id=info%3Adoi%2F10.1080%2F01621459.1958.10501480&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2281953%23id-name%3DJSTOR&rft.au=Good&rft.au=I%2C+J&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilsonD2019" class="citation journal cs1">Wilson, D J (2019). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6347718">"The harmonic mean <i>p</i>-value for combining dependent tests"</a>. <i>Proceedings of the National Academy of Sciences USA</i>. <b>116</b> (4): 1195–1200. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019PNAS..116.1195W">2019PNAS..116.1195W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.1814092116">10.1073/pnas.1814092116</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6347718">6347718</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/30610179">30610179</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences+USA&rft.atitle=The+harmonic+mean+p-value+for+combining+dependent+tests&rft.volume=116&rft.issue=4&rft.pages=1195-1200&rft.date=2019&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6347718%23id-name%3DPMC&rft_id=info%3Apmid%2F30610179&rft_id=info%3Adoi%2F10.1073%2Fpnas.1814092116&rft_id=info%3Abibcode%2F2019PNAS..116.1195W&rft.au=Wilson&rft.au=D%2C+J&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6347718&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSciences2019" class="citation journal cs1">Sciences, National Academy of (2019-10-22). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6815184">"Correction for Wilson, The harmonic mean p-value for combining dependent tests"</a>. <i>Proceedings of the National Academy of Sciences</i>. <b>116</b> (43): 21948. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019PNAS..11621948.">2019PNAS..11621948.</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.1914128116">10.1073/pnas.1914128116</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6815184">6815184</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/31591234">31591234</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences&rft.atitle=Correction+for+Wilson%2C+The+harmonic+mean+p-value+for+combining+dependent+tests&rft.volume=116&rft.issue=43&rft.pages=21948&rft.date=2019-10-22&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6815184%23id-name%3DPMC&rft_id=info%3Apmid%2F31591234&rft_id=info%3Adoi%2F10.1073%2Fpnas.1914128116&rft_id=info%3Abibcode%2F2019PNAS..11621948.&rft.aulast=Sciences&rft.aufirst=National+Academy+of&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6815184&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShaffer1995" class="citation journal cs1">Shaffer, J. P. (1995). "Multiple hypothesis testing". <i><a href="/wiki/Annual_Review_of_Psychology" title="Annual Review of Psychology">Annual Review of Psychology</a></i>. <b>46</b>: 561–584. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.ps.46.020195.003021">10.1146/annurev.ps.46.020195.003021</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F142950">10338.dmlcz/142950</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Psychology&rft.atitle=Multiple+hypothesis+testing&rft.volume=46&rft.pages=561-584&rft.date=1995&rft_id=info%3Ahdl%2F10338.dmlcz%2F142950&rft_id=info%3Adoi%2F10.1146%2Fannurev.ps.46.020195.003021&rft.aulast=Shaffer&rft.aufirst=J.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrane2015" class="citation journal cs1">Frane, Andrew (2015). "Are per-family Type I error rates relevant in social and behavioral science?". <i>Journal of Modern Applied Statistical Methods</i>. <b>14</b> (1): 12–23. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.22237%2Fjmasm%2F1430453040">10.22237/jmasm/1430453040</a></span> (inactive 2024-11-02).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Modern+Applied+Statistical+Methods&rft.atitle=Are+per-family+Type+I+error+rates+relevant+in+social+and+behavioral+science%3F&rft.volume=14&rft.issue=1&rft.pages=12-23&rft.date=2015&rft_id=info%3Adoi%2F10.22237%2Fjmasm%2F1430453040&rft.aulast=Frane&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFamily-wise+error+rate" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: DOI inactive as of November 2024 (<a href="/wiki/Category:CS1_maint:_DOI_inactive_as_of_November_2024" title="Category:CS1 maint: DOI inactive as of November 2024">link</a>)</span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Family-wise_error_rate&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://riffyn.com/riffyn-blog/2017/10/29/family-wise-error-rate">Understanding Family Wise Error Rate</a> - blog post including its utility relative to False Discovery Rate</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Family-wise_error_rate&oldid=1258017002">https://en.wikipedia.org/w/index.php?title=Family-wise_error_rate&oldid=1258017002</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Statistical_hypothesis_testing" title="Category:Statistical hypothesis testing">Statistical hypothesis testing</a></li><li><a href="/wiki/Category:Multiple_comparisons" title="Category:Multiple comparisons">Multiple comparisons</a></li><li><a href="/wiki/Category:Rates" title="Category:Rates">Rates</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_DOI_inactive_as_of_November_2024" title="Category:CS1 maint: DOI inactive as of November 2024">CS1 maint: DOI inactive as of November 2024</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_June_2016" title="Category:Articles needing additional references from June 2016">Articles needing additional references from June 2016</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_June_2016" title="Category:Articles with unsourced statements from June 2016">Articles with unsourced statements from June 2016</a></li><li><a href="/wiki/Category:Articles_to_be_expanded_from_February_2013" title="Category:Articles to be expanded from February 2013">Articles to be expanded from February 2013</a></li><li><a href="/wiki/Category:All_articles_to_be_expanded" title="Category:All articles to be expanded">All articles to be expanded</a></li><li><a href="/wiki/Category:Articles_with_empty_sections_from_February_2013" title="Category:Articles with empty sections from February 2013">Articles with empty sections from February 2013</a></li><li><a href="/wiki/Category:All_articles_with_empty_sections" title="Category:All articles with empty sections">All articles with empty sections</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 17 November 2024, at 18:37<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Family-wise_error_rate&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.canary-84779d6bf6-vqkvc","wgBackendResponseTime":164,"wgPageParseReport":{"limitreport":{"cputime":"0.476","walltime":"0.635","ppvisitednodes":{"value":3462,"limit":1000000},"postexpandincludesize":{"value":82499,"limit":2097152},"templateargumentsize":{"value":5641,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":17,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":88303,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 458.153 1 -total"," 44.47% 203.741 1 Template:Reflist"," 20.31% 93.064 4 Template:Cite_book"," 17.29% 79.200 17 Template:Cite_journal"," 14.28% 65.411 1 Template:Short_description"," 13.21% 60.518 8 Template:Citation_needed"," 11.79% 54.018 1 Template:More_citations_needed"," 11.49% 52.636 2 Template:Ambox"," 10.94% 50.099 8 Template:Fix"," 8.91% 40.839 2 Template:Pagetype"]},"scribunto":{"limitreport-timeusage":{"value":"0.268","limit":"10.000"},"limitreport-memusage":{"value":6629778,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-bnzx6","timestamp":"20241122145301","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Family-wise error rate","url":"https:\/\/en.wikipedia.org\/wiki\/Family-wise_error_rate","sameAs":"http:\/\/www.wikidata.org\/entity\/Q5433418","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q5433418","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2006-04-04T03:32:55Z","dateModified":"2024-11-17T18:37:41Z","headline":"probability of making one or more false discoveries among all the hypotheses when performing multiple hypotheses tests"}</script> </body> </html>