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<span>Approaches</span> </div> </a> <ul id="toc-Approaches-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistical_models_for_aggregate_data" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Statistical_models_for_aggregate_data"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Statistical models for aggregate data</span> </div> </a> <ul id="toc-Statistical_models_for_aggregate_data-sublist" class="vector-toc-list"> <li id="toc-Fixed_effect_model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fixed_effect_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Fixed effect model</span> </div> </a> <ul id="toc-Fixed_effect_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Random_effects_model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Random_effects_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Random effects model</span> </div> </a> <ul id="toc-Random_effects_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quality_effects_model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quality_effects_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Quality effects model</span> </div> </a> <ul id="toc-Quality_effects_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Network_meta-analysis_methods" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Network_meta-analysis_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4</span> <span>Network meta-analysis methods</span> </div> </a> <ul id="toc-Network_meta-analysis_methods-sublist" class="vector-toc-list"> <li id="toc-Bayesian_framework" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Bayesian_framework"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4.1</span> <span>Bayesian framework</span> </div> </a> <ul id="toc-Bayesian_framework-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frequentist_multivariate_framework" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Frequentist_multivariate_framework"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4.2</span> <span>Frequentist multivariate framework</span> </div> </a> <ul id="toc-Frequentist_multivariate_framework-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalized_pairwise_modelling_framework" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Generalized_pairwise_modelling_framework"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4.3</span> <span>Generalized pairwise modelling framework</span> </div> </a> <ul id="toc-Generalized_pairwise_modelling_framework-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tailored_meta-analysis" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Tailored_meta-analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4.4</span> <span>Tailored meta-analysis</span> </div> </a> <ul id="toc-Tailored_meta-analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Aggregating_IPD_and_AD" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Aggregating_IPD_and_AD"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.5</span> <span>Aggregating IPD and AD</span> </div> </a> <ul id="toc-Aggregating_IPD_and_AD-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Validation_of_meta-analysis_results" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Validation_of_meta-analysis_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Validation of meta-analysis results</span> </div> </a> <ul id="toc-Validation_of_meta-analysis_results-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Challenges" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Challenges"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Challenges</span> </div> </a> <button aria-controls="toc-Challenges-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 Challenges subsection</span> </button> <ul id="toc-Challenges-sublist" class="vector-toc-list"> <li id="toc-Publication_bias:_the_file_drawer_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Publication_bias:_the_file_drawer_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Publication bias: the file drawer problem</span> </div> </a> <ul id="toc-Publication_bias:_the_file_drawer_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_related_to_studies_not_reporting_non-statistically_significant_effects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problems_related_to_studies_not_reporting_non-statistically_significant_effects"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Problems related to studies not reporting non-statistically significant effects</span> </div> </a> <ul id="toc-Problems_related_to_studies_not_reporting_non-statistically_significant_effects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_related_to_the_statistical_approach" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problems_related_to_the_statistical_approach"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Problems related to the statistical approach</span> </div> </a> <ul id="toc-Problems_related_to_the_statistical_approach-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_arising_from_agenda-driven_bias" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problems_arising_from_agenda-driven_bias"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Problems arising from agenda-driven bias</span> </div> </a> <ul id="toc-Problems_arising_from_agenda-driven_bias-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparability_and_validity_of_included_studies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Comparability_and_validity_of_included_studies"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Comparability and validity of included studies</span> </div> </a> <ul id="toc-Comparability_and_validity_of_included_studies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weak_inclusion_standards_lead_to_misleading_conclusions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weak_inclusion_standards_lead_to_misleading_conclusions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Weak inclusion standards lead to misleading conclusions</span> </div> </a> <ul id="toc-Weak_inclusion_standards_lead_to_misleading_conclusions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications_in_modern_science" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_in_modern_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications in modern science</span> </div> </a> <ul id="toc-Applications_in_modern_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-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">8</span> <span>References</span> </div> </a> <ul id="toc-References-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">Meta-analysis</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 37 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-37" 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">37 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%AA%D9%84%D9%88%D9%8A" title="تحليل تلوي – Arabic" lang="ar" hreflang="ar" data-title="تحليل تلوي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Metaan%C3%A0lisi" title="Metaanàlisi – Catalan" lang="ca" hreflang="ca" data-title="Metaanàlisi" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Metaanal%C3%BDza" title="Metaanalýza – Czech" lang="cs" hreflang="cs" data-title="Metaanalýza" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Metaanalyse" title="Metaanalyse – Danish" lang="da" hreflang="da" data-title="Metaanalyse" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Metaanalyse" title="Metaanalyse – German" lang="de" hreflang="de" data-title="Metaanalyse" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Metaanal%C3%BC%C3%BCs" title="Metaanalüüs – Estonian" lang="et" hreflang="et" data-title="Metaanalüüs" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Metaan%C3%A1lisis" title="Metaanálisis – Spanish" lang="es" hreflang="es" data-title="Metaanálisis" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Metaanalisi" title="Metaanalisi – Basque" lang="eu" hreflang="eu" data-title="Metaanalisi" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B1%D8%A7%D8%AA%D8%AD%D9%84%DB%8C%D9%84" title="فراتحلیل – Persian" lang="fa" hreflang="fa" data-title="فراتحلیل" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9ta-analyse" title="Méta-analyse – French" lang="fr" hreflang="fr" data-title="Méta-analyse" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Metaan%C3%A1lise" title="Metaanálise – Galician" lang="gl" hreflang="gl" data-title="Metaanálise" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A9%94%ED%83%80%EB%B6%84%EC%84%9D" title="메타분석 – Korean" lang="ko" hreflang="ko" data-title="메타분석" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A5%D5%BF%D5%A1%D5%A1%D5%B6%D5%A1%D5%AC%D5%AB%D5%A6" title="Մետաանալիզ – Armenian" lang="hy" hreflang="hy" data-title="Մետաանալիզ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Metaanalisis" title="Metaanalisis – Indonesian" lang="id" hreflang="id" data-title="Metaanalisis" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Meta-analisi" title="Meta-analisi – Italian" lang="it" hreflang="it" data-title="Meta-analisi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%98%D7%90-%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94" title="מטא-אנליזה – Hebrew" lang="he" hreflang="he" data-title="מטא-אנליזה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Metaanal%C3%AEz" title="Metaanalîz – Kurdish" lang="ku" hreflang="ku" data-title="Metaanalîz" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Metaanal%C4%ABze" title="Metaanalīze – Latvian" lang="lv" hreflang="lv" data-title="Metaanalīze" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Metaanal%C3%ADzis" title="Metaanalízis – Hungarian" lang="hu" hreflang="hu" data-title="Metaanalízis" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%B0-%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Мета-анализа – Macedonian" lang="mk" hreflang="mk" data-title="Мета-анализа" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Meta-analyse" title="Meta-analyse – Dutch" lang="nl" hreflang="nl" data-title="Meta-analyse" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A1%E3%82%BF%E3%82%A2%E3%83%8A%E3%83%AA%E3%82%B7%E3%82%B9" title="メタアナリシス – Japanese" lang="ja" hreflang="ja" data-title="メタアナリシス" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Metaanalyse" title="Metaanalyse – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Metaanalyse" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Metaanaliza" title="Metaanaliza – Polish" lang="pl" hreflang="pl" data-title="Metaanaliza" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Metan%C3%A1lise" title="Metanálise – Portuguese" lang="pt" hreflang="pt" data-title="Metanálise" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%B0-%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Мета-анализ – Russian" lang="ru" hreflang="ru" data-title="Мета-анализ" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Metaanaliza" title="Metaanaliza – Slovenian" lang="sl" hreflang="sl" data-title="Metaanaliza" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Metaanaliza" title="Metaanaliza – Serbian" lang="sr" hreflang="sr" data-title="Metaanaliza" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Metaanaliza" title="Metaanaliza – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Metaanaliza" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Meta-analyysi" title="Meta-analyysi – Finnish" lang="fi" hreflang="fi" data-title="Meta-analyysi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Metaanalys" title="Metaanalys – Swedish" lang="sv" hreflang="sv" data-title="Metaanalys" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%A7%E0%B8%B4%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B2%E0%B8%B0%E0%B8%AB%E0%B9%8C%E0%B8%AD%E0%B8%A0%E0%B8%B4%E0%B8%A1%E0%B8%B2%E0%B8%99" title="การวิเคราะห์อภิมาน – Thai" lang="th" hreflang="th" data-title="การวิเคราะห์อภิมาน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Meta-analiz" title="Meta-analiz – Turkish" lang="tr" hreflang="tr" data-title="Meta-analiz" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%B0%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Метааналіз – Ukrainian" lang="uk" hreflang="uk" data-title="Метааналіз" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_t%C3%ADch_t%E1%BB%95ng_h%E1%BB%A3p" title="Phân tích tổng hợp – Vietnamese" lang="vi" hreflang="vi" data-title="Phân tích tổng hợp" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%85%83%E5%88%86%E6%9E%90" title="元分析 – Cantonese" lang="yue" hreflang="yue" data-title="元分析" data-language-autonym="粵語" data-language-local-name="Cantonese" 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.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">For the process in historical linguistics known as metanalysis, see <a href="/wiki/Rebracketing" title="Rebracketing">Rebracketing</a>.</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output 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.sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Research" title="Category:Research">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Research" title="Research">Research</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size mw-image-border" typeof="mw:File/Frameless"><a href="/wiki/File:Humanit%C3%A9s_Num%C3%A9riques.JPG" class="mw-file-description"><img alt="A laptop computer next to archival materials" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Humanit%C3%A9s_Num%C3%A9riques.JPG/220px-Humanit%C3%A9s_Num%C3%A9riques.JPG" decoding="async" 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href="/wiki/Category:GIS_software" title="Category:GIS software">Geographic information system software</a></li> <li><a href="/wiki/Category:Library_and_information_science_software" title="Category:Library and information science software">Library and information science software</a> <ul><li><a href="/wiki/Category:Bibliometrics_software" title="Category:Bibliometrics software">Bibliometrics</a></li> <li><a href="/wiki/Category:Reference_management_software" title="Category:Reference management software">Reference management</a></li></ul></li> <li><a href="/wiki/Category:Science_software" title="Category:Science software">Science software</a> <ul><li><a href="/wiki/Category:QDA_software" title="Category:QDA software">Qualitative data analysis</a></li> <li><a href="/wiki/Category:Scientific_simulation_software" title="Category:Scientific simulation software">Simulation</a></li> <li><a href="/wiki/Category:Statistical_software" title="Category:Statistical software">Statistics</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="padding-top: 0.1em; font-weight: bold;"> <a href="/wiki/Portal:Philosophy" title="Portal:Philosophy">Philosophy portal</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Research_sidebar" title="Template:Research sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Research_sidebar" title="Template talk:Research sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Research_sidebar" title="Special:EditPage/Template:Research sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table><p><b>Meta-analysis</b> is a method of synthesis of quantitative data from multiple independent studies addressing a common research question. An important part of this method involves computing a combined effect size across all of the studies. As such, this statistical approach involves extracting <a href="/wiki/Effect_size" title="Effect size">effect sizes</a> and variance measures from various studies. By combining these effect sizes the statistical power is improved and can resolve uncertainties or discrepancies found in individual studies. Meta-analyses are integral in supporting research grant proposals, shaping treatment guidelines, and influencing health policies. They are also pivotal in summarizing existing research to guide future studies, thereby cementing their role as a fundamental methodology in <a href="/wiki/Metascience" title="Metascience">metascience</a>. Meta-analyses are often, but not always, important components of a <a href="/wiki/Systematic_review" title="Systematic review">systematic review</a>. </p><meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term "meta-analysis" was coined in 1976 by the statistician <a href="/wiki/Gene_V._Glass" title="Gene V. Glass">Gene Glass</a>,<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_2-0" class="reference"><a href="#cite_note-:2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> who stated <i>"Meta-analysis refers to the analysis of analyses"</i>.<sup id="cite_ref-Glass_3-0" class="reference"><a href="#cite_note-Glass-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Glass's work aimed at describing aggregated measures of relationships and effects.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> While Glass is credited with authoring the first modern meta-analysis, a paper published in 1904 by the statistician <a href="/wiki/Karl_Pearson" title="Karl Pearson">Karl Pearson</a> in the <i><a href="/wiki/British_Medical_Journal" class="mw-redirect" title="British Medical Journal">British Medical Journal</a></i><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> collated data from several studies of typhoid inoculation and is seen as the first time a meta-analytic approach was used to aggregate the outcomes of multiple clinical studies.<sup id="cite_ref-Nordmann2012_6-0" class="reference"><a href="#cite_note-Nordmann2012-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Numerous other examples of early meta-analyses can be found including occupational aptitude testing,<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><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> and agriculture.<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> </p><p>The first model meta-analysis was published in 1978 on the effectiveness of psychotherapy outcomes by <a href="/wiki/Mary_Lee_Smith" title="Mary Lee Smith">Mary Lee Smith</a> and <a href="/wiki/Gene_V._Glass" title="Gene V. Glass">Gene Glass</a>.<sup id="cite_ref-:2_2-1" class="reference"><a href="#cite_note-:2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><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> After publication of their article there was pushback on the usefulness and validity of meta-analysis as a tool for evidence synthesis. The first example of this was by <a href="/wiki/Hans_Eysenck" title="Hans Eysenck">Han Eysenck</a> who in a 1978 article in response to the work done by Mary Lee Smith and Gene Glass called meta-analysis an "exercise in mega-silliness".<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><sup id="cite_ref-:4_13-0" class="reference"><a href="#cite_note-:4-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Later Eysenck would refer to meta-analysis as "statistical alchemy".<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> Despite these criticisms the use of meta-analysis has only grown since its modern introduction. By 1991 there were 334 published meta-analyses;<sup id="cite_ref-:4_13-1" class="reference"><a href="#cite_note-:4-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> this number grew to 9,135 by 2014.<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>The field of meta-analysis expanded greatly since the 1970s and touches multiple disciplines including psychology, medicine, and ecology.<sup id="cite_ref-:0_1-2" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Further the more recent creation of evidence synthesis communities has increased the cross pollination of ideas, methods, and the creation of software tools across disciplines.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><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> </p> <div class="mw-heading mw-heading2"><h2 id="Literature_Search">Literature Search</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=2" title="Edit section: Literature Search"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One of the most important steps of a meta-analysis is data collection. For an efficient database search, appropriate keywords and search limits need to be identified.<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> The use of Boolean operators and search limits can assist the literature search.<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><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> A number of databases are available (e.g., PubMed, Embase, PsychInfo), however, it is up to the researcher to choose the most appropriate sources for their research area.<sup id="cite_ref-:1_22-0" class="reference"><a href="#cite_note-:1-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Indeed, many scientists use duplicate search terms within two or more databases to cover multiple sources.<sup id="cite_ref-:6_23-0" class="reference"><a href="#cite_note-:6-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The reference lists of eligible studies can also be searched for eligible studies (i.e., snowballing).<sup id="cite_ref-:7_24-0" class="reference"><a href="#cite_note-:7-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> The initial search may return a large volume of studies.<sup id="cite_ref-:7_24-1" class="reference"><a href="#cite_note-:7-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Quite often, the abstract or the title of the manuscript reveals that the study is not eligible for inclusion, based on the pre-specified criteria.<sup id="cite_ref-:1_22-1" class="reference"><a href="#cite_note-:1-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> These studies can be discarded. However, if it appears that the study may be eligible (or even if there is some doubt) the full paper can be retained for closer inspection. The references lists of eligible articles can also be searched for any relevant articles.<sup id="cite_ref-:6_23-1" class="reference"><a href="#cite_note-:6-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> These search results need to be detailed in a PRIMSA flow diagram<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> which details the flow of information through all stages of the review. Thus, it is important to note how many studies were returned after using the specified search terms and how many of these studies were discarded, and for what reason.<sup id="cite_ref-:1_22-2" class="reference"><a href="#cite_note-:1-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> The search terms and strategy should be specific enough for a reader to reproduce the search.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> The date range of studies, along with the date (or date period) the search was conducted should also be provided.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p>A data collection form provides a standardized means of collecting data from eligible studies.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> For a meta-analysis of correlational data, effect size information is usually collected as Pearson's <i>r</i> statistic.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Partial correlations are often reported in research, however, these may inflate relationships in comparison to zero-order correlations.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Moreover, the partialed out variables will likely vary from study-to-study. As a consequence, many meta-analyses exclude partial correlations from their analysis.<sup id="cite_ref-:1_22-3" class="reference"><a href="#cite_note-:1-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> As a final resort, plot digitizers can be used to scrape data points from scatterplots (if available) for the calculation of Pearson's <i>r</i>.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Data reporting important study characteristics that may moderate effects, such as the mean age of participants, should also be collected.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> A measure of study quality can also be included in these forms to assess the quality of evidence from each study.<sup id="cite_ref-:3_36-0" class="reference"><a href="#cite_note-:3-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> There are more than 80 tools available to assess the quality and risk of bias in observational studies reflecting the diversity of research approaches between fields.<sup id="cite_ref-:3_36-1" class="reference"><a href="#cite_note-:3-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> These tools usually include an assessment of how dependent variables were measured, appropriate selection of participants, and appropriate control for confounding factors. Other quality measures that may be more relevant for correlational studies include sample size, psychometric properties, and reporting of methods.<sup id="cite_ref-:1_22-4" class="reference"><a href="#cite_note-:1-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>A final consideration is whether to include studies from the gray literature,<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> which is defined as research that has not been formally published.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> This type of literature includes conference abstracts,<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> dissertations,<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> and pre-prints.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> While the inclusion of gray literature reduces the risk of publication bias, the methodological quality of the work is often (but not always) lower than formally published work.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> Reports from conference proceedings, which are the most common source of gray literature,<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> are poorly reported<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> and data in the subsequent publication is often inconsistent, with differences observed in almost 20% of published studies.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Methods_and_assumptions">Methods and assumptions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=3" title="Edit section: Methods and assumptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Approaches">Approaches</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=4" title="Edit section: Approaches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general, two types of evidence can be distinguished when performing a meta-analysis: <a href="/wiki/Individual_participant_data" title="Individual participant data">individual participant data</a> (IPD), and aggregate data (AD).<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> The aggregate data can be direct or indirect. </p><p>AD is more commonly available (e.g. from the literature) and typically represents summary estimates such as odds ratios<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> or relative risks.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> This can be directly synthesized across conceptually similar studies using several approaches. On the other hand, indirect aggregate data measures the effect of two treatments that were each compared against a similar control group in a meta-analysis. For example, if treatment A and treatment B were directly compared vs placebo in separate meta-analyses, we can use these two pooled results to get an estimate of the effects of A vs B in an indirect comparison as effect A vs Placebo minus effect B vs Placebo. </p><p>IPD evidence represents raw data as collected by the study centers. This distinction has raised the need for different meta-analytic methods when evidence synthesis is desired, and has led to the development of one-stage and two-stage methods.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> In one-stage methods the IPD from all studies are modeled simultaneously whilst accounting for the clustering of participants within studies. Two-stage methods first compute summary statistics for AD from each study and then calculate overall statistics as a weighted average of the study statistics. By reducing IPD to AD, two-stage methods can also be applied when IPD is available; this makes them an appealing choice when performing a meta-analysis. Although it is conventionally believed that one-stage and two-stage methods yield similar results, recent studies have shown that they may occasionally lead to different conclusions.<sup id="cite_ref-pmid23585842_53-0" class="reference"><a href="#cite_note-pmid23585842-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Statistical_models_for_aggregate_data">Statistical models for aggregate data</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=5" title="Edit section: Statistical models for aggregate data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Fixed_effect_model">Fixed effect model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=6" title="Edit section: Fixed effect model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Generic_forest_plot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Generic_forest_plot.png/220px-Generic_forest_plot.png" decoding="async" width="220" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Generic_forest_plot.png/330px-Generic_forest_plot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Generic_forest_plot.png/440px-Generic_forest_plot.png 2x" data-file-width="1125" data-file-height="727" /></a><figcaption>Forest Plot of Effect Sizes</figcaption></figure> <p>The fixed effect model provides a weighted average of a series of study estimates.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> The inverse of the estimates' variance is commonly used as study weight, so that larger studies tend to contribute more than smaller studies to the weighted average.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> Consequently, when studies within a meta-analysis are dominated by a very large study, the findings from smaller studies are practically ignored.<sup id="cite_ref-pmid11884693_57-0" class="reference"><a href="#cite_note-pmid11884693-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> Most importantly, the fixed effects model assumes that all included studies investigate the same population, use the same variable and outcome definitions, etc.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> This assumption is typically unrealistic as research is often prone to several sources of <a href="/wiki/Study_heterogeneity" title="Study heterogeneity">heterogeneity</a>.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p><p>If we start with a collection of independent effect size estimates, each estimate a corresponding effect size <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 i=1,\ldots ,k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\ldots ,k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dd4cb0548150c5ac4440b8a1e3b4f6218bdfd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.453ex; height:2.509ex;" alt="{\displaystyle i=1,\ldots ,k}"></span> we can assume 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="{\textstyle y_{i}=\theta _{i}+e_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle y_{i}=\theta _{i}+e_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b9c9c8ae5f50ac030b26b16281a254f44ec6e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.651ex; height:2.509ex;" alt="{\textstyle y_{i}=\theta _{i}+e_{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="{\displaystyle y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d30d30b6c2dbe4d6f150d699de040937ecc95f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}"></span> denotes the observed effect in 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="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th study, <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 \theta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302b19204ed378e99ff4575341a67eebdbe5a555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.509ex;" alt="{\displaystyle \theta _{i}}"></span> the corresponding (unknown) true effect, <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 e_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}"></span> is the sampling error, 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 e_{i}\thicksim N(0,v_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo class="MJX-variant">∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}\thicksim N(0,v_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6feb8d0820346aaf0ae1bf1c5de47d206bdb829f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.978ex; height:2.843ex;" alt="{\displaystyle e_{i}\thicksim N(0,v_{i})}"></span>. Therefore, 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="{\displaystyle y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d30d30b6c2dbe4d6f150d699de040937ecc95f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}"></span>’s are assumed to be unbiased and <a href="/wiki/Normal_distribution" title="Normal distribution">normally distributed</a> estimates of their corresponding true effects. The sampling variances (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="{\displaystyle v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}"></span> values) are assumed to be known.<sup id="cite_ref-:5_61-0" class="reference"><a href="#cite_note-:5-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Random_effects_model">Random effects model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=7" title="Edit section: Random effects model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most meta-analyses are based on sets of studies that are not exactly identical in their methods and/or the characteristics of the included samples.<sup id="cite_ref-:5_61-1" class="reference"><a href="#cite_note-:5-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> Differences in the methods and sample characteristics may introduce variability (“heterogeneity”) among the true effects.<sup id="cite_ref-:5_61-2" class="reference"><a href="#cite_note-:5-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> One way to model the heterogeneity is to treat it as purely random. The weight that is applied in this process of weighted averaging with a random effects meta-analysis is achieved in two steps:<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Step 1: Inverse variance weighting</li> <li>Step 2: Un-weighting of this inverse variance weighting by applying a random effects variance component (REVC) that is simply derived from the extent of variability of the effect sizes of the underlying studies.</li></ol> <p>This means that the greater this variability in effect sizes (otherwise known as <a href="/wiki/Study_heterogeneity" title="Study heterogeneity">heterogeneity</a>), the greater the un-weighting and this can reach a point when the random effects meta-analysis result becomes simply the un-weighted average effect size across the studies. At the other extreme, when all effect sizes are similar (or variability does not exceed sampling error), no REVC is applied and the random effects meta-analysis defaults to simply a fixed effect meta-analysis (only inverse variance weighting). </p><p>The extent of this reversal is solely dependent on two factors:<sup id="cite_ref-ReferenceA_64-0" class="reference"><a href="#cite_note-ReferenceA-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Heterogeneity of precision</li> <li>Heterogeneity of effect size</li></ol> <p>Since neither of these factors automatically indicates a faulty larger study or more reliable smaller studies, the re-distribution of weights under this model will not bear a relationship to what these studies actually might offer. Indeed, it has been demonstrated that redistribution of weights is simply in one direction from larger to smaller studies as heterogeneity increases until eventually all studies have equal weight and no more redistribution is possible.<sup id="cite_ref-ReferenceA_64-1" class="reference"><a href="#cite_note-ReferenceA-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> Another issue with the random effects model is that the most commonly used confidence intervals generally do not retain their coverage probability above the specified nominal level and thus substantially underestimate the statistical error and are potentially overconfident in their conclusions.<sup id="cite_ref-Brockwell2001_65-0" class="reference"><a href="#cite_note-Brockwell2001-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Noma2011_66-0" class="reference"><a href="#cite_note-Noma2011-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> Several fixes have been suggested<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> but the debate continues on.<sup id="cite_ref-Noma2011_66-1" class="reference"><a href="#cite_note-Noma2011-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-pmid19016302_69-0" class="reference"><a href="#cite_note-pmid19016302-69"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> A further concern is that the average treatment effect can sometimes be even less conservative compared to the fixed effect model<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> and therefore misleading in practice. One interpretational fix that has been suggested is to create a prediction interval around the random effects estimate to portray the range of possible effects in practice.<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> However, an assumption behind the calculation of such a prediction interval is that trials are considered more or less homogeneous entities and that included patient populations and comparator treatments should be considered exchangeable<sup id="cite_ref-pmid23494781_72-0" class="reference"><a href="#cite_note-pmid23494781-72"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> and this is usually unattainable in practice. </p><p>There are many methods used to estimate between studies variance with restricted maximum likelihood estimator being the least prone to bias and one of the most commonly used.<sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> Several advanced iterative techniques for computing the between studies variance exist including both maximum likelihood and restricted maximum likelihood methods and random effects models using these methods can be run with multiple software platforms including Excel,<sup id="cite_ref-Manual_74-0" class="reference"><a href="#cite_note-Manual-74"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> Stata,<sup id="cite_ref-metaan_75-0" class="reference"><a href="#cite_note-metaan-75"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> SPSS,<sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> and R.<sup id="cite_ref-:5_61-3" class="reference"><a href="#cite_note-:5-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> </p><p>Most meta-analyses include between 2 and 4 studies and such a sample is more often than not inadequate to accurately estimate <a href="/wiki/Study_heterogeneity" title="Study heterogeneity">heterogeneity</a>. Thus it appears that in small meta-analyses, an incorrect zero between study variance estimate is obtained, leading to a false homogeneity assumption. Overall, it appears that heterogeneity is being consistently underestimated in meta-analyses and sensitivity analyses in which high heterogeneity levels are assumed could be informative.<sup id="cite_ref-KontopantelisP1_77-0" class="reference"><a href="#cite_note-KontopantelisP1-77"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup> These random effects models and software packages mentioned above relate to study-aggregate meta-analyses and researchers wishing to conduct individual patient data (IPD) meta-analyses need to consider mixed-effects modelling approaches.<sup id="cite_ref-ipdforest_78-0" class="reference"><a href="#cite_note-ipdforest-78"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup>/ </p> <div class="mw-heading mw-heading4"><h4 id="Quality_effects_model">Quality effects model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=8" title="Edit section: Quality effects model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Doi and Thalib originally introduced the quality effects model.<sup id="cite_ref-Doi_2008_79-0" class="reference"><a href="#cite_note-Doi_2008-79"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> They<sup id="cite_ref-80" class="reference"><a href="#cite_note-80"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> introduced a new approach to adjustment for inter-study variability by incorporating the contribution of variance due to a relevant component (quality) in addition to the contribution of variance due to random error that is used in any fixed effects meta-analysis model to generate weights for each study. The strength of the quality effects meta-analysis is that it allows available methodological evidence to be used over subjective random effects, and thereby helps to close the damaging gap which has opened up between methodology and statistics in clinical research. To do this a synthetic bias variance is computed based on quality information to adjust inverse variance weights and the quality adjusted weight of the <i>i</i>th study is introduced.<sup id="cite_ref-Doi_2008_79-1" class="reference"><a href="#cite_note-Doi_2008-79"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> These adjusted weights are then used in meta-analysis. In other words, if study <i>i</i> is of good quality and other studies are of poor quality, a proportion of their quality adjusted weights is mathematically redistributed to study <i>i</i> giving it more weight towards the overall effect size. As studies become increasingly similar in terms of quality, re-distribution becomes progressively less and ceases when all studies are of equal quality (in the case of equal quality, the quality effects model defaults to the IVhet model – see previous section). A recent evaluation of the quality effects model (with some updates) demonstrates that despite the subjectivity of quality assessment, the performance (MSE and true variance under simulation) is superior to that achievable with the random effects model.<sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> This model thus replaces the untenable interpretations that abound in the literature and a software is available to explore this method further.<sup id="cite_ref-Epigear_83-0" class="reference"><a href="#cite_note-Epigear-83"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Network_meta-analysis_methods">Network meta-analysis methods</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=9" title="Edit section: Network meta-analysis methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Indirekt_j%C3%A4mf%C3%B6relse.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Indirekt_j%C3%A4mf%C3%B6relse.jpg/300px-Indirekt_j%C3%A4mf%C3%B6relse.jpg" decoding="async" width="300" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Indirekt_j%C3%A4mf%C3%B6relse.jpg/450px-Indirekt_j%C3%A4mf%C3%B6relse.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Indirekt_j%C3%A4mf%C3%B6relse.jpg/600px-Indirekt_j%C3%A4mf%C3%B6relse.jpg 2x" data-file-width="717" data-file-height="373" /></a><figcaption>A network meta-analysis looks at indirect comparisons. In the image, A has been analyzed in relation to C and C has been analyzed in relation to B. However the relation between A and B is only known indirectly, and a network meta-analysis looks at such indirect evidence of differences between methods and interventions using statistical method.</figcaption></figure> <p>Indirect comparison meta-analysis methods (also called network meta-analyses, in particular when multiple treatments are assessed simultaneously) generally use two main methodologies.<sup id="cite_ref-84" class="reference"><a href="#cite_note-84"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-85" class="reference"><a href="#cite_note-85"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> First, is the Bucher method<sup id="cite_ref-86" class="reference"><a href="#cite_note-86"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup> which is a single or repeated comparison of a closed loop of three-treatments such that one of them is common to the two studies and forms the node where the loop begins and ends. Therefore, multiple two-by-two comparisons (3-treatment loops) are needed to compare multiple treatments. This methodology requires that trials with more than two arms have two arms only selected as independent pair-wise comparisons are required. The alternative methodology uses complex <a href="/wiki/Statistical_model" title="Statistical model">statistical modelling</a> to include the multiple arm trials and comparisons simultaneously between all competing treatments. These have been executed using Bayesian methods, mixed linear models and meta-regression approaches.<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 2018)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading5"><h5 id="Bayesian_framework">Bayesian framework</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=10" title="Edit section: Bayesian framework"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Specifying a Bayesian network meta-analysis model involves writing a directed acyclic graph (DAG) model for general-purpose <a href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a> (MCMC) software such as WinBUGS.<sup id="cite_ref-Valkenhoef,_G._2012_87-0" class="reference"><a href="#cite_note-Valkenhoef,_G._2012-87"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> In addition, prior distributions have to be specified for a number of the parameters, and the data have to be supplied in a specific format.<sup id="cite_ref-Valkenhoef,_G._2012_87-1" class="reference"><a href="#cite_note-Valkenhoef,_G._2012-87"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> Together, the DAG, priors, and data form a Bayesian hierarchical model. To complicate matters further, because of the nature of MCMC estimation, overdispersed starting values have to be chosen for a number of independent chains so that convergence can be assessed.<sup id="cite_ref-88" class="reference"><a href="#cite_note-88"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup> Recently, multiple <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a> software packages were developed to simplify the model fitting (e.g., <i>metaBMA</i><sup id="cite_ref-89" class="reference"><a href="#cite_note-89"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup> and <i>RoBMA</i><sup id="cite_ref-90" class="reference"><a href="#cite_note-90"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup>) and even implemented in statistical software with graphical user interface (<a href="/wiki/Graphical_user_interface" title="Graphical user interface">GUI</a>): <a href="/wiki/JASP" title="JASP">JASP</a>. Although the complexity of the Bayesian approach limits usage of this methodology, recent tutorial papers are trying to increase accessibility of the methods.<sup id="cite_ref-91" class="reference"><a href="#cite_note-91"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup> Methodology for automation of this method has been suggested<sup id="cite_ref-Valkenhoef,_G._2012_87-2" class="reference"><a href="#cite_note-Valkenhoef,_G._2012-87"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> but requires that arm-level outcome data are available, and this is usually unavailable. Great claims are sometimes made for the inherent ability of the Bayesian framework to handle network meta-analysis and its greater flexibility. However, this choice of implementation of framework for inference, Bayesian or frequentist, may be less important than other choices regarding the modeling of effects<sup id="cite_ref-ReferenceC_93-0" class="reference"><a href="#cite_note-ReferenceC-93"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup> (see discussion on models above). </p> <div class="mw-heading mw-heading5"><h5 id="Frequentist_multivariate_framework">Frequentist multivariate framework</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=11" title="Edit section: Frequentist multivariate framework"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On the other hand, the frequentist multivariate methods involve approximations and assumptions that are not stated explicitly or verified when the methods are applied (see discussion on meta-analysis models above). For example, the mvmeta package for Stata enables network meta-analysis in a frequentist framework.<sup id="cite_ref-94" class="reference"><a href="#cite_note-94"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup> However, if there is no common comparator in the network, then this has to be handled by augmenting the dataset with fictional arms with high variance, which is not very objective and requires a decision as to what constitutes a sufficiently high variance.<sup id="cite_ref-Valkenhoef,_G._2012_87-3" class="reference"><a href="#cite_note-Valkenhoef,_G._2012-87"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> The other issue is use of the random effects model in both this frequentist framework and the Bayesian framework. Senn advises analysts to be cautious about interpreting the 'random effects' analysis since only one random effect is allowed for but one could envisage many.<sup id="cite_ref-ReferenceC_93-1" class="reference"><a href="#cite_note-ReferenceC-93"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup> Senn goes on to say that it is rather naıve, even in the case where only two treatments are being compared to assume that random-effects analysis accounts for all uncertainty about the way effects can vary from trial to trial. Newer models of meta-analysis such as those discussed above would certainly help alleviate this situation and have been implemented in the next framework. </p> <div class="mw-heading mw-heading5"><h5 id="Generalized_pairwise_modelling_framework">Generalized pairwise modelling framework</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=12" title="Edit section: Generalized pairwise modelling framework"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An approach that has been tried since the late 1990s is the implementation of the multiple three-treatment closed-loop analysis. This has not been popular because the process rapidly becomes overwhelming as network complexity increases. Development in this area was then abandoned in favor of the Bayesian and multivariate frequentist methods which emerged as alternatives. Very recently, automation of the three-treatment closed loop method has been developed for complex networks by some researchers<sup id="cite_ref-Manual_74-1" class="reference"><a href="#cite_note-Manual-74"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> as a way to make this methodology available to the mainstream research community. This proposal does restrict each trial to two interventions, but also introduces a workaround for multiple arm trials: a different fixed control node can be selected in different runs. It also utilizes robust meta-analysis methods so that many of the problems highlighted above are avoided. Further research around this framework is required to determine if this is indeed superior to the Bayesian or multivariate frequentist frameworks. Researchers willing to try this out have access to this framework through a free software.<sup id="cite_ref-Epigear_83-1" class="reference"><a href="#cite_note-Epigear-83"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading5"><h5 id="Tailored_meta-analysis">Tailored meta-analysis</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=13" title="Edit section: Tailored meta-analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another form of additional information comes from the intended setting. If the target setting for applying the meta-analysis results is known then it may be possible to use data from the setting to tailor the results thus producing a 'tailored meta-analysis'.,<sup id="cite_ref-95" class="reference"><a href="#cite_note-95"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-96" class="reference"><a href="#cite_note-96"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup> This has been used in test accuracy meta-analyses, where empirical knowledge of the test positive rate and the prevalence have been used to derive a region in <a href="/wiki/Receiver_Operating_Characteristic" class="mw-redirect" title="Receiver Operating Characteristic">Receiver Operating Characteristic</a> (ROC) space known as an 'applicable region'. Studies are then selected for the target setting based on comparison with this region and aggregated to produce a summary estimate which is tailored to the target setting. </p> <div class="mw-heading mw-heading4"><h4 id="Aggregating_IPD_and_AD">Aggregating IPD and AD</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=14" title="Edit section: Aggregating IPD and AD"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Meta-analysis can also be applied to combine IPD and AD. This is convenient when the researchers who conduct the analysis have their own raw data while collecting aggregate or summary data from the literature. The generalized integration model (GIM)<sup id="cite_ref-Zhang2020_97-0" class="reference"><a href="#cite_note-Zhang2020-97"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup> is a generalization of the meta-analysis. It allows that the model fitted on the individual participant data (IPD) is different from the ones used to compute the aggregate data (AD). GIM can be viewed as a model calibration method for integrating information with more flexibility. </p> <div class="mw-heading mw-heading3"><h3 id="Validation_of_meta-analysis_results">Validation of meta-analysis results</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=15" title="Edit section: Validation of meta-analysis results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The meta-analysis estimate represents a weighted average across studies and when there is <a href="/wiki/Study_heterogeneity" title="Study heterogeneity">heterogeneity</a> this may result in the summary estimate not being representative of individual studies. Qualitative appraisal of the primary studies using established tools can uncover potential biases,<sup id="cite_ref-98" class="reference"><a href="#cite_note-98"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-99" class="reference"><a href="#cite_note-99"><span class="cite-bracket">[</span>99<span class="cite-bracket">]</span></a></sup> but does not quantify the aggregate effect of these biases on the summary estimate. Although the meta-analysis result could be compared with an independent prospective primary study, such external validation is often impractical. This has led to the development of methods that exploit a form of <a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">leave-one-out cross validation</a>, sometimes referred to as internal-external cross validation (IOCV).<sup id="cite_ref-100" class="reference"><a href="#cite_note-100"><span class="cite-bracket">[</span>100<span class="cite-bracket">]</span></a></sup> Here each of the k included studies in turn is omitted and compared with the summary estimate derived from aggregating the remaining k- 1 studies. A general <b>validation statistic, Vn</b> based on IOCV has been developed to measure the statistical validity of meta-analysis results.<sup id="cite_ref-101" class="reference"><a href="#cite_note-101"><span class="cite-bracket">[</span>101<span class="cite-bracket">]</span></a></sup> For test accuracy and prediction, particularly when there are multivariate effects, other approaches which seek to estimate the prediction error have also been proposed.<sup id="cite_ref-102" class="reference"><a href="#cite_note-102"><span class="cite-bracket">[</span>102<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Challenges">Challenges</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=16" title="Edit section: Challenges"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A meta-analysis of several small studies does not always predict the results of a single large study.<sup id="cite_ref-103" class="reference"><a href="#cite_note-103"><span class="cite-bracket">[</span>103<span class="cite-bracket">]</span></a></sup> Some have argued that a weakness of the method is that sources of bias are not controlled by the method: a good meta-analysis cannot correct for poor design or bias in the original studies.<sup id="cite_ref-Slavin_104-0" class="reference"><a href="#cite_note-Slavin-104"><span class="cite-bracket">[</span>104<span class="cite-bracket">]</span></a></sup> This would mean that only methodologically sound studies should be included in a meta-analysis, a practice called 'best evidence synthesis'.<sup id="cite_ref-Slavin_104-1" class="reference"><a href="#cite_note-Slavin-104"><span class="cite-bracket">[</span>104<span class="cite-bracket">]</span></a></sup> Other meta-analysts would include weaker studies, and add a study-level predictor variable that reflects the methodological quality of the studies to examine the effect of study quality on the effect size.<sup id="cite_ref-105" class="reference"><a href="#cite_note-105"><span class="cite-bracket">[</span>105<span class="cite-bracket">]</span></a></sup> However, others have argued that a better approach is to preserve information about the variance in the study sample, casting as wide a net as possible, and that methodological selection criteria introduce unwanted subjectivity, defeating the purpose of the approach.<sup id="cite_ref-106" class="reference"><a href="#cite_note-106"><span class="cite-bracket">[</span>106<span class="cite-bracket">]</span></a></sup> More recently, and under the influence of a push for open practices in science, tools to develop "crowd-sourced" living meta-analyses that are updated by communities of scientists <sup id="cite_ref-107" class="reference"><a href="#cite_note-107"><span class="cite-bracket">[</span>107<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-108" class="reference"><a href="#cite_note-108"><span class="cite-bracket">[</span>108<span class="cite-bracket">]</span></a></sup> in hopes of making all the subjective choices more explicit. </p> <div class="mw-heading mw-heading3"><h3 id="Publication_bias:_the_file_drawer_problem">Publication bias: the file drawer problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=17" title="Edit section: Publication bias: the file drawer problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Example_of_a_symmetrical_funnel_plot_created_with_MetaXL_Sept_2015.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Example_of_a_symmetrical_funnel_plot_created_with_MetaXL_Sept_2015.jpg/220px-Example_of_a_symmetrical_funnel_plot_created_with_MetaXL_Sept_2015.jpg" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Example_of_a_symmetrical_funnel_plot_created_with_MetaXL_Sept_2015.jpg/330px-Example_of_a_symmetrical_funnel_plot_created_with_MetaXL_Sept_2015.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Example_of_a_symmetrical_funnel_plot_created_with_MetaXL_Sept_2015.jpg/440px-Example_of_a_symmetrical_funnel_plot_created_with_MetaXL_Sept_2015.jpg 2x" data-file-width="587" data-file-height="360" /></a><figcaption>A funnel plot expected without the file drawer problem. The largest studies converge at the tip while smaller studies show more or less symmetrical scatter at the base.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Funnel_plot_depicting_asymmetry_Sept_2015.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Funnel_plot_depicting_asymmetry_Sept_2015.jpg/220px-Funnel_plot_depicting_asymmetry_Sept_2015.jpg" decoding="async" width="220" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Funnel_plot_depicting_asymmetry_Sept_2015.jpg/330px-Funnel_plot_depicting_asymmetry_Sept_2015.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Funnel_plot_depicting_asymmetry_Sept_2015.jpg/440px-Funnel_plot_depicting_asymmetry_Sept_2015.jpg 2x" data-file-width="578" data-file-height="351" /></a><figcaption>A funnel plot expected with the file drawer problem. The largest studies still cluster around the tip, but the bias against publishing negative studies has caused the smaller studies as a whole to have an unjustifiably favorable result to the hypothesis.</figcaption></figure> <p>Another potential pitfall is the reliance on the available body of published studies, which may create exaggerated outcomes due to <a href="/wiki/Publication_bias" title="Publication bias">publication bias</a>,<sup id="cite_ref-109" class="reference"><a href="#cite_note-109"><span class="cite-bracket">[</span>109<span class="cite-bracket">]</span></a></sup> as studies which show <a href="/wiki/Null_result" title="Null result">negative results</a> or <a href="/wiki/Statistically_insignificant" class="mw-redirect" title="Statistically insignificant">insignificant</a> results are less likely to be published.<sup id="cite_ref-110" class="reference"><a href="#cite_note-110"><span class="cite-bracket">[</span>110<span class="cite-bracket">]</span></a></sup> For example, pharmaceutical companies have been known to hide negative studies<sup id="cite_ref-111" class="reference"><a href="#cite_note-111"><span class="cite-bracket">[</span>111<span class="cite-bracket">]</span></a></sup> and researchers may have overlooked unpublished studies such as dissertation studies or conference abstracts that did not reach publication.<sup id="cite_ref-112" class="reference"><a href="#cite_note-112"><span class="cite-bracket">[</span>112<span class="cite-bracket">]</span></a></sup> This is not easily solved, as one cannot know how many studies have gone unreported.<sup id="cite_ref-Rosenthal1979_113-0" class="reference"><a href="#cite_note-Rosenthal1979-113"><span class="cite-bracket">[</span>113<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Publication_bias_in_research_synthe_114-0" class="reference"><a href="#cite_note-Publication_bias_in_research_synthe-114"><span class="cite-bracket">[</span>114<span class="cite-bracket">]</span></a></sup> </p><p>This <a href="/wiki/File_drawer_problem" class="mw-redirect" title="File drawer problem">file drawer problem</a> characterized by negative or non-significant results being tucked away in a cabinet, can result in a biased distribution of effect sizes thus creating a serious <a href="/wiki/Base_rate_fallacy" title="Base rate fallacy">base rate fallacy</a>, in which the significance of the published studies is overestimated, as other studies were either not submitted for publication or were rejected. This should be seriously considered when interpreting the outcomes of a meta-analysis.<sup id="cite_ref-Rosenthal1979_113-1" class="reference"><a href="#cite_note-Rosenthal1979-113"><span class="cite-bracket">[</span>113<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hunter&Schmidt1990_115-0" class="reference"><a href="#cite_note-Hunter&Schmidt1990-115"><span class="cite-bracket">[</span>115<span class="cite-bracket">]</span></a></sup> </p><p>The distribution of effect sizes can be visualized with a <a href="/wiki/Funnel_plot" title="Funnel plot">funnel plot</a> which (in its most common version) is a scatter plot of standard error versus the effect size.<sup id="cite_ref-116" class="reference"><a href="#cite_note-116"><span class="cite-bracket">[</span>116<span class="cite-bracket">]</span></a></sup> It makes use of the fact that the smaller studies (thus larger standard errors) have more scatter of the magnitude of effect (being less precise) while the larger studies have less scatter and form the tip of the funnel. If many negative studies were not published, the remaining positive studies give rise to a funnel plot in which the base is skewed to one side (asymmetry of the funnel plot). In contrast, when there is no publication bias, the effect of the smaller studies has no reason to be skewed to one side and so a symmetric funnel plot results. This also means that if no publication bias is present, there would be no relationship between standard error and effect size.<sup id="cite_ref-117" class="reference"><a href="#cite_note-117"><span class="cite-bracket">[</span>117<span class="cite-bracket">]</span></a></sup> A negative or positive relation between standard error and effect size would imply that smaller studies that found effects in one direction only were more likely to be published and/or to be submitted for publication. </p><p>Apart from the visual funnel plot, statistical methods for detecting publication bias have also been proposed.<sup id="cite_ref-Publication_bias_in_research_synthe_114-1" class="reference"><a href="#cite_note-Publication_bias_in_research_synthe-114"><span class="cite-bracket">[</span>114<span class="cite-bracket">]</span></a></sup> These are controversial because they typically have low power for detection of bias, but also may make false positives under some circumstances.<sup id="cite_ref-Ioannidis_and_Trikalinos_118-0" class="reference"><a href="#cite_note-Ioannidis_and_Trikalinos-118"><span class="cite-bracket">[</span>118<span class="cite-bracket">]</span></a></sup> For instance small study effects (biased smaller studies), wherein methodological differences between smaller and larger studies exist, may cause asymmetry in effect sizes that resembles publication bias. However, small study effects may be just as problematic for the interpretation of meta-analyses, and the imperative is on meta-analytic authors to investigate potential sources of bias.<sup id="cite_ref-119" class="reference"><a href="#cite_note-119"><span class="cite-bracket">[</span>119<span class="cite-bracket">]</span></a></sup> </p><p>The problem of publication bias is not trivial as it is suggested that 25% of meta-analyses in the psychological sciences may have suffered from publication bias.<sup id="cite_ref-Ferguson_and_Brannick_120-0" class="reference"><a href="#cite_note-Ferguson_and_Brannick-120"><span class="cite-bracket">[</span>120<span class="cite-bracket">]</span></a></sup> However, low power of existing tests and problems with the visual appearance of the funnel plot remain an issue, and estimates of publication bias may remain lower than what truly exists. </p><p>Most discussions of publication bias focus on journal practices favoring publication of statistically significant findings. However, questionable research practices, such as reworking statistical models until significance is achieved, may also favor statistically significant findings in support of researchers' hypotheses.<sup id="cite_ref-Simmons_121-0" class="reference"><a href="#cite_note-Simmons-121"><span class="cite-bracket">[</span>121<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-LeBel_122-0" class="reference"><a href="#cite_note-LeBel-122"><span class="cite-bracket">[</span>122<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Problems_related_to_studies_not_reporting_non-statistically_significant_effects">Problems related to studies not reporting non-statistically significant effects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=18" title="Edit section: Problems related to studies not reporting non-statistically significant effects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Studies often do not report the effects when they do not reach statistical significance.<sup id="cite_ref-123" class="reference"><a href="#cite_note-123"><span class="cite-bracket">[</span>123<span class="cite-bracket">]</span></a></sup> For example, they may simply say that the groups did not show statistically significant differences, without reporting any other information (e.g. a statistic or p-value).<sup id="cite_ref-124" class="reference"><a href="#cite_note-124"><span class="cite-bracket">[</span>124<span class="cite-bracket">]</span></a></sup> Exclusion of these studies would lead to a situation similar to publication bias, but their inclusion (assuming null effects) would also bias the meta-analysis. </p> <div class="mw-heading mw-heading3"><h3 id="Problems_related_to_the_statistical_approach">Problems related to the statistical approach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=19" title="Edit section: Problems related to the statistical approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Other weaknesses are that it has not been determined if the statistically most accurate method for combining results is the fixed, IVhet, random or quality effect models, though the criticism against the random effects model is mounting because of the perception that the new random effects (used in meta-analysis) are essentially formal devices to facilitate smoothing or shrinkage and prediction may be impossible or ill-advised.<sup id="cite_ref-125" class="reference"><a href="#cite_note-125"><span class="cite-bracket">[</span>125<span class="cite-bracket">]</span></a></sup> The main problem with the random effects approach is that it uses the classic statistical thought of generating a "compromise estimator" that makes the weights close to the naturally weighted estimator if <a href="/wiki/Study_heterogeneity" title="Study heterogeneity">heterogeneity</a> across studies is large but close to the inverse variance weighted estimator if the between study heterogeneity is small. However, what has been ignored is the distinction between the model <i>we choose</i> to analyze a given dataset, and the <i>mechanism by which the data came into being</i>.<sup id="cite_ref-ReferenceB_126-0" class="reference"><a href="#cite_note-ReferenceB-126"><span class="cite-bracket">[</span>126<span class="cite-bracket">]</span></a></sup> A random effect can be present in either of these roles, but the two roles are quite distinct. There's no reason to think the analysis model and data-generation mechanism (model) are similar in form, but many sub-fields of statistics have developed the habit of assuming, for theory and simulations, that the data-generation mechanism (model) is identical to the analysis model we choose (or would like others to choose). As a hypothesized mechanisms for producing the data, the random effect model for meta-analysis is silly and it is more appropriate to think of this model as a superficial description and something we choose as an analytical tool – but this choice for meta-analysis may not work because the study effects are a fixed feature of the respective meta-analysis and the probability distribution is only a descriptive tool.<sup id="cite_ref-ReferenceB_126-1" class="reference"><a href="#cite_note-ReferenceB-126"><span class="cite-bracket">[</span>126<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Problems_arising_from_agenda-driven_bias">Problems arising from agenda-driven bias</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=20" title="Edit section: Problems arising from agenda-driven bias"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The most severe fault in meta-analysis often occurs when the person or persons doing the meta-analysis have an <a href="/wiki/Economic" class="mw-redirect" title="Economic">economic</a>, <a href="/wiki/Social_issues" class="mw-redirect" title="Social issues">social</a>, or <a href="/wiki/Political" class="mw-redirect" title="Political">political</a> agenda such as the passage or defeat of <a href="/wiki/Legislation" title="Legislation">legislation</a>.<sup id="cite_ref-127" class="reference"><a href="#cite_note-127"><span class="cite-bracket">[</span>127<span class="cite-bracket">]</span></a></sup> People with these types of agendas may be more likely to abuse meta-analysis due to personal <a href="/wiki/Bias" title="Bias">bias</a>. For example, researchers favorable to the author's agenda are likely to have their studies <a href="/wiki/Cherry_picking_(fallacy)" class="mw-redirect" title="Cherry picking (fallacy)">cherry-picked</a> while those not favorable will be ignored or labeled as "not credible". In addition, the favored authors may themselves be biased or paid to produce results that support their overall political, social, or economic goals in ways such as selecting small favorable data sets and not incorporating larger unfavorable data sets. The influence of such biases on the results of a meta-analysis is possible because the methodology of meta-analysis is highly malleable.<sup id="cite_ref-Stegenga_128-0" class="reference"><a href="#cite_note-Stegenga-128"><span class="cite-bracket">[</span>128<span class="cite-bracket">]</span></a></sup> </p><p>A 2011 study done to disclose possible conflicts of interests in underlying research studies used for medical meta-analyses reviewed 29 meta-analyses and found that conflicts of interests in the studies underlying the meta-analyses were rarely disclosed. The 29 meta-analyses included 11 from general medicine journals, 15 from specialty medicine journals, and three from the <a href="/wiki/Cochrane_Database_of_Systematic_Reviews" class="mw-redirect" title="Cochrane Database of Systematic Reviews">Cochrane Database of Systematic Reviews</a>. The 29 meta-analyses reviewed a total of 509 <a href="/wiki/Randomized_controlled_trials" class="mw-redirect" title="Randomized controlled trials">randomized controlled trials</a> (RCTs). Of these, 318 RCTs reported funding sources, with 219 (69%) receiving funding from industry (i.e. one or more authors having financial ties to the pharmaceutical industry). Of the 509 RCTs, 132 reported author conflict of interest disclosures, with 91 studies (69%) disclosing one or more authors having financial ties to industry. The information was, however, seldom reflected in the meta-analyses. Only two (7%) reported RCT funding sources and none reported RCT author-industry ties. The authors concluded "without acknowledgment of COI due to industry funding or author industry financial ties from RCTs included in meta-analyses, readers' understanding and appraisal of the evidence from the meta-analysis may be compromised."<sup id="cite_ref-129" class="reference"><a href="#cite_note-129"><span class="cite-bracket">[</span>129<span class="cite-bracket">]</span></a></sup> </p><p>For example, in 1998, a US federal judge found that the United States <a href="/wiki/Environmental_Protection_Agency" class="mw-redirect" title="Environmental Protection Agency">Environmental Protection Agency</a> had abused the meta-analysis process to produce a study claiming cancer risks to non-smokers from environmental tobacco smoke (ETS) with the intent to influence policy makers to pass smoke-free–workplace laws.<sup id="cite_ref-130" class="reference"><a href="#cite_note-130"><span class="cite-bracket">[</span>130<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-131" class="reference"><a href="#cite_note-131"><span class="cite-bracket">[</span>131<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-132" class="reference"><a href="#cite_note-132"><span class="cite-bracket">[</span>132<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Comparability_and_validity_of_included_studies">Comparability and validity of included studies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=21" title="Edit section: Comparability and validity of included studies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Meta-analysis may often not be a substitute for an adequately powered primary study, particularly in the biological sciences.<sup id="cite_ref-133" class="reference"><a href="#cite_note-133"><span class="cite-bracket">[</span>133<span class="cite-bracket">]</span></a></sup> </p><p>Heterogeneity of methods used may lead to faulty conclusions.<sup id="cite_ref-134" class="reference"><a href="#cite_note-134"><span class="cite-bracket">[</span>134<span class="cite-bracket">]</span></a></sup> For instance, differences in the forms of an intervention or the cohorts that are thought to be minor or are unknown to the scientists could lead to substantially different results, including results that distort the meta-analysis' results or are not adequately considered in its data. Vice versa, results from meta-analyses may also make certain hypothesis or interventions seem nonviable and preempt further research or approvals, despite certain modifications – such as intermittent administration, <a href="/wiki/Personalized_medicine" title="Personalized medicine">personalized criteria</a> and <a href="/wiki/Combination_therapy" title="Combination therapy">combination measures</a> – leading to substantially different results, including in cases where such have been successfully identified and applied in small-scale studies that were considered in the meta-analysis.<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. (January 2022)">citation needed</span></a></i>]</sup> <a href="/wiki/Standardization" title="Standardization">Standardization</a>, <a href="/wiki/Reproducibility" title="Reproducibility">reproduction of experiments</a>, <a href="/wiki/Open_science" title="Open science">open data and open protocols</a> may often not mitigate such problems, for instance as relevant factors and criteria could be unknown or not be recorded.<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. (January 2022)">citation needed</span></a></i>]</sup> </p><p>There is a debate about the appropriate balance between testing with as few animals or humans as possible and the need to obtain robust, reliable findings. It has been argued that unreliable research is inefficient and wasteful and that studies are not just wasteful when they stop too late but also when they stop too early. In large clinical trials, planned, sequential analyses are sometimes used if there is considerable expense or potential harm associated with testing participants.<sup id="cite_ref-135" class="reference"><a href="#cite_note-135"><span class="cite-bracket">[</span>135<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Applied_science" title="Applied science">applied</a> behavioural science, "megastudies" have been proposed to investigate the efficacy of many different interventions designed in an interdisciplinary manner by separate teams.<sup id="cite_ref-10.1038/s41586-021-04128-4_136-0" class="reference"><a href="#cite_note-10.1038/s41586-021-04128-4-136"><span class="cite-bracket">[</span>136<span class="cite-bracket">]</span></a></sup> One such study used a fitness chain to recruit a large number participants. It has been suggested that behavioural interventions are often hard to compare [in meta-analyses and reviews], as "different scientists test different intervention ideas in different samples using different outcomes over different time intervals", causing a lack of comparability of such individual investigations which limits "their potential to inform <a href="/wiki/Policy" title="Policy">policy</a>".<sup id="cite_ref-10.1038/s41586-021-04128-4_136-1" class="reference"><a href="#cite_note-10.1038/s41586-021-04128-4-136"><span class="cite-bracket">[</span>136<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Weak_inclusion_standards_lead_to_misleading_conclusions">Weak inclusion standards lead to misleading conclusions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=22" title="Edit section: Weak inclusion standards lead to misleading conclusions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Meta-analyses in education are often not restrictive enough in regards to the methodological quality of the studies they include. For example, studies that include small samples or researcher-made measures lead to inflated effect size estimates.<sup id="cite_ref-137" class="reference"><a href="#cite_note-137"><span class="cite-bracket">[</span>137<span class="cite-bracket">]</span></a></sup> However, this problem also troubles meta-analysis of clinical trials. The use of different quality assessment tools (QATs) lead to including different studies and obtaining conflicting estimates of average treatment effects.<sup id="cite_ref-138" class="reference"><a href="#cite_note-138"><span class="cite-bracket">[</span>138<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-139" class="reference"><a href="#cite_note-139"><span class="cite-bracket">[</span>139<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications_in_modern_science">Applications in modern science</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=23" title="Edit section: Applications in modern science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Integrated_Molecular_Meta-Analysis_of_1,000_Pediatric_High-Grade_and_Diffuse_Intrinsic_Pontine_Glioma_-_graphical_abstract.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Integrated_Molecular_Meta-Analysis_of_1%2C000_Pediatric_High-Grade_and_Diffuse_Intrinsic_Pontine_Glioma_-_graphical_abstract.jpg/300px-Integrated_Molecular_Meta-Analysis_of_1%2C000_Pediatric_High-Grade_and_Diffuse_Intrinsic_Pontine_Glioma_-_graphical_abstract.jpg" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Integrated_Molecular_Meta-Analysis_of_1%2C000_Pediatric_High-Grade_and_Diffuse_Intrinsic_Pontine_Glioma_-_graphical_abstract.jpg/450px-Integrated_Molecular_Meta-Analysis_of_1%2C000_Pediatric_High-Grade_and_Diffuse_Intrinsic_Pontine_Glioma_-_graphical_abstract.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Integrated_Molecular_Meta-Analysis_of_1%2C000_Pediatric_High-Grade_and_Diffuse_Intrinsic_Pontine_Glioma_-_graphical_abstract.jpg/600px-Integrated_Molecular_Meta-Analysis_of_1%2C000_Pediatric_High-Grade_and_Diffuse_Intrinsic_Pontine_Glioma_-_graphical_abstract.jpg 2x" data-file-width="664" data-file-height="664" /></a><figcaption>Graphical summary of a meta-analysis of over 1,000 cases of <a href="/wiki/Diffuse_intrinsic_pontine_glioma" class="mw-redirect" title="Diffuse intrinsic pontine glioma">diffuse intrinsic pontine glioma</a> and other pediatric gliomas, in which information about the <a href="/wiki/Mutation" title="Mutation">mutations</a> involved as well as generic outcomes were distilled from the underlying <a href="/wiki/Primary_literature" class="mw-redirect" title="Primary literature">primary literature</a></figcaption></figure><p>Modern statistical meta-analysis does more than just combine the effect sizes of a set of studies using a weighted average. It can test if the outcomes of studies show more variation than the variation that is expected because of the sampling of different numbers of research participants. Additionally, study characteristics such as measurement instrument used, population sampled, or aspects of the studies' design can be coded and used to reduce variance of the estimator (see statistical models above). Thus some methodological weaknesses in studies can be corrected statistically. Other uses of meta-analytic methods include the development and validation of clinical prediction models, where meta-analysis may be used to combine individual participant data from different research centers and to assess the model's generalisability,<sup id="cite_ref-Individual_participant_data_IPD_m_140-0" class="reference"><a href="#cite_note-Individual_participant_data_IPD_m-140"><span class="cite-bracket">[</span>140<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-141" class="reference"><a href="#cite_note-141"><span class="cite-bracket">[</span>141<span class="cite-bracket">]</span></a></sup> or even to aggregate existing prediction models.<sup id="cite_ref-142" class="reference"><a href="#cite_note-142"><span class="cite-bracket">[</span>142<span class="cite-bracket">]</span></a></sup> </p><p>Meta-analysis can be done with <a href="/wiki/Single-subject_design" title="Single-subject design">single-subject design</a> as well as group research designs.<sup id="cite_ref-143" class="reference"><a href="#cite_note-143"><span class="cite-bracket">[</span>143<span class="cite-bracket">]</span></a></sup> This is important because much research has been done with <a href="/wiki/Single-subject_research" title="Single-subject research">single-subject research</a> designs.<sup id="cite_ref-144" class="reference"><a href="#cite_note-144"><span class="cite-bracket">[</span>144<span class="cite-bracket">]</span></a></sup> Considerable dispute exists for the most appropriate meta-analytic technique for single subject research.<sup id="cite_ref-145" class="reference"><a href="#cite_note-145"><span class="cite-bracket">[</span>145<span class="cite-bracket">]</span></a></sup> </p><p>Meta-analysis leads to a shift of emphasis from single studies to multiple studies. It emphasizes the practical importance of the effect size instead of the statistical significance of individual studies. This shift in thinking has been termed "meta-analytic thinking". The results of a meta-analysis are often shown in a <a href="/wiki/Forest_plot" title="Forest plot">forest plot</a>. </p><p>Results from studies are combined using different approaches. One approach frequently used in meta-analysis in health care research is termed '<a href="/wiki/Inverse-variance_weighting" title="Inverse-variance weighting">inverse variance method</a>'. The average <a href="/wiki/Effect_size" title="Effect size">effect size</a> across all studies is computed as a <i>weighted mean</i>, whereby the weights are equal to the inverse variance of each study's effect estimator. Larger studies and studies with less random variation are given greater weight than smaller studies. Other common approaches include the <a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Mantel–Haenszel method</a><sup id="cite_ref-146" class="reference"><a href="#cite_note-146"><span class="cite-bracket">[</span>146<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Richard_Peto" title="Richard Peto">Peto method</a>.<sup id="cite_ref-147" class="reference"><a href="#cite_note-147"><span class="cite-bracket">[</span>147<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Seed-based_d_mapping" title="Seed-based d mapping">Seed-based d mapping</a> (formerly signed differential mapping, SDM) is a statistical technique for meta-analyzing studies on differences in brain activity or structure which used neuroimaging techniques such as fMRI, VBM or PET. </p><p>Different high throughput techniques such as <a href="/wiki/DNA_microarray" title="DNA microarray">microarrays</a> have been used to understand <a href="/wiki/Gene_expression" title="Gene expression">Gene expression</a>. <a href="/wiki/MicroRNA" title="MicroRNA">MicroRNA</a> expression profiles have been used to identify differentially expressed microRNAs in particular cell or tissue type or disease conditions or to check the effect of a treatment. A meta-analysis of such expression profiles was performed to derive novel conclusions and to validate the known findings.<sup id="cite_ref-148" class="reference"><a href="#cite_note-148"><span class="cite-bracket">[</span>148<span class="cite-bracket">]</span></a></sup> </p><p>Meta-analysis of whole genome sequencing studies provides an attractive solution to the problem of collecting large sample sizes for discovering rare variants associated with complex phenotypes. Some methods have been developed to enable functionally informed rare variant association meta-analysis in biobank-scale cohorts using efficient approaches for summary statistic storage.<sup id="cite_ref-149" class="reference"><a href="#cite_note-149"><span class="cite-bracket">[</span>149<span class="cite-bracket">]</span></a></sup> </p><p>Sweeping meta-analyses can also be used to estimate a network of effects. This allows researchers to examine patterns in the fuller panorama of more accurately estimated results and draw conclusions that consider the broader context (e.g., how personality-intelligence relations vary by trait family).<sup id="cite_ref-150" class="reference"><a href="#cite_note-150"><span class="cite-bracket">[</span>150<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=24" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid 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plot">Galbraith plot</a></li> <li><a href="/wiki/Data_aggregation" title="Data aggregation">Data aggregation</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Meta-analysis&action=edit&section=25" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i><span typeof="mw:File"><a href="//creativecommons.org/licenses/by/4.0/" title="creativecommons:by/4.0/"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Creative_Commons_by_small.svg/80px-Creative_Commons_by_small.svg.png" decoding="async" width="80" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Creative_Commons_by_small.svg/120px-Creative_Commons_by_small.svg.png 1.5x, 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Quintana available under the <a href="//creativecommons.org/licenses/by/4.0/" class="extiw" title="creativecommons:by/4.0/">CC BY 4.0</a> license.</i> </p><p><i><span typeof="mw:File"><a href="//creativecommons.org/licenses/by/3.0/" title="creativecommons:by/3.0/"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Creative_Commons_by_small.svg/80px-Creative_Commons_by_small.svg.png" decoding="async" width="80" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Creative_Commons_by_small.svg/120px-Creative_Commons_by_small.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Creative_Commons_by_small.svg/160px-Creative_Commons_by_small.svg.png 2x" data-file-width="80" data-file-height="15" /></a></span> This article incorporates <a rel="nofollow" class="external text" href="https://www.jstatsoft.org/article/view/v036i03">text</a> by Wolfgang Viechtbauer available under the <a href="//creativecommons.org/licenses/by/3.0/" class="extiw" title="creativecommons:by/3.0/">CC BY 3.0</a> license.</i> </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=Meta-analysis&action=edit&section=26" 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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_1-2"><sup><i><b>c</b></i></sup></a></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="CITEREFShadishLecy2015" class="citation journal cs1">Shadish WR, Lecy JD (September 2015). 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Minneapolis, Minnesota, United States: University of Minnesota Libraries Publishing. pp. Chapters 4–7. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.24926%2F9781946135988">10.24926/9781946135988</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-946135-98-8" title="Special:BookSources/978-1-946135-98-8"><bdi>978-1-946135-98-8</bdi></a>. <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:265335858">265335858</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Of+Anchors+%26+Sails%3A+Personality-ability+trait+constellations&rft.place=Minneapolis%2C+Minnesota%2C+United+States&rft.pages=Chapters+4-7&rft.pub=University+of+Minnesota+Libraries+Publishing&rft.date=2023&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A265335858%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.24926%2F9781946135988&rft.isbn=978-1-946135-98-8&rft.aulast=Stanek&rft.aufirst=Kevin+C.&rft.au=Ones%2C+Deniz+S.&rft_id=https%3A%2F%2Fumnlibraries.manifoldapp.org%2Fprojects%2Fof-anchors-and-sails&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMeta-analysis" class="Z3988"></span></span> </li> </ol></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Statistics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Statistics" title="Template:Statistics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Statistics" title="Template talk:Statistics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Statistics" title="Special:EditPage/Template:Statistics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Outline_of_statistics" title="Outline of statistics">Outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">Heteroscedasticity</a></li> <li><a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">Homoscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li> <li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li> <li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications" style="font-size:114%;margin:0 4em"><a href="/wiki/List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bioinformatics" title="Bioinformatics">Bioinformatics</a></li> <li><a href="/wiki/Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="/wiki/Clinical_study_design" title="Clinical study design">studies</a></li> <li><a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a></li> <li><a href="/wiki/Medical_statistics" title="Medical 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Statistics">WikiProject</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Clinical_research_and_experimental_design" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Medical_research_studies" title="Template:Medical research studies"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Medical_research_studies" title="Template talk:Medical research studies"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Medical_research_studies" title="Special:EditPage/Template:Medical research studies"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Clinical_research_and_experimental_design" style="font-size:114%;margin:0 4em"><a href="/wiki/Clinical_research" title="Clinical research">Clinical research</a> and <a href="/wiki/Design_of_experiments" title="Design of experiments">experimental design</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Clinical_trial" title="Clinical trial">Clinical trial</a> <ul><li><a href="/wiki/Protocol_(science)" title="Protocol (science)">Trial protocols</a></li> <li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li></ul></li> <li><a href="/wiki/Academic_clinical_trials" class="mw-redirect" title="Academic clinical trials">Academic clinical trials</a></li> <li><a href="/wiki/Clinical_study_design" title="Clinical study design">Clinical study design</a></li> <li><a href="/wiki/Evidence-based_medicine" title="Evidence-based medicine">Evidence-based medicine</a></li> <li><a href="/wiki/Real_world_evidence" title="Real world evidence">Real world evidence</a></li> <li><a href="/wiki/Patient_and_public_involvement" title="Patient and public involvement">Patient and public involvement</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Scientific_control" title="Scientific control">Controlled study</a><br />(<a href="/wiki/Hierarchy_of_evidence#USA" title="Hierarchy of evidence">EBM I to II-1</a>)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a> <ul><li><a href="/wiki/Experiment" title="Experiment">Scientific experiment</a></li> <li><a href="/wiki/Blind_experiment" class="mw-redirect" title="Blind experiment">Blind experiment</a></li> <li><a href="/wiki/Open-label_trial" title="Open-label trial">Open-label trial</a></li></ul></li> <li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a> <ul><li><a href="/wiki/Platform_trial" title="Platform trial">Platform trial</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Observational_study" title="Observational study">Observational study</a><br />(<a href="/wiki/Hierarchy_of_evidence#USA" title="Hierarchy of evidence">EBM II-2 to II-3</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a> vs. <a href="/wiki/Longitudinal_study" title="Longitudinal study">Longitudinal study</a>, <a href="/wiki/Ecological_study" title="Ecological study">Ecological study</a></li> <li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a> <ul><li><a href="/wiki/Retrospective_cohort_study" title="Retrospective cohort study">Retrospective</a></li> <li><a href="/wiki/Prospective_cohort_study" title="Prospective cohort study">Prospective</a></li></ul></li> <li><a href="/wiki/Case%E2%80%93control_study" title="Case–control study">Case–control study</a> (<a href="/wiki/Nested_case%E2%80%93control_study" title="Nested case–control study">Nested case–control study</a>)</li> <li><a href="/wiki/Case_series" title="Case series">Case series</a></li> <li><a href="/wiki/Case_study" title="Case study">Case study</a></li> <li><a href="/wiki/Case_report" title="Case report">Case report</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Outcome_measure" title="Outcome measure">Measures</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Epidemiological_methods#Measures_of_occurrence" class="mw-redirect" title="Epidemiological methods">Occurrence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="/wiki/Incidence_(epidemiology)" title="Incidence (epidemiology)">Incidence</a>, <a href="/wiki/Cumulative_incidence" class="mw-redirect" title="Cumulative incidence">Cumulative incidence</a>, <a href="/wiki/Prevalence" title="Prevalence">Prevalence</a>, <a href="/wiki/Point_prevalence" class="mw-redirect" title="Point prevalence">Point prevalence</a>, <a href="/wiki/Period_prevalence" class="mw-redirect" title="Period prevalence">Period prevalence</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Epidemiological_methods#Measures_of_association" class="mw-redirect" title="Epidemiological methods">Association</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="/wiki/Risk_difference" title="Risk difference">Risk difference</a>, <a href="/wiki/Number_needed_to_treat" title="Number needed to treat">Number needed to treat</a>, <a href="/wiki/Number_needed_to_harm" title="Number needed to harm">Number needed to harm</a>, <a href="/wiki/Risk_ratio" class="mw-redirect" title="Risk ratio">Risk ratio</a>, <a href="/wiki/Relative_risk_reduction" title="Relative risk reduction">Relative risk reduction</a>, <a href="/wiki/Odds_ratio" title="Odds ratio">Odds ratio</a>, <a href="/wiki/Hazard_ratio" title="Hazard ratio">Hazard ratio</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Population_impact_measure" title="Population impact measure">Population impact</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="/wiki/Attributable_fraction_among_the_exposed" title="Attributable fraction among the exposed">Attributable fraction among the exposed</a>, <a href="/wiki/Attributable_fraction_for_the_population" title="Attributable fraction for the population">Attributable fraction for the population</a>, <a href="/wiki/Preventable_fraction_among_the_unexposed" title="Preventable fraction among the unexposed">Preventable fraction among the unexposed</a>, <a href="/wiki/Preventable_fraction_for_the_population" title="Preventable fraction for the population">Preventable fraction for the population</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Epidemiological_methods#Other_measures" class="mw-redirect" title="Epidemiological methods">Other</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="/wiki/Clinical_endpoint" title="Clinical endpoint">Clinical endpoint</a>, <a href="/wiki/Virulence" title="Virulence">Virulence</a>, <a href="/wiki/Infectivity" title="Infectivity">Infectivity</a>, <a href="/wiki/Mortality_rate" title="Mortality rate">Mortality rate</a>, <a href="/wiki/Morbidity" class="mw-redirect" title="Morbidity">Morbidity</a>, <a href="/wiki/Case_fatality_rate" title="Case fatality rate">Case fatality rate</a>, <a href="/wiki/Specificity_and_sensitivity" class="mw-redirect" title="Specificity and sensitivity">Specificity and sensitivity</a>, <a href="/wiki/Likelihood_ratios_in_diagnostic_testing" title="Likelihood ratios in diagnostic testing">Likelihood-ratios</a>, <a href="/wiki/Pre-_and_post-test_probability" title="Pre- and post-test probability">Pre- and post-test probability</a></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Trial/test types</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/In_vitro" title="In vitro">In vitro</a></li> <li><a href="/wiki/In_vivo" title="In vivo">In vivo</a></li> <li><a href="/wiki/Animal_testing" title="Animal testing">Animal testing</a></li> <li><a href="/wiki/Animal_testing_on_non-human_primates" title="Animal testing on non-human primates">Animal testing on non-human primates</a></li> <li><a href="/wiki/First-in-man_study" class="mw-redirect" title="First-in-man study">First-in-man study</a></li> <li><a href="/wiki/Multicenter_trial" title="Multicenter trial">Multicenter trial</a></li> <li><a href="/wiki/Seeding_trial" title="Seeding trial">Seeding trial</a></li> <li><a href="/wiki/Vaccine_trial" title="Vaccine trial">Vaccine trial</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Analysis_of_clinical_trials" title="Analysis of clinical trials">Analysis of clinical trials</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Risk%E2%80%93benefit_ratio" title="Risk–benefit ratio">Risk–benefit ratio</a></li> <li><a href="/wiki/Systematic_review" title="Systematic review">Systematic review</a></li> <li><a href="/wiki/Reproducibility" title="Reproducibility">Replication</a></li> <li><a class="mw-selflink selflink">Meta-analysis</a></li> <li><a href="/wiki/Intention-to-treat_analysis" title="Intention-to-treat analysis">Intention-to-treat analysis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Interpretation of results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Selection_bias" title="Selection bias">Selection bias</a></li> <li><a href="/wiki/Survivorship_bias" title="Survivorship bias">Survivorship bias</a></li> <li><a href="/wiki/Correlation_does_not_imply_causation" title="Correlation does not imply causation">Correlation does not imply causation</a></li> <li><a href="/wiki/Null_result" title="Null result">Null result</a></li> <li><a href="/wiki/Sex_as_a_biological_variable" title="Sex as a biological variable">Sex as a biological variable</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><a href="/wiki/Category:Clinical_research" title="Category:Clinical research">Category</a></b></li> <li><b><a href="/wiki/Glossary_of_clinical_research" title="Glossary of clinical research">Glossary</a></b></li> <li><b><a href="/wiki/List_of_clinical_research_topics" class="mw-redirect" title="List of clinical research topics">List of topics</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Meta-" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li 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