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class="vector-toc-list"> <li id="toc-Relation_to_unexplained_variance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_unexplained_variance"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Relation to unexplained variance</span> </div> </a> <ul id="toc-Relation_to_unexplained_variance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_explained_variance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_explained_variance"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>As explained variance</span> </div> </a> <ul id="toc-As_explained_variance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_squared_correlation_coefficient" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_squared_correlation_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>As squared correlation coefficient</span> </div> </a> <ul id="toc-As_squared_correlation_coefficient-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Interpretation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Interpretation</span> </div> </a> <button aria-controls="toc-Interpretation-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 Interpretation subsection</span> </button> <ul id="toc-Interpretation-sublist" class="vector-toc-list"> <li id="toc-In_a_multiple_linear_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_a_multiple_linear_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>In a multiple linear model</span> </div> </a> <ul id="toc-In_a_multiple_linear_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inflation_of_R2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inflation_of_R2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Inflation of <i>R</i><sup>2</sup></span> </div> </a> <ul id="toc-Inflation_of_R2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Caveats" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Caveats"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Caveats</span> </div> </a> <ul id="toc-Caveats-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Extensions</span> </div> </a> <button aria-controls="toc-Extensions-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 Extensions subsection</span> </button> <ul id="toc-Extensions-sublist" class="vector-toc-list"> <li id="toc-Adjusted_R2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Adjusted_R2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Adjusted <i>R</i><sup>2</sup></span> </div> </a> <ul id="toc-Adjusted_R2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coefficient_of_partial_determination" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coefficient_of_partial_determination"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Coefficient of partial determination</span> </div> </a> <ul id="toc-Coefficient_of_partial_determination-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizing_and_decomposing_R2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalizing_and_decomposing_R2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Generalizing and decomposing <i>R</i><sup>2</sup></span> </div> </a> <ul id="toc-Generalizing_and_decomposing_R2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-R2_in_logistic_regression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#R2_in_logistic_regression"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span><i>R</i><sup>2</sup> in logistic regression</span> </div> </a> <ul id="toc-R2_in_logistic_regression-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Comparison_with_residual_statistics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comparison_with_residual_statistics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Comparison with residual statistics</span> </div> </a> <ul id="toc-Comparison_with_residual_statistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History</span> </div> </a> <ul id="toc-History-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-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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" 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Available in 19 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-19" 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">19 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/%D9%85%D8%B9%D8%A7%D9%85%D9%84_%D8%A7%D9%84%D8%AA%D8%AD%D8%AF%D9%8A%D8%AF" title="معامل التحديد – Arabic" lang="ar" hreflang="ar" data-title="معامل التحديد" 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/Coeficient_de_determinaci%C3%B3" title="Coeficient de determinació – Catalan" lang="ca" hreflang="ca" data-title="Coeficient de determinació" 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/Koeficient_determinace" title="Koeficient determinace – Czech" lang="cs" hreflang="cs" data-title="Koeficient determinace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://de.wikipedia.org/wiki/Bestimmtheitsma%C3%9F" title="Bestimmtheitsmaß – German" lang="de" hreflang="de" data-title="Bestimmtheitsmaß" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Coeficiente_de_determinaci%C3%B3n" title="Coeficiente de determinación – Spanish" lang="es" hreflang="es" data-title="Coeficiente de determinación" 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/Mugatze-koefiziente" title="Mugatze-koefiziente – Basque" lang="eu" hreflang="eu" data-title="Mugatze-koefiziente" 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/%D8%B6%D8%B1%DB%8C%D8%A8_%D8%AA%D8%B9%DB%8C%DB%8C%D9%86" title="ضریب تعیین – Persian" lang="fa" hreflang="fa" data-title="ضریب تعیین" 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/Coefficient_de_d%C3%A9termination" title="Coefficient de détermination – French" lang="fr" hreflang="fr" data-title="Coefficient de détermination" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B2%B0%EC%A0%95%EA%B3%84%EC%88%98" title="결정계수 – Korean" lang="ko" hreflang="ko" data-title="결정계수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Coefficiente_di_determinazione" title="Coefficiente di determinazione – Italian" lang="it" hreflang="it" data-title="Coefficiente di determinazione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Determinatieco%C3%ABffici%C3%ABnt" title="Determinatiecoëfficiënt – Dutch" lang="nl" hreflang="nl" data-title="Determinatiecoëfficiënt" 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/%E6%B1%BA%E5%AE%9A%E4%BF%82%E6%95%B0" title="決定係数 – Japanese" lang="ja" hreflang="ja" data-title="決定係数" 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/Determinasjonskoeffisient" title="Determinasjonskoeffisient – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Determinasjonskoeffisient" 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/Wsp%C3%B3%C5%82czynnik_determinacji" title="Współczynnik determinacji – Polish" lang="pl" hreflang="pl" data-title="Współczynnik determinacji" 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/Coeficiente_de_determina%C3%A7%C3%A3o" title="Coeficiente de determinação – Portuguese" lang="pt" hreflang="pt" data-title="Coeficiente de determinação" 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%9A%D0%BE%D1%8D%D1%84%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82_%D0%B4%D0%B5%D1%82%D0%B5%D1%80%D0%BC%D0%B8%D0%BD%D0%B0%D1%86%D0%B8%D0%B8" title="Коэффициент детерминации – Russian" lang="ru" hreflang="ru" data-title="Коэффициент детерминации" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82_%D0%B4%D0%B5%D1%82%D0%B5%D1%80%D0%BC%D1%96%D0%BD%D0%B0%D1%86%D1%96%D1%97" title="Коефіцієнт детермінації – Ukrainian" lang="uk" hreflang="uk" data-title="Коефіцієнт детермінації" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%B1%BA%E5%AE%9A%E7%B3%BB%E6%95%B8" title="決定系數 – Cantonese" lang="yue" hreflang="yue" data-title="決定系數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%86%B3%E5%AE%9A%E7%B3%BB%E6%95%B0" title="决定系数 – Chinese" lang="zh" hreflang="zh" data-title="决定系数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q192830#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Indicator for how well data points fit a line or curve</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a> or <a href="/wiki/Coefficient_of_correlation" class="mw-redirect" title="Coefficient of correlation">Coefficient of correlation</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid 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padding-right: 3em; font-weight: normal; text-align: left">You can help <b>expand this article with text translated from <a href="https://de.wikipedia.org/wiki/Bestimmtheitsma%C3%9F" class="extiw" title="de:Bestimmtheitsmaß">the corresponding article</a> in German</b>.  <i><small>(September 2019)</small></i> <small>Click [show] for important translation instructions.</small></div><div class="hidden-content mw-collapsible-content" style=""> <ul><li><a rel="nofollow" class="external text" href="https://translate.google.com/translate?&u=https%3A%2F%2Fde.wikipedia.org%2Fwiki%2FBestimmtheitsma%C3%9F&sl=de&tl=en&prev=_t&hl=en">View</a> a machine-translated version of the German article.</li> <li>Machine translation, like <a rel="nofollow" class="external text" href="https://deepl.com">DeepL</a> or <a rel="nofollow" class="external text" href="https://translate.google.com/">Google Translate</a>, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Wikipedia.</li> <li>Consider <b><a href="/wiki/Template:Expand_German#Topics_and_categorization" title="Template:Expand German">adding a topic</a></b> to this template: there are already 2,073 articles in the <a href="/wiki/Category:Articles_needing_translation_from_German_Wikipedia" title="Category:Articles needing translation from German Wikipedia">main category</a>, and specifying<code class="tpl-para" style="word-break:break-word;">|topic=</code> will aid in categorization.</li> <li>Do not translate text that appears unreliable or low-quality. If possible, verify the text with references provided in the foreign-language article.</li> <li>You <b>must</b> provide <a href="/wiki/Wikipedia:Copying_within_Wikipedia" title="Wikipedia:Copying within Wikipedia">copyright attribution</a> in the <a href="/wiki/Help:Edit_summary" title="Help:Edit summary">edit summary</a> accompanying your translation by providing an <a href="/wiki/Help:Interlanguage_links" title="Help:Interlanguage links">interlanguage link</a> to the source of your translation. A model attribution edit summary is <code>Content in this edit is translated from the existing German Wikipedia article at [[:de:Bestimmtheitsmaß]]; see its history for attribution.</code></li> <li>You may also add the template <code>{{Translated|de|Bestimmtheitsmaß}}</code> to the <a href="/wiki/Talk:Coefficient_of_determination" title="Talk:Coefficient of determination">talk page</a>.</li> <li>For more guidance, see <a href="/wiki/Wikipedia:Translation" title="Wikipedia:Translation">Wikipedia:Translation</a>.</li></ul></div></div></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Okuns_law_quarterly_differences.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Okuns_law_quarterly_differences.svg/300px-Okuns_law_quarterly_differences.svg.png" decoding="async" width="300" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Okuns_law_quarterly_differences.svg/450px-Okuns_law_quarterly_differences.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Okuns_law_quarterly_differences.svg/600px-Okuns_law_quarterly_differences.svg.png 2x" data-file-width="301" data-file-height="199" /></a><figcaption><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a> regression of <a href="/wiki/Okun%27s_law" title="Okun's law">Okun's law</a>. Since the regression line does not miss any of the points by very much, the <i>R</i><sup>2</sup> of the regression is relatively high.</figcaption></figure> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, the <b>coefficient of determination</b>, denoted <i>R</i><sup>2</sup> or <i>r</i><sup>2</sup> and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). </p><p>It is a <a href="/wiki/Statistic" title="Statistic">statistic</a> used in the context of <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a> whose main purpose is either the <a href="/wiki/Prediction#Statistics" title="Prediction">prediction</a> of future outcomes or the testing of <a href="/wiki/Hypotheses" class="mw-redirect" title="Hypotheses">hypotheses</a>, on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>There are several definitions of <i>R</i><sup>2</sup> that are only sometimes equivalent. One class of such cases includes that of <a href="/wiki/Simple_linear_regression" title="Simple linear regression">simple linear regression</a> where <i>r</i><sup>2</sup> is used instead of <i>R</i><sup>2</sup>. When only an <a href="/wiki/Regression_intercept" class="mw-redirect" title="Regression intercept">intercept</a> is included, then <i>r</i><sup>2</sup> is simply the square of the sample <a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">correlation coefficient</a> (i.e., <i>r</i>) between the observed outcomes and the observed predictor values.<sup id="cite_ref-Devore_4-0" class="reference"><a href="#cite_note-Devore-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> If additional <a href="/wiki/Regressor" class="mw-redirect" title="Regressor">regressors</a> are included, <i>R</i><sup>2</sup> is the square of the <a href="/wiki/Coefficient_of_multiple_correlation" title="Coefficient of multiple correlation">coefficient of multiple correlation</a>. In both such cases, the coefficient of determination normally ranges from 0 to 1. </p><p>There are cases where <i>R</i><sup>2</sup> can yield negative values. This can arise when the predictions that are being compared to the corresponding outcomes have not been derived from a model-fitting procedure using those data. Even if a model-fitting procedure has been used, <i>R</i><sup>2</sup> may still be negative, for example when linear regression is conducted without including an intercept,<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> or when a non-linear function is used to fit the data.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In cases where negative values arise, the mean of the data provides a better fit to the outcomes than do the fitted function values, according to this particular criterion. </p><p>The coefficient of determination can be more intuitively informative than <a href="/wiki/Mean_absolute_error" title="Mean absolute error">MAE</a>, <a href="/wiki/MAPE" class="mw-redirect" title="MAPE">MAPE</a>, <a href="/wiki/Mean_square_error" class="mw-redirect" title="Mean square error">MSE</a>, and <a href="/wiki/RMSE" class="mw-redirect" title="RMSE">RMSE</a> in <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a> evaluation, as the former can be expressed as a percentage, whereas the latter measures have arbitrary ranges. It also proved more robust for poor fits compared to <a href="/wiki/SMAPE" class="mw-redirect" title="SMAPE">SMAPE</a> on certain test datasets.<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> </p><p>When evaluating the goodness-of-fit of simulated (<i>Y</i><sub>pred</sub>) versus measured (<i>Y</i><sub>obs</sub>) values, it is not appropriate to base this on the <i>R</i><sup>2</sup> of the linear regression (i.e., <i>Y</i><sub>obs</sub>= <i>m</i>·<i>Y</i><sub>pred</sub> + b).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2021)">citation needed</span></a></i>]</sup> The <i>R</i><sup>2</sup> quantifies the degree of any linear correlation between <i>Y</i><sub>obs</sub> and <i>Y</i><sub>pred</sub>, while for the goodness-of-fit evaluation only one specific linear correlation should be taken into consideration: <i>Y</i><sub>obs</sub> = 1·<i>Y</i><sub>pred</sub> + 0 (i.e., the 1:1 line).<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Coefficient_of_Determination.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Coefficient_of_Determination.svg/400px-Coefficient_of_Determination.svg.png" decoding="async" width="400" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Coefficient_of_Determination.svg/600px-Coefficient_of_Determination.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/Coefficient_of_Determination.svg/800px-Coefficient_of_Determination.svg.png 2x" data-file-width="1000" data-file-height="500" /></a><figcaption><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{2}=1-{\frac {\color {blue}{SS_{\text{res}}}}{\color {red}{SS_{\text{tot}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> </mrow> </mstyle> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}=1-{\frac {\color {blue}{SS_{\text{res}}}}{\color {red}{SS_{\text{tot}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b863cb70dd04b45984983cb6ed00801d5eddc94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.013ex; height:5.843ex;" alt="{\displaystyle R^{2}=1-{\frac {\color {blue}{SS_{\text{res}}}}{\color {red}{SS_{\text{tot}}}}}}"></span><br /> The better the linear regression (on the right) fits the data in comparison to the simple average (on the left graph), the closer the value of <i>R</i><sup>2</sup> is to 1. The areas of the blue squares represent the squared residuals with respect to the linear regression. The areas of the red squares represent the squared residuals with respect to the average value.</figcaption></figure> <p>A <a href="/wiki/Data_set" title="Data set">data set</a> has <i>n</i> values marked <i>y</i><sub>1</sub>, ..., <i>y</i><sub><i>n</i></sub> (collectively known as <i>y</i><sub><i>i</i></sub> or as a vector <i><b>y</b></i> = [<i>y</i><sub>1</sub>, ..., <i>y</i><sub><i>n</i></sub>]<sup>T</sup>), each associated with a fitted (or modeled, or predicted) value <i>f</i><sub>1</sub>, ..., <i>f</i><sub><i>n</i></sub> (known as <i>f</i><sub><i>i</i></sub>, or sometimes <i>ŷ</i><sub><i>i</i></sub>, as a vector <i><b>f</b></i>). </p><p>Define the <a href="/wiki/Residuals_(statistics)" class="mw-redirect" title="Residuals (statistics)">residuals</a> as <span class="nowrap"><i>e</i><sub><i>i</i></sub> = <i>y</i><sub><i>i</i></sub> − <i>f</i><sub><i>i</i></sub></span> (forming a vector <i><b>e</b></i>). </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b298744237368f34e61ff7dc90b34016a7037af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.343ex;" alt="{\displaystyle {\bar {y}}}"></span> is the mean of the observed data: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {y}}={\frac {1}{n}}\sum _{i=1}^{n}y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {y}}={\frac {1}{n}}\sum _{i=1}^{n}y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44a8b57e2a4335f02faa2bd5003d94979af4f408" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.7ex; height:6.843ex;" alt="{\displaystyle {\bar {y}}={\frac {1}{n}}\sum _{i=1}^{n}y_{i}}"></span> then the variability of the data set can be measured with two <a href="/wiki/Mean_squared_error" title="Mean squared error">sums of squares</a> formulas: </p> <ul><li>The sum of squares of residuals, also called the <a href="/wiki/Residual_sum_of_squares" title="Residual sum of squares">residual sum of squares</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SS_{\text{res}}=\sum _{i}(y_{i}-f_{i})^{2}=\sum _{i}e_{i}^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SS_{\text{res}}=\sum _{i}(y_{i}-f_{i})^{2}=\sum _{i}e_{i}^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2669c9340581d55b274d3b8ea67a7deb2225510b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.579ex; height:5.509ex;" alt="{\displaystyle SS_{\text{res}}=\sum _{i}(y_{i}-f_{i})^{2}=\sum _{i}e_{i}^{2}\,}"></span></li> <li>The <a href="/wiki/Total_sum_of_squares" title="Total sum of squares">total sum of squares</a> (proportional to the <a href="/wiki/Variance" title="Variance">variance</a> of the data): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SS_{\text{tot}}=\sum _{i}(y_{i}-{\bar {y}})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SS_{\text{tot}}=\sum _{i}(y_{i}-{\bar {y}})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1f55d7e84c24299917fb3fec4d0439b81e728d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.656ex; height:5.509ex;" alt="{\displaystyle SS_{\text{tot}}=\sum _{i}(y_{i}-{\bar {y}})^{2}}"></span></li></ul> <p>The most general definition of the coefficient of determination is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{2}=1-{SS_{\rm {res}} \over SS_{\rm {tot}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}=1-{SS_{\rm {res}} \over SS_{\rm {tot}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42b3cd78531d8b2f2590c7fda76acff7caeb643a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.013ex; height:5.843ex;" alt="{\displaystyle R^{2}=1-{SS_{\rm {res}} \over SS_{\rm {tot}}}}"></span> </p><p>In the best case, the modeled values exactly match the observed values, which results in <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 SS_{\text{res}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SS_{\text{res}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d346fa0b307220ee37e11e0545b6a400e3b7006" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.44ex; height:2.509ex;" alt="{\displaystyle SS_{\text{res}}=0}"></span> and <span class="nowrap"><i>R</i><sup>2</sup> = 1</span>. A baseline model, which always predicts <span style="text-decoration:overline;"><i>y</i></span>, will have <span class="nowrap"><i>R</i><sup>2</sup> = 0</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_unexplained_variance">Relation to unexplained variance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=2" title="Edit section: Relation to unexplained variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fraction_of_variance_unexplained" title="Fraction of variance unexplained">Fraction of variance unexplained</a></div> <p>In a general form, <i>R</i><sup>2</sup> can be seen to be related to the fraction of variance unexplained (FVU), since the second term compares the unexplained variance (variance of the model's errors) with the total variance (of the data): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{2}=1-{\text{FVU}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>FVU</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}=1-{\text{FVU}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3a91b19063a8783a00a4faf55ab6740c11b51b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.924ex; height:2.843ex;" alt="{\displaystyle R^{2}=1-{\text{FVU}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="As_explained_variance">As explained variance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=3" title="Edit section: As explained variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A larger value of <i>R</i><sup>2</sup> implies a more successful regression model.<sup id="cite_ref-Devore_4-1" class="reference"><a href="#cite_note-Devore-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 463">: 463 </span></sup> Suppose <span class="nowrap"><i>R</i><sup>2</sup> = 0.49</span>. This implies that 49% of the variability of the dependent variable in the data set has been accounted for, and the remaining 51% of the variability is still unaccounted for. For regression models, the regression sum of squares, also called the <a href="/wiki/Explained_sum_of_squares" title="Explained sum of squares">explained sum of squares</a>, is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SS_{\text{reg}}=\sum _{i}(f_{i}-{\bar {y}})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>reg</mtext> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SS_{\text{reg}}=\sum _{i}(f_{i}-{\bar {y}})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c592f4737ce8fc69472f60bb75deb1ab1d9a799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.752ex; height:5.509ex;" alt="{\displaystyle SS_{\text{reg}}=\sum _{i}(f_{i}-{\bar {y}})^{2}}"></span></dd></dl> <p>In some cases, as in <a href="/wiki/Simple_linear_regression" title="Simple linear regression">simple linear regression</a>, the <a href="/wiki/Total_sum_of_squares" title="Total sum of squares">total sum of squares</a> equals the sum of the two other sums of squares defined above: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SS_{\text{res}}+SS_{\text{reg}}=SS_{\text{tot}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> <mo>+</mo> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>reg</mtext> </mrow> </msub> <mo>=</mo> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SS_{\text{res}}+SS_{\text{reg}}=SS_{\text{tot}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0d94955df2eeb90bd1885b05ea76a8408800f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.729ex; height:2.843ex;" alt="{\displaystyle SS_{\text{res}}+SS_{\text{reg}}=SS_{\text{tot}}}"></span></dd></dl> <p>See <a href="/wiki/Explained_sum_of_squares#Partitioning_in_the_general_ordinary_least_squares_model" title="Explained sum of squares">Partitioning in the general OLS model</a> for a derivation of this result for one case where the relation holds. When this relation does hold, the above definition of <i>R</i><sup>2</sup> is equivalent to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{2}={\frac {SS_{\text{reg}}}{SS_{\text{tot}}}}={\frac {SS_{\text{reg}}/n}{SS_{\text{tot}}/n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>reg</mtext> </mrow> </msub> </mrow> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>reg</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}={\frac {SS_{\text{reg}}}{SS_{\text{tot}}}}={\frac {SS_{\text{reg}}/n}{SS_{\text{tot}}/n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55776df55201c4bf54ba05eb9a2d63a64fb40528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.951ex; height:6.509ex;" alt="{\displaystyle R^{2}={\frac {SS_{\text{reg}}}{SS_{\text{tot}}}}={\frac {SS_{\text{reg}}/n}{SS_{\text{tot}}/n}}}"></span></dd></dl> <p>where <i>n</i> is the number of observations (cases) on the variables. </p><p>In this form <i>R</i><sup>2</sup> is expressed as the ratio of the <a href="/wiki/Explained_variation" title="Explained variation">explained variance</a> (variance of the model's predictions, which is <span class="nowrap"><i>SS</i><sub>reg</sub> / <i>n</i></span>) to the total variance (sample variance of the dependent variable, which is <span class="nowrap"><i>SS</i><sub>tot</sub> / <i>n</i></span>). </p><p>This partition of the sum of squares holds for instance when the model values <i>ƒ</i><sub><i>i</i></sub> have been obtained by <a href="/wiki/Linear_regression" title="Linear regression">linear regression</a>. A milder <a href="/wiki/Sufficient_condition" class="mw-redirect" title="Sufficient condition">sufficient condition</a> reads as follows: The model has the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}={\widehat {\alpha }}+{\widehat {\beta }}q_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}={\widehat {\alpha }}+{\widehat {\beta }}q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5386280bbe75b1e23a65ab13698a2f24da39cf2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.718ex; height:3.343ex;" alt="{\displaystyle f_{i}={\widehat {\alpha }}+{\widehat {\beta }}q_{i}}"></span></dd></dl> <p>where the <i>q</i><sub><i>i</i></sub> are arbitrary values that may or may not depend on <i>i</i> or on other free parameters (the common choice <i>q</i><sub><i>i</i></sub> = <i>x</i><sub><i>i</i></sub> is just one special case), and the coefficient estimates <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 {\widehat {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ca8a0303970de1e7553ee5eccd19e8ab5d2007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.343ex;" alt="{\displaystyle {\widehat {\alpha }}}"></span> 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 {\widehat {\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21fd425a5a1a245a101aae3ff48df531b4dc96ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\displaystyle {\widehat {\beta }}}"></span> are obtained by minimizing the residual sum of squares. </p><p>This set of conditions is an important one and it has a number of implications for the properties of the fitted <a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">residuals</a> and the modelled values. In particular, under these conditions: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {f}}={\bar {y}}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {f}}={\bar {y}}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c14c80f4f4e03541606e0d113a3094cdd3d49910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.134ex; height:3.009ex;" alt="{\displaystyle {\bar {f}}={\bar {y}}.\,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="As_squared_correlation_coefficient">As squared correlation coefficient</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=4" title="Edit section: As squared correlation coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In linear least squares <a href="/wiki/Multiple_regression" class="mw-redirect" title="Multiple regression">multiple regression</a> (with fitted intercept and slope), <i>R</i><sup>2</sup> equals <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 \rho ^{2}(y,f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{2}(y,f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/472e44fe51a924e525e30ecc61363262bc319213" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.533ex; height:3.176ex;" alt="{\displaystyle \rho ^{2}(y,f)}"></span> the square of the <a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson correlation coefficient</a> between the observed <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> and modeled (predicted) <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> data values of the dependent variable. </p><p>In a <a href="/wiki/Simple_regression" class="mw-redirect" title="Simple regression">linear least squares regression with a single explanator</a> (with fitted intercept and slope), this is also equal to <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 \rho ^{2}(y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{2}(y,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3755a8420f206551ec4f9be3caa368ed11a4fe00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.585ex; height:3.176ex;" alt="{\displaystyle \rho ^{2}(y,x)}"></span> the squared Pearson correlation coefficient between the dependent variable <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> and explanatory variable <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. </p><p>It should not be confused with the correlation coefficient between two <a href="/wiki/Explanatory_variable" class="mw-redirect" title="Explanatory variable">explanatory variables</a>, defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{{\widehat {\alpha }},{\widehat {\beta }}}={\operatorname {cov} \left({\widehat {\alpha }},{\widehat {\beta }}\right) \over \sigma _{\widehat {\alpha }}\sigma _{\widehat {\beta }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>^</mo> </mover> </mrow> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{{\widehat {\alpha }},{\widehat {\beta }}}={\operatorname {cov} \left({\widehat {\alpha }},{\widehat {\beta }}\right) \over \sigma _{\widehat {\alpha }}\sigma _{\widehat {\beta }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2bec9aaa7e4be77d61b3c8718a6a23650be86f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.89ex; height:8.843ex;" alt="{\displaystyle \rho _{{\widehat {\alpha }},{\widehat {\beta }}}={\operatorname {cov} \left({\widehat {\alpha }},{\widehat {\beta }}\right) \over \sigma _{\widehat {\alpha }}\sigma _{\widehat {\beta }}},}"></span></dd></dl> <p>where the covariance between two coefficient estimates, as well as their <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviations</a>, are obtained from the <a href="/wiki/Ordinary_least_squares#Covariance_matrix" title="Ordinary least squares">covariance matrix</a> of the coefficient estimates, <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 (X^{T}X)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>X</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X^{T}X)^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbff6be7ba8dd7435c6bccafe14e0a0ae8909eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.508ex; height:3.176ex;" alt="{\displaystyle (X^{T}X)^{-1}}"></span>. </p><p>Under more general modeling conditions, where the predicted values might be generated from a model different from linear least squares regression, an <i>R</i><sup>2</sup> value can be calculated as the square of the <a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">correlation coefficient</a> between the original <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> and modeled <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> data values. In this case, the value is not directly a measure of how good the modeled values are, but rather a measure of how good a predictor might be constructed from the modeled values (by creating a revised predictor of the form <span class="nowrap"><i>α</i> + <i>βƒ</i><sub><i>i</i></sub></span>).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="The citation for the next sentence does not discuss the information in this sentence. (March 2017)">citation needed</span></a></i>]</sup> According to Everitt,<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> this usage is specifically the definition of the term "coefficient of determination": the square of the correlation between two (general) variables. </p> <div class="mw-heading mw-heading2"><h2 id="Interpretation">Interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=5" title="Edit section: Interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>R</i><sup>2</sup> is a measure of the <a href="/wiki/Goodness_of_fit" title="Goodness of fit">goodness of fit</a> of a model.<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> In regression, the <i>R</i><sup>2</sup> coefficient of determination is a statistical measure of how well the regression predictions approximate the real data points. An <i>R</i><sup>2</sup> of 1 indicates that the regression predictions perfectly fit the data. </p><p>Values of <i>R</i><sup>2</sup> outside the range 0 to 1 occur when the model fits the data worse than the worst possible <a href="/wiki/Least-squares" class="mw-redirect" title="Least-squares">least-squares</a> predictor (equivalent to a horizontal hyperplane at a height equal to the mean of the observed data). This occurs when a wrong model was chosen, or nonsensical constraints were applied by mistake. If equation 1 of Kvålseth<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> is used (this is the equation used most often), <i>R</i><sup>2</sup> can be less than zero. If equation 2 of Kvålseth is used, <i>R</i><sup>2</sup> can be greater than one. </p><p>In all instances where <i>R</i><sup>2</sup> is used, the predictors are calculated by ordinary least-squares regression: that is, by minimizing <i>SS</i><sub>res</sub>. In this case, <i>R</i><sup>2</sup> increases as the number of variables in the model is increased (<i>R</i><sup>2</sup> is <a href="/wiki/Monotonic_function" title="Monotonic function">monotone increasing</a> with the number of variables included—it will never decrease). This illustrates a drawback to one possible use of <i>R</i><sup>2</sup>, where one might keep adding variables (<a href="/wiki/Kitchen_sink_regression" title="Kitchen sink regression">kitchen sink regression</a>) to increase the <i>R</i><sup>2</sup> value. For example, if one is trying to predict the sales of a model of car from the car's gas mileage, price, and engine power, one can include probably irrelevant factors such as the first letter of the model's name or the height of the lead engineer designing the car because the <i>R</i><sup>2</sup> will never decrease as variables are added and will likely experience an increase due to chance alone. </p><p>This leads to the alternative approach of looking at the <a href="#Adjusted_R2">adjusted <i>R</i><sup>2</sup></a>. The explanation of this statistic is almost the same as <i>R</i><sup>2</sup> but it penalizes the statistic as extra variables are included in the model. For cases other than fitting by ordinary least squares, the <i>R</i><sup>2</sup> statistic can be calculated as above and may still be a useful measure. If fitting is by <a href="/wiki/Weighted_least_squares" title="Weighted least squares">weighted least squares</a> or <a href="/wiki/Generalized_least_squares" title="Generalized least squares">generalized least squares</a>, alternative versions of <i>R</i><sup>2</sup> can be calculated appropriate to those statistical frameworks, while the "raw" <i>R</i><sup>2</sup> may still be useful if it is more easily interpreted. Values for <i>R</i><sup>2</sup> can be calculated for any type of predictive model, which need not have a statistical basis. </p> <div class="mw-heading mw-heading3"><h3 id="In_a_multiple_linear_model">In a multiple linear model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=6" title="Edit section: In a multiple linear model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a linear model with <a href="/wiki/Multiple_regression" class="mw-redirect" title="Multiple regression">more than a single explanatory variable</a>, of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}=\beta _{0}+\sum _{j=1}^{p}\beta _{j}X_{i,j}+\varepsilon _{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> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}=\beta _{0}+\sum _{j=1}^{p}\beta _{j}X_{i,j}+\varepsilon _{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59bf17d9a20fae5e903e3e444e8a17027358d8d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.656ex; height:7.343ex;" alt="{\displaystyle Y_{i}=\beta _{0}+\sum _{j=1}^{p}\beta _{j}X_{i,j}+\varepsilon _{i},}"></span></dd></dl> <p>where, for the <i>i</i>th case, <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"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </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/00e3ac799afe4689d537cdb2c01d56f43275ff98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle {Y_{i}}}"></span> is the response variable, <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 X_{i,1},\dots ,X_{i,p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i,1},\dots ,X_{i,p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3618bd6b70d01eecb44e97e6564601eeb04d8451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.19ex; height:2.843ex;" alt="{\displaystyle X_{i,1},\dots ,X_{i,p}}"></span> are <i>p</i> regressors, 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 \varepsilon _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e1b6ad3cbad4af49bf21a3ad2dc379ff045079" 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 \varepsilon _{i}}"></span> is a mean zero <a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">error</a> term. The quantities <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 \beta _{0},\dots ,\beta _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0},\dots ,\beta _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702dc960429decdafe7ff833e862173b363bbbc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.923ex; height:2.843ex;" alt="{\displaystyle \beta _{0},\dots ,\beta _{p}}"></span> are unknown coefficients, whose values are estimated by <a href="/wiki/Least_squares" title="Least squares">least squares</a>. The coefficient of determination <i>R</i><sup>2</sup> is a measure of the global fit of the model. Specifically, <i>R</i><sup>2</sup> is an element of [0, 1] and represents the proportion of variability in <i>Y</i><sub><i>i</i></sub> that may be attributed to some linear combination of the regressors (<a href="/wiki/Explanatory_variable" class="mw-redirect" title="Explanatory variable">explanatory variables</a>) in <i>X</i>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p><i>R</i><sup>2</sup> is often interpreted as the proportion of response variation "explained" by the regressors in the model. Thus, <i>R</i><sup>2</sup> = 1 indicates that the fitted model explains all variability in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>, while <i>R</i><sup>2</sup> = 0 indicates no 'linear' relationship (for straight line regression, this means that the straight line model is a constant line (slope = 0, intercept = <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 {\bar {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b298744237368f34e61ff7dc90b34016a7037af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.343ex;" alt="{\displaystyle {\bar {y}}}"></span>) between the response variable and regressors). An interior value such as <i>R</i><sup>2</sup> = 0.7 may be interpreted as follows: "Seventy percent of the variance in the response variable can be explained by the explanatory variables. The remaining thirty percent can be attributed to unknown, <a href="/wiki/Lurking_variable" class="mw-redirect" title="Lurking variable">lurking variables</a> or inherent variability." </p><p>A caution that applies to <i>R</i><sup>2</sup>, as to other statistical descriptions of <a href="/wiki/Correlation" title="Correlation">correlation</a> and association is that "<a href="/wiki/Correlation_does_not_imply_causation" title="Correlation does not imply causation">correlation does not imply causation</a>." In other words, while correlations may sometimes provide valuable clues in uncovering causal relationships among variables, a non-zero estimated correlation between two variables is not, on its own, evidence that changing the value of one variable would result in changes in the values of other variables. For example, the practice of carrying matches (or a lighter) is correlated with incidence of lung cancer, but carrying matches does not cause cancer (in the standard sense of "cause"). </p><p>In case of a single regressor, fitted by least squares, <i>R</i><sup>2</sup> is the square of the <a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment correlation coefficient</a> relating the regressor and the response variable. More generally, <i>R</i><sup>2</sup> is the square of the correlation between the constructed predictor and the response variable. With more than one regressor, the <i>R</i><sup>2</sup> can be referred to as the <a href="/wiki/Coefficient_of_multiple_determination" class="mw-redirect" title="Coefficient of multiple determination">coefficient of multiple determination</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Inflation_of_R2">Inflation of <i>R</i><sup>2</sup></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=7" title="Edit section: Inflation of R2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Least_squares" title="Least squares">least squares</a> regression using typical data, <i>R</i><sup>2</sup> is at least weakly increasing with an increase in number of regressors in the model. Because increases in the number of regressors increase the value of <i>R</i><sup>2</sup>, <i>R</i><sup>2</sup> alone cannot be used as a meaningful comparison of models with very different numbers of independent variables. For a meaningful comparison between two models, an <a href="/wiki/F-test" title="F-test">F-test</a> can be performed on the <a href="/wiki/Residual_sum_of_squares" title="Residual sum of squares">residual sum of squares</a> <sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2021)">citation needed</span></a></i>]</sup>, similar to the F-tests in <a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a>, though this is not always appropriate<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag needs further explanation. (October 2021)">further explanation needed</span></a></i>]</sup>. As a reminder of this, some authors denote <i>R</i><sup>2</sup> by <i>R</i><sub><i>q</i></sub><sup>2</sup>, where <i>q</i> is the number of columns in <i>X</i> (the number of explanators including the constant). </p><p>To demonstrate this property, first recall that the objective of least squares linear regression is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min _{b}SS_{\text{res}}(b)\Rightarrow \min _{b}\sum _{i}(y_{i}-X_{i}b)^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munder> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min _{b}SS_{\text{res}}(b)\Rightarrow \min _{b}\sum _{i}(y_{i}-X_{i}b)^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b44446bc9a77453e6712c7b846d22509c969962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.231ex; height:5.509ex;" alt="{\displaystyle \min _{b}SS_{\text{res}}(b)\Rightarrow \min _{b}\sum _{i}(y_{i}-X_{i}b)^{2}\,}"></span></dd></dl> <p>where <i>X<sub>i</sub></i> is a row vector of values of explanatory variables for case <i>i</i> and <i>b</i> is a column vector of coefficients of the respective elements of <i>X<sub>i</sub></i>. </p><p>The optimal value of the objective is weakly smaller as more explanatory variables are added and hence additional columns of <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> (the explanatory data matrix whose <i>i</i>th row is <i>X<sub>i</sub></i>) are added, by the fact that less constrained minimization leads to an optimal cost which is weakly smaller than more constrained minimization does. Given the previous conclusion and noting that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SS_{tot}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SS_{tot}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030cf35df7e4317862e1760212bef487e7ed59ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.141ex; height:2.509ex;" alt="{\displaystyle SS_{tot}}"></span> depends only on <i>y</i>, the non-decreasing property of <i>R</i><sup>2</sup> follows directly from the definition above. </p><p>The intuitive reason that using an additional explanatory variable cannot lower the <i>R</i><sup>2</sup> is this: Minimizing <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 SS_{\text{res}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SS_{\text{res}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08186db3b6f8ccedb13fa37ea1c2e59478948cce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.179ex; height:2.509ex;" alt="{\displaystyle SS_{\text{res}}}"></span> is equivalent to maximizing <i>R</i><sup>2</sup>. When the extra variable is included, the data always have the option of giving it an estimated coefficient of zero, leaving the predicted values and the <i>R</i><sup>2</sup> unchanged. The only way that the optimization problem will give a non-zero coefficient is if doing so improves the <i>R</i><sup>2</sup>. </p><p>The above gives an analytical explanation of the inflation of <i>R</i><sup>2</sup>. Next, an example based on ordinary least square from a geometric perspective is shown below. <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> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Screen_shot_proj_fig.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Screen_shot_proj_fig.jpg/400px-Screen_shot_proj_fig.jpg" decoding="async" width="400" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Screen_shot_proj_fig.jpg/600px-Screen_shot_proj_fig.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Screen_shot_proj_fig.jpg/800px-Screen_shot_proj_fig.jpg 2x" data-file-width="868" data-file-height="433" /></a><figcaption>This is an example of residuals of regression models in smaller and larger spaces based on ordinary least square regression.</figcaption></figure> <p>A simple case to be considered first: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\beta _{0}+\beta _{1}\cdot X_{1}+\varepsilon \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\beta _{0}+\beta _{1}\cdot X_{1}+\varepsilon \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c38bf264d05942c2e868fecbd413428d4098b64b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.421ex; height:2.509ex;" alt="{\displaystyle Y=\beta _{0}+\beta _{1}\cdot X_{1}+\varepsilon \,}"></span></dd></dl> <p>This equation describes the <a href="/wiki/Ordinary_least_squares_regression" class="mw-redirect" title="Ordinary least squares regression">ordinary least squares regression</a> model with one regressor. The prediction is shown as the red vector in the figure on the right. Geometrically, it is the projection of true value onto a model space in <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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> (without intercept). The residual is shown as the red line. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\beta _{0}+\beta _{1}\cdot X_{1}+\beta _{2}\cdot X_{2}+\varepsilon \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\beta _{0}+\beta _{1}\cdot X_{1}+\beta _{2}\cdot X_{2}+\varepsilon \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a77ebd9e09837bd620bbc5d09a3203aa46700ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.289ex; height:2.509ex;" alt="{\displaystyle Y=\beta _{0}+\beta _{1}\cdot X_{1}+\beta _{2}\cdot X_{2}+\varepsilon \,}"></span></dd></dl> <p>This equation corresponds to the ordinary least squares regression model with two regressors. The prediction is shown as the blue vector in the figure on the right. Geometrically, it is the projection of true value onto a larger model space in <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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> (without intercept). Noticeably, the values of <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 \beta _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b42f71f244103a8fca3c76885c7580a92831c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{0}}"></span> 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 \beta _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b42f71f244103a8fca3c76885c7580a92831c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{0}}"></span> are not the same as in the equation for smaller model space as long as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}"></span> 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 X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}"></span> are not zero vectors. Therefore, the equations are expected to yield different predictions (i.e., the blue vector is expected to be different from the red vector). The least squares regression criterion ensures that the residual is minimized. In the figure, the blue line representing the residual is orthogonal to the model space in <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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>, giving the minimal distance from the space. </p><p>The smaller model space is a subspace of the larger one, and thereby the residual of the smaller model is guaranteed to be larger. Comparing the red and blue lines in the figure, the blue line is orthogonal to the space, and any other line would be larger than the blue one. Considering the calculation for <i>R</i><sup>2</sup>, a smaller value of <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 SS_{tot}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SS_{tot}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030cf35df7e4317862e1760212bef487e7ed59ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.141ex; height:2.509ex;" alt="{\displaystyle SS_{tot}}"></span> will lead to a larger value of <i>R</i><sup>2</sup>, meaning that adding regressors will result in inflation of <i>R</i><sup>2</sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Caveats">Caveats</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=8" title="Edit section: Caveats"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>R</i><sup>2</sup> does not indicate whether: </p> <ul><li>the independent variables are a cause of the changes in the <a href="/wiki/Dependent_variable" class="mw-redirect" title="Dependent variable">dependent variable</a>;</li> <li><a href="/wiki/Omitted-variable_bias" title="Omitted-variable bias">omitted-variable bias</a> exists;</li> <li>the correct <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a> was used;</li> <li>the most appropriate set of independent variables has been chosen;</li> <li>there is <a href="/wiki/Multicollinearity" title="Multicollinearity">collinearity</a> present in the data on the explanatory variables;</li> <li>the model might be improved by using transformed versions of the existing set of independent variables;</li> <li>there are enough data points to make a solid conclusion;</li> <li>there are a few <a href="/wiki/Outlier" title="Outlier">outliers</a> in an otherwise good sample.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Thiel-Sen_estimator.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Thiel-Sen_estimator.svg/220px-Thiel-Sen_estimator.svg.png" decoding="async" width="220" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Thiel-Sen_estimator.svg/330px-Thiel-Sen_estimator.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Thiel-Sen_estimator.svg/440px-Thiel-Sen_estimator.svg.png 2x" data-file-width="2936" data-file-height="3075" /></a><figcaption>Comparison of the <a href="/wiki/Theil%E2%80%93Sen_estimator" title="Theil–Sen estimator">Theil–Sen estimator</a> (black) and <a href="/wiki/Simple_linear_regression" title="Simple linear regression">simple linear regression</a> (blue) for a set of points with <a href="/wiki/Outlier" title="Outlier">outliers</a>. Because of the many outliers, neither of the regression lines fits the data well, as measured by the fact that neither gives a very high <i>R</i><sup>2</sup>.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=9" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Adjusted_R2">Adjusted <i>R</i><sup>2</sup></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=10" title="Edit section: Adjusted R2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Effect_size#Omega-squared_(ω2)" title="Effect size">Omega-squared (ω<sup>2</sup>)</a></div> <p>The use of an adjusted <i>R</i><sup>2</sup> (one common notation is <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 {\bar {R}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {R}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73a485e4d18c5d6ad8b3ebec4e35470684c9afb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:3.009ex;" alt="{\displaystyle {\bar {R}}^{2}}"></span>, pronounced "R bar squared"; another is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\text{a}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\text{a}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c809f585d68a07497363968d42fcfbb50d73a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.843ex;" alt="{\displaystyle R_{\text{a}}^{2}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\text{adj}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>adj</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\text{adj}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/274617a246146dd8a6f3e0cdd32c0c30b95d3af9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.236ex; height:3.509ex;" alt="{\displaystyle R_{\text{adj}}^{2}}"></span>) is an attempt to account for the phenomenon of the <i>R</i><sup>2</sup> automatically increasing when extra explanatory variables are added to the model. There are many different ways of adjusting.<sup id="cite_ref-raju_15-0" class="reference"><a href="#cite_note-raju-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> By far the most used one, to the point that it is typically just referred to as adjusted <i>R</i>, is the correction proposed by <a href="/wiki/Mordecai_Ezekiel" title="Mordecai Ezekiel">Mordecai Ezekiel</a>.<sup id="cite_ref-raju_15-1" class="reference"><a href="#cite_note-raju-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><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> The adjusted <i>R</i><sup>2</sup> is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {R}}^{2}={1-{SS_{\text{res}}/{\text{df}}_{\text{res}} \over SS_{\text{tot}}/{\text{df}}_{\text{tot}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>df</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> </mrow> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>df</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {R}}^{2}={1-{SS_{\text{res}}/{\text{df}}_{\text{res}} \over SS_{\text{tot}}/{\text{df}}_{\text{tot}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0ce04f5eeb594a40630419791cb987b4469141" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.514ex; height:6.509ex;" alt="{\displaystyle {\bar {R}}^{2}={1-{SS_{\text{res}}/{\text{df}}_{\text{res}} \over SS_{\text{tot}}/{\text{df}}_{\text{tot}}}}}"></span></dd></dl> <p>where df<sub><i>res</i></sub> is the <a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">degrees of freedom</a> of the estimate of the population variance around the model, and df<sub><i>tot</i></sub> is the degrees of freedom of the estimate of the population variance around the mean. df<sub><i>res</i></sub> is given in terms of the sample size <i>n</i> and the number of variables <i>p</i> in the model, <span class="nowrap">df<sub><i>res</i></sub> = <i>n</i> − <i>p</i> − 1</span>. df<sub><i>tot</i></sub> is given in the same way, but with <i>p</i> being unity for the mean, i.e. <span class="nowrap">df<sub><i>tot</i></sub> = <i>n</i> − 1</span>. </p><p>Inserting the degrees of freedom and using the definition of <i>R</i><sup>2</sup>, it can be rewritten as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {R}}^{2}=1-(1-R^{2}){n-1 \over n-p-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {R}}^{2}=1-(1-R^{2}){n-1 \over n-p-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a082d105dfbb4339e40cf7898950ce748743e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.793ex; height:5.676ex;" alt="{\displaystyle {\bar {R}}^{2}=1-(1-R^{2}){n-1 \over n-p-1}}"></span></dd></dl> <p>where <i>p</i> is the total number of explanatory variables in the model (excluding the intercept), and <i>n</i> is the sample size. </p><p>The adjusted <i>R</i><sup>2</sup> can be negative, and its value will always be less than or equal to that of <i>R</i><sup>2</sup>. Unlike <i>R</i><sup>2</sup>, the adjusted <i>R</i><sup>2</sup> increases only when the increase in <i>R</i><sup>2</sup> (due to the inclusion of a new explanatory variable) is more than one would expect to see by chance. If a set of explanatory variables with a predetermined hierarchy of importance are introduced into a regression one at a time, with the adjusted <i>R</i><sup>2</sup> computed each time, the level at which adjusted <i>R</i><sup>2</sup> reaches a maximum, and decreases afterward, would be the regression with the ideal combination of having the best fit without excess/unnecessary terms. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Bias_and_variance_contributing_to_total_error.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Bias_and_variance_contributing_to_total_error.svg/640px-Bias_and_variance_contributing_to_total_error.svg.png" decoding="async" width="640" height="403" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Bias_and_variance_contributing_to_total_error.svg/960px-Bias_and_variance_contributing_to_total_error.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Bias_and_variance_contributing_to_total_error.svg/1280px-Bias_and_variance_contributing_to_total_error.svg.png 2x" data-file-width="461" data-file-height="290" /></a><figcaption>Schematic of the bias and variance contribution into the total error</figcaption></figure> <p>The adjusted <i>R</i><sup>2</sup> can be interpreted as an instance of the <a href="/wiki/Bias-variance_tradeoff" class="mw-redirect" title="Bias-variance tradeoff">bias-variance tradeoff</a>. When we consider the performance of a model, a lower error represents a better performance. When the model becomes more complex, the variance will increase whereas the square of bias will decrease, and these two metrices add up to be the total error. Combining these two trends, the bias-variance tradeoff describes a relationship between the performance of the model and its complexity, which is shown as a u-shape curve on the right. For the adjusted <i>R</i><sup>2</sup> specifically, the model complexity (i.e. number of parameters) affects the <i>R</i><sup>2</sup> and the term / frac and thereby captures their attributes in the overall performance of the model. </p><p><i>R</i><sup>2</sup> can be interpreted as the variance of the model, which is influenced by the model complexity. A high <i>R</i><sup>2</sup> indicates a lower bias error because the model can better explain the change of Y with predictors. For this reason, we make fewer (erroneous) assumptions, and this results in a lower bias error. Meanwhile, to accommodate fewer assumptions, the model tends to be more complex. Based on bias-variance tradeoff, a higher complexity will lead to a decrease in bias and a better performance (below the optimal line). In <span style="text-decoration:overline;"><i>R</i></span><sup>2</sup>, the term (<span class="nowrap">1 − <i>R</i><sup>2</sup></span>) will be lower with high complexity and resulting in a higher <span style="text-decoration:overline;"><i>R</i></span><sup>2</sup>, consistently indicating a better performance. </p><p>On the other hand, the term/frac term is reversely affected by the model complexity. The term/frac will increase when adding regressors (i.e. increased model complexity) and lead to worse performance. Based on bias-variance tradeoff, a higher model complexity (beyond the optimal line) leads to increasing errors and a worse performance. </p><p>Considering the calculation of <span style="text-decoration:overline;"><i>R</i></span><sup>2</sup>, more parameters will increase the <i>R</i><sup>2</sup> and lead to an increase in <span style="text-decoration:overline;"><i>R</i></span><sup>2</sup>. Nevertheless, adding more parameters will increase the term/frac and thus decrease <span style="text-decoration:overline;"><i>R</i></span><sup>2</sup>. These two trends construct a reverse u-shape relationship between model complexity and <span style="text-decoration:overline;"><i>R</i></span><sup>2</sup>, which is in consistent with the u-shape trend of model complexity versus overall performance. Unlike <i>R</i><sup>2</sup>, which will always increase when model complexity increases, <span style="text-decoration:overline;"><i>R</i></span><sup>2</sup> will increase only when the bias eliminated by the added regressor is greater than the variance introduced simultaneously. Using <span style="text-decoration:overline;"><i>R</i></span><sup>2</sup> instead of <i>R</i><sup>2</sup> could thereby prevent overfitting. </p><p>Following the same logic, adjusted <i>R</i><sup>2</sup> can be interpreted as a less biased estimator of the population <i>R</i><sup>2</sup>, whereas the observed sample <i>R</i><sup>2</sup> is a positively biased estimate of the population value.<sup id="cite_ref-:0_18-0" class="reference"><a href="#cite_note-:0-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Adjusted <i>R</i><sup>2</sup> is more appropriate when evaluating model fit (the variance in the dependent variable accounted for by the independent variables) and in comparing alternative models in the <a href="/wiki/Feature_selection" title="Feature selection">feature selection</a> stage of model building.<sup id="cite_ref-:0_18-1" class="reference"><a href="#cite_note-:0-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The principle behind the adjusted <i>R</i><sup>2</sup> statistic can be seen by rewriting the ordinary <i>R</i><sup>2</sup> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{2}={1-{{\text{VAR}}_{\text{res}} \over {\text{VAR}}_{\text{tot}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>VAR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>VAR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> </mfrac> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}={1-{{\text{VAR}}_{\text{res}} \over {\text{VAR}}_{\text{tot}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed0972c845f03d59aeb853e3cfab0b98723e80d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.286ex; height:5.843ex;" alt="{\displaystyle R^{2}={1-{{\text{VAR}}_{\text{res}} \over {\text{VAR}}_{\text{tot}}}}}"></span></dd></dl> <p>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 {\text{VAR}}_{\text{res}}=SS_{\text{res}}/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>VAR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> <mo>=</mo> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{VAR}}_{\text{res}}=SS_{\text{res}}/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9528c427112a8c0c4c526aaade3e0eb8a5fae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.286ex; height:2.843ex;" alt="{\displaystyle {\text{VAR}}_{\text{res}}=SS_{\text{res}}/n}"></span> 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 {\text{VAR}}_{\text{tot}}=SS_{\text{tot}}/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>VAR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> <mo>=</mo> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{VAR}}_{\text{tot}}=SS_{\text{tot}}/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c57c33540e59dcdd1ef87021ee1a35ee02e1d94b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.444ex; height:2.843ex;" alt="{\displaystyle {\text{VAR}}_{\text{tot}}=SS_{\text{tot}}/n}"></span> are the sample variances of the estimated residuals and the dependent variable respectively, which can be seen as biased estimates of the population variances of the errors and of the dependent variable. These estimates are replaced by statistically <a href="/wiki/Bias_of_an_estimator#Sample_variance" title="Bias of an estimator">unbiased</a> versions: <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 {\text{VAR}}_{\text{res}}=SS_{\text{res}}/(n-p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>VAR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> <mo>=</mo> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{VAR}}_{\text{res}}=SS_{\text{res}}/(n-p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/011273b9c02016870aa56c83003f47c0fa5be825" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.105ex; height:2.843ex;" alt="{\displaystyle {\text{VAR}}_{\text{res}}=SS_{\text{res}}/(n-p)}"></span> 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 {\text{VAR}}_{\text{tot}}=SS_{\text{tot}}/(n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>VAR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> <mo>=</mo> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{VAR}}_{\text{tot}}=SS_{\text{tot}}/(n-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba13bc9c0f18209ef1291908a16f620b920c003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.256ex; height:2.843ex;" alt="{\displaystyle {\text{VAR}}_{\text{tot}}=SS_{\text{tot}}/(n-1)}"></span>. </p><p>Despite using unbiased estimators for the population variances of the error and the dependent variable, adjusted <i>R</i><sup>2</sup> is not an unbiased estimator of the population <i>R</i><sup>2</sup>,<sup id="cite_ref-:0_18-2" class="reference"><a href="#cite_note-:0-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> which results by using the population variances of the errors and the dependent variable instead of estimating them. <a href="/wiki/Ingram_Olkin" title="Ingram Olkin">Ingram Olkin</a> and <a href="/wiki/John_W._Pratt" title="John W. Pratt">John W. Pratt</a> derived the <a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">minimum-variance unbiased estimator</a> for the population <i>R</i><sup>2</sup>,<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> which is known as Olkin–Pratt estimator. Comparisons of different approaches for adjusting <i>R</i><sup>2</sup> concluded that in most situations either an approximate version of the Olkin–Pratt estimator <sup id="cite_ref-:0_18-3" class="reference"><a href="#cite_note-:0-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> or the exact Olkin–Pratt estimator <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> should be preferred over (Ezekiel) adjusted <i>R</i><sup>2</sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Coefficient_of_partial_determination">Coefficient of partial determination</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=11" title="Edit section: Coefficient of partial determination"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></div> <p>The coefficient of partial determination can be defined as the proportion of variation that cannot be explained in a reduced model, but can be explained by the predictors specified in a full(er) model.<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><sup id="cite_ref-Nagelkerke_1991_22-0" class="reference"><a href="#cite_note-Nagelkerke_1991-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> This coefficient is used to provide insight into whether or not one or more additional predictors may be useful in a more fully specified regression model. </p><p>The calculation for the partial <i>R</i><sup>2</sup> is relatively straightforward after estimating two models and generating the <a href="/wiki/ANOVA" class="mw-redirect" title="ANOVA">ANOVA</a> tables for them. The calculation for the partial <i>R</i><sup>2</sup> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {SS_{\text{ res, reduced}}-SS_{\text{ res, full}}}{SS_{\text{ res, reduced}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> res, reduced</mtext> </mrow> </msub> <mo>−</mo> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> res, full</mtext> </mrow> </msub> </mrow> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> res, reduced</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {SS_{\text{ res, reduced}}-SS_{\text{ res, full}}}{SS_{\text{ res, reduced}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49425f019d3277d2165259645c1e6beae526630f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.827ex; height:6.509ex;" alt="{\displaystyle {\frac {SS_{\text{ res, reduced}}-SS_{\text{ res, full}}}{SS_{\text{ res, reduced}}}},}"></span></dd></dl> <p>which is analogous to the usual coefficient of determination: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {SS_{\text{tot}}-SS_{\text{res}}}{SS_{\text{tot}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> <mo>−</mo> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> </mrow> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {SS_{\text{tot}}-SS_{\text{res}}}{SS_{\text{tot}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74fd98b3113ef10bf35e38791d33021ddc30b82c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.76ex; height:5.843ex;" alt="{\displaystyle {\frac {SS_{\text{tot}}-SS_{\text{res}}}{SS_{\text{tot}}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Generalizing_and_decomposing_R2">Generalizing and decomposing <i>R</i><sup>2</sup></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=12" title="Edit section: Generalizing and decomposing R2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As explained above, model selection heuristics such as the adjusted <i>R</i><sup>2</sup> criterion and the <a href="/wiki/F-test" title="F-test">F-test</a> examine whether the total <i>R</i><sup>2</sup> sufficiently increases to determine if a new regressor should be added to the model. If a regressor is added to the model that is highly correlated with other regressors which have already been included, then the total <i>R</i><sup>2</sup> will hardly increase, even if the new regressor is of relevance. As a result, the above-mentioned heuristics will ignore relevant regressors when cross-correlations are high.<sup id="cite_ref-Hoornweg2018SUS_24-0" class="reference"><a href="#cite_note-Hoornweg2018SUS-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geometric_R_squared_.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Geometric_R_squared_.svg/220px-Geometric_R_squared_.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Geometric_R_squared_.svg/330px-Geometric_R_squared_.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Geometric_R_squared_.svg/440px-Geometric_R_squared_.svg.png 2x" data-file-width="560" data-file-height="420" /></a><figcaption>Geometric representation of <i>r</i><sup>2</sup>.</figcaption></figure> <p>Alternatively, one can decompose a generalized version of <i>R</i><sup>2</sup> to quantify the relevance of deviating from a hypothesis.<sup id="cite_ref-Hoornweg2018SUS_24-1" class="reference"><a href="#cite_note-Hoornweg2018SUS-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> As Hoornweg (2018) shows, several <a href="/wiki/Shrinkage_(statistics)" title="Shrinkage (statistics)">shrinkage estimators</a> – such as <a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian linear regression</a>, <a href="/wiki/Ridge_regression" title="Ridge regression">ridge regression</a>, and the (adaptive) <a href="/wiki/Lasso_(statistics)#Lasso_method" title="Lasso (statistics)">lasso</a> – make use of this decomposition of <i>R</i><sup>2</sup> when they gradually shrink parameters from the unrestricted OLS solutions towards the hypothesized values. Let us first define the linear regression model as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=X\beta +\varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>X</mi> <mi>β</mi> <mo>+</mo> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=X\beta +\varepsilon .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69a51d0e5f0a36459e16208f4c0b158841fc1557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.137ex; height:2.509ex;" alt="{\displaystyle y=X\beta +\varepsilon .}"></span></dd></dl> <p>It is assumed that the matrix <i>X</i> is standardized with Z-scores and that the column vector <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> is centered to have a mean of zero. Let the column vector <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 \beta _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b42f71f244103a8fca3c76885c7580a92831c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{0}}"></span> refer to the hypothesized regression parameters and let the column vector <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> denote the estimated parameters. We can then define </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{2}=1-{\frac {(y-Xb)'(y-Xb)}{(y-X\beta _{0})'(y-X\beta _{0})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>X</mi> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>X</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>X</mi> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>X</mi> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}=1-{\frac {(y-Xb)'(y-Xb)}{(y-X\beta _{0})'(y-X\beta _{0})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd2782ff2171d82f0d3807e0289ae9632b88342f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.398ex; height:6.509ex;" alt="{\displaystyle R^{2}=1-{\frac {(y-Xb)'(y-Xb)}{(y-X\beta _{0})'(y-X\beta _{0})}}.}"></span></dd></dl> <p>An <i>R</i><sup>2</sup> of 75% means that the in-sample accuracy improves by 75% if the data-optimized <i>b</i> solutions are used instead of the hypothesized <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 \beta _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b42f71f244103a8fca3c76885c7580a92831c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{0}}"></span> values. In the special case that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b42f71f244103a8fca3c76885c7580a92831c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{0}}"></span> is a vector of zeros, we obtain the traditional <i>R</i><sup>2</sup> again. </p><p>The individual effect on <i>R</i><sup>2</sup> of deviating from a hypothesis can be computed with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\otimes }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\otimes }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f174aee39d9980758ad54016db862604e297624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.275ex; height:2.509ex;" alt="{\displaystyle R^{\otimes }}"></span> ('R-outer'). This <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> times <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> matrix is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\otimes }=(X'{\tilde {y}}_{0})(X'{\tilde {y}}_{0})'(X'X)^{-1}({\tilde {y}}_{0}'{\tilde {y}}_{0})^{-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊗</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mi>X</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\otimes }=(X'{\tilde {y}}_{0})(X'{\tilde {y}}_{0})'(X'X)^{-1}({\tilde {y}}_{0}'{\tilde {y}}_{0})^{-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f14da42e50b661826235b919030a742a137b7aec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.058ex; height:3.176ex;" alt="{\displaystyle R^{\otimes }=(X'{\tilde {y}}_{0})(X'{\tilde {y}}_{0})'(X'X)^{-1}({\tilde {y}}_{0}'{\tilde {y}}_{0})^{-1},}"></span></dd></dl> <p>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 {\tilde {y}}_{0}=y-X\beta _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>y</mi> <mo>−</mo> <mi>X</mi> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {y}}_{0}=y-X\beta _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81bcc40c86fc1d15f4882bc6e918c1e428616b0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.801ex; height:2.676ex;" alt="{\displaystyle {\tilde {y}}_{0}=y-X\beta _{0}}"></span>. The diagonal elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\otimes }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\otimes }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f174aee39d9980758ad54016db862604e297624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.275ex; height:2.509ex;" alt="{\displaystyle R^{\otimes }}"></span> exactly add up to <i>R</i><sup>2</sup>. If regressors are uncorrelated 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 \beta _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b42f71f244103a8fca3c76885c7580a92831c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{0}}"></span> is a vector of zeros, then 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 j^{\text{th}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>th</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j^{\text{th}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8170c26debcb1b3326c93ce9527b34af28581b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.771ex; height:3.009ex;" alt="{\displaystyle j^{\text{th}}}"></span> diagonal element of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\otimes }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\otimes }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f174aee39d9980758ad54016db862604e297624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.275ex; height:2.509ex;" alt="{\displaystyle R^{\otimes }}"></span> simply corresponds to the <i>r</i><sup>2</sup> value between <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 x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db47cb3d2f9496205a17a6856c91c1d3d363ccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}"></span> 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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. When regressors <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 x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> 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 x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db47cb3d2f9496205a17a6856c91c1d3d363ccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}"></span> are correlated, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{ii}^{\otimes }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{ii}^{\otimes }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/129c3aa02b02558ce64a4e4653c61fbf133cd43f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.275ex; height:3.176ex;" alt="{\displaystyle R_{ii}^{\otimes }}"></span> might increase at the cost of a decrease in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{jj}^{\otimes }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{jj}^{\otimes }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2766f8284db86d9b7d98adbc7e508c53c1727ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.351ex; height:3.509ex;" alt="{\displaystyle R_{jj}^{\otimes }}"></span>. As a result, the diagonal elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\otimes }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\otimes }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f174aee39d9980758ad54016db862604e297624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.275ex; height:2.509ex;" alt="{\displaystyle R^{\otimes }}"></span> may be smaller than 0 and, in more exceptional cases, larger than 1. To deal with such uncertainties, several shrinkage estimators implicitly take a weighted average of the diagonal elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\otimes }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\otimes }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f174aee39d9980758ad54016db862604e297624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.275ex; height:2.509ex;" alt="{\displaystyle R^{\otimes }}"></span> to quantify the relevance of deviating from a hypothesized value.<sup id="cite_ref-Hoornweg2018SUS_24-2" class="reference"><a href="#cite_note-Hoornweg2018SUS-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Click on the <a href="/wiki/Lasso_(statistics)#Interpretations_of_lasso" title="Lasso (statistics)">lasso</a> for an example. </p> <div class="mw-heading mw-heading3"><h3 id="R2_in_logistic_regression"><i>R</i><sup>2</sup> in logistic regression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=13" title="Edit section: R2 in logistic regression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the case of <a href="/wiki/Logistic_regression" title="Logistic regression">logistic regression</a>, usually fit by <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a>, there are several choices of <a href="/wiki/Logistic_regression#Pseudo-R-squared" title="Logistic regression">pseudo-<i>R</i><sup>2</sup></a>. </p><p>One is the generalized <i>R</i><sup>2</sup> originally proposed by Cox & Snell,<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> and independently by Magee:<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> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{2}=1-\left({{\mathcal {L}}(0) \over {\mathcal {L}}({\widehat {\theta }})}\right)^{2/n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}=1-\left({{\mathcal {L}}(0) \over {\mathcal {L}}({\widehat {\theta }})}\right)^{2/n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed020ef63446fbf7c4c3762aa08782a55e9f43b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.068ex; height:8.009ex;" alt="{\displaystyle R^{2}=1-\left({{\mathcal {L}}(0) \over {\mathcal {L}}({\widehat {\theta }})}\right)^{2/n}}"></span></dd></dl> <p>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 {\mathcal {L}}(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05c346122ed6751d44cb9dee3b5d2c12e1ed5017" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}(0)}"></span> is the likelihood of the model with only the intercept, <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 {{\mathcal {L}}({\widehat {\theta }})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{\mathcal {L}}({\widehat {\theta }})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06dbd67dcd3bb7bcb665766675866896a6319099" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.769ex; height:3.343ex;" alt="{\displaystyle {{\mathcal {L}}({\widehat {\theta }})}}"></span> is the likelihood of the estimated model (i.e., the model with a given set of parameter estimates) and <i>n</i> is the sample size. It is easily rewritten to: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{2}=1-e^{{\frac {2}{n}}(\ln({\mathcal {L}}(0))-\ln({\mathcal {L}}({\widehat {\theta }})))}=1-e^{-D/n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}=1-e^{{\frac {2}{n}}(\ln({\mathcal {L}}(0))-\ln({\mathcal {L}}({\widehat {\theta }})))}=1-e^{-D/n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29a3d03bd36b97a9e206465a8e86b89163143921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:40.26ex; height:3.509ex;" alt="{\displaystyle R^{2}=1-e^{{\frac {2}{n}}(\ln({\mathcal {L}}(0))-\ln({\mathcal {L}}({\widehat {\theta }})))}=1-e^{-D/n}}"></span></dd></dl> <p>where <i>D</i> is the test statistic of the <a href="/wiki/Likelihood_ratio_test" class="mw-redirect" title="Likelihood ratio test">likelihood ratio test</a>. </p><p><a href="/wiki/Nico_Nagelkerke" title="Nico Nagelkerke">Nico Nagelkerke</a> noted that it had the following properties:<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><sup id="cite_ref-Nagelkerke_1991_22-1" class="reference"><a href="#cite_note-Nagelkerke_1991-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <ol><li>It is consistent with the classical coefficient of determination when both can be computed;</li> <li>Its value is maximised by the maximum likelihood estimation of a model;</li> <li>It is asymptotically independent of the sample size;</li> <li>The interpretation is the proportion of the variation explained by the model;</li> <li>The values are between 0 and 1, with 0 denoting that model does not explain any variation and 1 denoting that it perfectly explains the observed variation;</li> <li>It does not have any unit.</li></ol> <p>However, in the case of a logistic model, 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 {\mathcal {L}}({\widehat {\theta }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}({\widehat {\theta }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae89c0e871f8bcf2e3337a446e53bf9900e88bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.769ex; height:3.343ex;" alt="{\displaystyle {\mathcal {L}}({\widehat {\theta }})}"></span> cannot be greater than 1, <i>R</i><sup>2</sup> is between 0 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 R_{\max }^{2}=1-({\mathcal {L}}(0))^{2/n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\max }^{2}=1-({\mathcal {L}}(0))^{2/n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188614a19d2a22ae00e12c20fc1bd26469daad6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.404ex; height:3.343ex;" alt="{\displaystyle R_{\max }^{2}=1-({\mathcal {L}}(0))^{2/n}}"></span>: thus, Nagelkerke suggested the possibility to define a scaled <i>R</i><sup>2</sup> as <i>R</i><sup>2</sup>/<i>R</i><sup>2</sup><sub>max</sub>.<sup id="cite_ref-Nagelkerke_1991_22-2" class="reference"><a href="#cite_note-Nagelkerke_1991-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Comparison_with_residual_statistics">Comparison with residual statistics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=14" title="Edit section: Comparison with residual statistics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Occasionally, residual statistics are used for indicating goodness of fit. The <a href="/wiki/Norm_(mathematics)#Examples" title="Norm (mathematics)">norm</a> of residuals is calculated as the square-root of the <a href="/wiki/Sum_of_squares_of_residuals" class="mw-redirect" title="Sum of squares of residuals">sum of squares of residuals</a> (SSR): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{norm of residuals}}={\sqrt {SS_{\text{res}}}}=\|e\|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>norm of residuals</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>e</mi> <mo fence="false" stretchy="false">‖</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{norm of residuals}}={\sqrt {SS_{\text{res}}}}=\|e\|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b53cd635193da52b721698b395a38617fffe59c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.911ex; height:3.343ex;" alt="{\displaystyle {\text{norm of residuals}}={\sqrt {SS_{\text{res}}}}=\|e\|.}"></span></dd></dl> <p>Similarly, the <a href="/wiki/Reduced_chi-square" class="mw-redirect" title="Reduced chi-square">reduced chi-square</a> is calculated as the SSR divided by the degrees of freedom. </p><p>Both <i>R</i><sup>2</sup> and the norm of residuals have their relative merits. For <a href="/wiki/Least_squares" title="Least squares">least squares</a> analysis <i>R</i><sup>2</sup> varies between 0 and 1, with larger numbers indicating better fits and 1 representing a perfect fit. The norm of residuals varies from 0 to infinity with smaller numbers indicating better fits and zero indicating a perfect fit. One advantage and disadvantage of <i>R</i><sup>2</sup> is 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 SS_{\text{tot}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SS_{\text{tot}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7174e7bb411271b3ddcaeee4b85ea6fd0e69f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.258ex; height:2.509ex;" alt="{\displaystyle SS_{\text{tot}}}"></span> term acts to <a href="/wiki/Normalization_(statistics)" title="Normalization (statistics)">normalize</a> the value. If the <i>y<sub>i</sub></i> values are all multiplied by a constant, the norm of residuals will also change by that constant but <i>R</i><sup>2</sup> will stay the same. As a basic example, for the linear least squares fit to the set of data: </p> <dl><dd><table class="wikitable"> <tbody><tr> <th>x </th> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5 </td></tr> <tr> <th>y </th> <td>1.9</td> <td>3.7</td> <td>5.8</td> <td>8.0</td> <td>9.6 </td></tr></tbody></table></dd></dl> <p><i>R</i><sup>2</sup> = 0.998, and norm of residuals = 0.302. If all values of <i>y</i> are multiplied by 1000 (for example, in an <a href="/wiki/Metric_prefix" title="Metric prefix">SI prefix</a> change), then <i>R</i><sup>2</sup> remains the same, but norm of residuals = 302. </p><p>Another single-parameter indicator of fit is the <a href="/wiki/Root-mean-square_deviation" class="mw-redirect" title="Root-mean-square deviation">RMSE</a> of the residuals, or standard deviation of the residuals. This would have a value of 0.135 for the above example given that the fit was linear with an unforced intercept.<sup id="cite_ref-origin_wp_28-0" class="reference"><a href="#cite_note-origin_wp-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <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=Coefficient_of_determination&action=edit&section=15" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The creation of the coefficient of determination has been attributed to the geneticist <a href="/wiki/Sewall_Wright" title="Sewall Wright">Sewall Wright</a> and was first published in 1921.<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> </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=Coefficient_of_determination&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Anscombe%27s_quartet" title="Anscombe's quartet">Anscombe's quartet</a></li> <li><a href="/wiki/Fraction_of_variance_unexplained" title="Fraction of variance unexplained">Fraction of variance unexplained</a></li> <li><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></li> <li><a href="/wiki/Nash%E2%80%93Sutcliffe_model_efficiency_coefficient" title="Nash–Sutcliffe model efficiency coefficient">Nash–Sutcliffe model efficiency coefficient</a> (<a href="/wiki/Hydrology" title="Hydrology">hydrological applications</a>)</li> <li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment correlation coefficient</a></li> <li><a href="/wiki/Proportional_reduction_in_loss" title="Proportional reduction in loss">Proportional reduction in loss</a></li> <li><a href="/wiki/Regression_model_validation" class="mw-redirect" title="Regression model validation">Regression model validation</a></li> <li><a href="/wiki/Root_mean_square_deviation" title="Root mean square deviation">Root mean square deviation</a></li> <li><a href="/wiki/Stepwise_regression" title="Stepwise regression">Stepwise regression</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=17" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output 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J. D. (September 1991). <a rel="nofollow" class="external text" href="http://www.cesarzamudio.com/uploads/1/7/9/1/17916581/nagelkerke_n.j.d._1991_-_a_note_on_a_general_definition_of_the_coefficient_of_determination.pdf">"A Note on a General Definition of the Coefficient of Determination"</a> <span class="cs1-format">(PDF)</span>. <i>Biometrika</i>. <b>78</b> (3): 691–692. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F78.3.691">10.1093/biomet/78.3.691</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2337038">2337038</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=A+Note+on+a+General+Definition+of+the+Coefficient+of+Determination&rft.volume=78&rft.issue=3&rft.pages=691-692&rft.date=1991-09&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F78.3.691&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2337038%23id-name%3DJSTOR&rft.aulast=Nagelkerke&rft.aufirst=N.+J.+D.&rft_id=http%3A%2F%2Fwww.cesarzamudio.com%2Fuploads%2F1%2F7%2F9%2F1%2F17916581%2Fnagelkerke_n.j.d._1991_-_a_note_on_a_general_definition_of_the_coefficient_of_determination.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoefficient+of+determination" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stats.stackexchange.com/q/7775">"regression – R implementation of coefficient of partial determination"</a>. <i>Cross Validated</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Cross+Validated&rft.atitle=regression+%E2%80%93+R+implementation+of+coefficient+of+partial+determination&rft_id=https%3A%2F%2Fstats.stackexchange.com%2Fq%2F7775&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoefficient+of+determination" class="Z3988"></span></span> </li> <li id="cite_note-Hoornweg2018SUS-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hoornweg2018SUS_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hoornweg2018SUS_24-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Hoornweg2018SUS_24-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoornweg2018" class="citation book cs1">Hoornweg, Victor (2018). <a rel="nofollow" class="external text" href="http://www.victorhoornweg.com">"Part II: On Keeping Parameters Fixed"</a>. <i>Science: Under Submission</i>. Hoornweg Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-829188-0-9" title="Special:BookSources/978-90-829188-0-9"><bdi>978-90-829188-0-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Part+II%3A+On+Keeping+Parameters+Fixed&rft.btitle=Science%3A+Under+Submission&rft.pub=Hoornweg+Press&rft.date=2018&rft.isbn=978-90-829188-0-9&rft.aulast=Hoornweg&rft.aufirst=Victor&rft_id=http%3A%2F%2Fwww.victorhoornweg.com&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoefficient+of+determination" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxSnell1989" class="citation book cs1">Cox, D. D.; <a href="/wiki/Joyce_Snell" title="Joyce Snell">Snell, E. J.</a> (1989). <i>The Analysis of Binary Data</i> (2nd ed.). Chapman and Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Analysis+of+Binary+Data&rft.edition=2nd&rft.pub=Chapman+and+Hall&rft.date=1989&rft.aulast=Cox&rft.aufirst=D.+D.&rft.au=Snell%2C+E.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoefficient+of+determination" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMagee1990" class="citation journal cs1">Magee, L. (1990). "<i>R</i><sup>2</sup> measures based on Wald and likelihood ratio joint significance tests". <i>The American Statistician</i>. <b>44</b> (3): 250–3. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.1990.10475731">10.1080/00031305.1990.10475731</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=R%3Csup%3E2%3C%2Fsup%3E+measures+based+on+Wald+and+likelihood+ratio+joint+significance+tests&rft.volume=44&rft.issue=3&rft.pages=250-3&rft.date=1990&rft_id=info%3Adoi%2F10.1080%2F00031305.1990.10475731&rft.aulast=Magee&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoefficient+of+determination" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNagelkerke1992" class="citation book cs1">Nagelkerke, Nico J. D. (1992). <i>Maximum Likelihood Estimation of Functional Relationships, Pays-Bas</i>. Lecture Notes in Statistics. Vol. 69. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97721-8" title="Special:BookSources/978-0-387-97721-8"><bdi>978-0-387-97721-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Maximum+Likelihood+Estimation+of+Functional+Relationships%2C+Pays-Bas&rft.series=Lecture+Notes+in+Statistics&rft.date=1992&rft.isbn=978-0-387-97721-8&rft.aulast=Nagelkerke&rft.aufirst=Nico+J.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoefficient+of+determination" class="Z3988"></span></span> </li> <li id="cite_note-origin_wp-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-origin_wp_28-0">^</a></b></span> <span class="reference-text">OriginLab webpage, <a rel="nofollow" class="external free" href="http://www.originlab.com/doc/Origin-Help/LR-Algorithm">http://www.originlab.com/doc/Origin-Help/LR-Algorithm</a>. Retrieved February 9, 2016.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWright1921" class="citation journal cs1">Wright, Sewall (January 1921). "Correlation and causation". <i>Journal of Agricultural Research</i>. <b>20</b>: 557–585.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Agricultural+Research&rft.atitle=Correlation+and+causation&rft.volume=20&rft.pages=557-585&rft.date=1921-01&rft.aulast=Wright&rft.aufirst=Sewall&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoefficient+of+determination" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coefficient_of_determination&action=edit&section=18" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGujaratiPorter2009" class="citation book cs1"><a href="/wiki/Damodar_N._Gujarati" title="Damodar N. Gujarati">Gujarati, Damodar N.</a>; <a href="/wiki/Dawn_C._Porter" title="Dawn C. Porter">Porter, Dawn C.</a> (2009). <i>Basic Econometrics</i> (Fifth ed.). New York: McGraw-Hill/Irwin. pp. 73–78. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-337577-9" title="Special:BookSources/978-0-07-337577-9"><bdi>978-0-07-337577-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Econometrics&rft.place=New+York&rft.pages=73-78&rft.edition=Fifth&rft.pub=McGraw-Hill%2FIrwin&rft.date=2009&rft.isbn=978-0-07-337577-9&rft.aulast=Gujarati&rft.aufirst=Damodar+N.&rft.au=Porter%2C+Dawn+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoefficient+of+determination" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHughesGrawoig1971" class="citation book cs1">Hughes, Ann; Grawoig, Dennis (1971). <a rel="nofollow" class="external text" href="https://archive.org/details/trent_0116302260611/page/344"><i>Statistics: A Foundation for Analysis</i></a>. Reading: Addison-Wesley. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/trent_0116302260611/page/344">344–348</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-03021-7" title="Special:BookSources/0-201-03021-7"><bdi>0-201-03021-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistics%3A+A+Foundation+for+Analysis&rft.place=Reading&rft.pages=344-348&rft.pub=Addison-Wesley&rft.date=1971&rft.isbn=0-201-03021-7&rft.aulast=Hughes&rft.aufirst=Ann&rft.au=Grawoig%2C+Dennis&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftrent_0116302260611%2Fpage%2F344&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoefficient+of+determination" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKmenta1986" class="citation book cs1"><a href="/wiki/Jan_Kmenta" title="Jan Kmenta">Kmenta, Jan</a> (1986). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/elementsofeconom0003kmen/page/240"><i>Elements of Econometrics</i></a></span> (Second ed.). New York: Macmillan. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/elementsofeconom0003kmen/page/240">240–243</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-02-365070-3" title="Special:BookSources/978-0-02-365070-3"><bdi>978-0-02-365070-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Econometrics&rft.place=New+York&rft.pages=240-243&rft.edition=Second&rft.pub=Macmillan&rft.date=1986&rft.isbn=978-0-02-365070-3&rft.aulast=Kmenta&rft.aufirst=Jan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementsofeconom0003kmen%2Fpage%2F240&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoefficient+of+determination" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLewis-BeckSkalaban1990" class="citation journal cs1"><a href="/wiki/Michael_Lewis-Beck" title="Michael Lewis-Beck">Lewis-Beck, Michael S.</a>; Skalaban, Andrew (1990). "The <i>R</i>-Squared: Some Straight Talk". <i><a href="/wiki/Political_Analysis_(journal)" title="Political Analysis (journal)">Political Analysis</a></i>. <b>2</b>: 153–171. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fpan%2F2.1.153">10.1093/pan/2.1.153</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/23317769">23317769</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Political+Analysis&rft.atitle=The+R-Squared%3A+Some+Straight+Talk&rft.volume=2&rft.pages=153-171&rft.date=1990&rft_id=info%3Adoi%2F10.1093%2Fpan%2F2.1.153&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F23317769%23id-name%3DJSTOR&rft.aulast=Lewis-Beck&rft.aufirst=Michael+S.&rft.au=Skalaban%2C+Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoefficient+of+determination" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChiccoWarrensJurman2021" class="citation journal cs1">Chicco, Davide; Warrens, Matthijs J.; Jurman, Giuseppe (2021). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8279135">"The coefficient of determination R-squared is more informative than SMAPE, MAE, MAPE, MSE and RMSE in regression analysis evaluation"</a>. <i>PeerJ Computer Science</i>. <b>7</b> (e623): e623. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.7717%2Fpeerj-cs.623">10.7717/peerj-cs.623</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8279135">8279135</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/34307865">34307865</a>.</cite><span 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href="/wiki/Template:Machine_learning_evaluation_metrics" title="Template:Machine learning evaluation metrics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Machine_learning_evaluation_metrics" title="Template talk:Machine learning evaluation metrics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Machine_learning_evaluation_metrics" title="Special:EditPage/Template:Machine learning evaluation metrics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Machine_learning_evaluation_metrics" style="font-size:114%;margin:0 4em"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a> evaluation metrics</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean_squared_error" title="Mean squared error">MSE</a></li> <li><a href="/wiki/Mean_absolute_error" title="Mean absolute error">MAE</a></li> <li><a href="/wiki/Symmetric_mean_absolute_percentage_error" title="Symmetric mean absolute percentage error">sMAPE</a></li> <li><a href="/wiki/Mean_absolute_percentage_error" title="Mean absolute percentage error">MAPE</a></li> <li><a href="/wiki/Mean_absolute_scaled_error" title="Mean absolute scaled error">MASE</a></li> <li><a href="/wiki/Mean_squared_prediction_error" title="Mean squared prediction error">MSPE</a></li> <li><a href="/wiki/Root_mean_square" title="Root mean square">RMS</a></li> <li><a href="/wiki/Root-mean-square_deviation" class="mw-redirect" title="Root-mean-square deviation">RMSE/RMSD</a></li> <li><a class="mw-selflink selflink">R<sup>2</sup></a></li> <li><a href="/wiki/Mean_directional_accuracy" title="Mean directional accuracy">MDA</a></li> <li><a href="/wiki/Median_absolute_deviation" title="Median absolute deviation">MAD</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/F-score" title="F-score">F-score</a></li> <li><a href="/wiki/P4-metric" title="P4-metric">P4</a></li> <li><a href="/wiki/Accuracy_and_precision" title="Accuracy and precision">Accuracy</a></li> <li><a href="/wiki/Precision_and_recall" title="Precision and recall">Precision</a></li> <li><a href="/wiki/Precision_and_recall" title="Precision and recall">Recall</a></li> <li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Kappa</a></li> <li><a href="/wiki/Phi_coefficient" title="Phi coefficient">MCC</a></li> <li><a href="/wiki/Receiver_operating_characteristic#Area_under_the_curve" title="Receiver operating characteristic">AUC</a></li> <li><a href="/wiki/Receiver_operating_characteristic" title="Receiver operating characteristic">ROC</a></li> <li><a href="/wiki/Sensitivity_and_specificity" title="Sensitivity and specificity">Sensitivity and specificity</a></li> <li><a href="/wiki/Cross-entropy#Cross-entropy_loss_function_and_logistic_regression" title="Cross-entropy">Logarithmic Loss</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Silhouette_(clustering)" title="Silhouette (clustering)">Silhouette</a></li> <li><a href="/wiki/Calinski-Harabasz_index" class="mw-redirect" title="Calinski-Harabasz index">Calinski-Harabasz index</a></li> <li><a href="/wiki/Davies%E2%80%93Bouldin_index" title="Davies–Bouldin index">Davies-Bouldin</a></li> <li><a href="/wiki/Dunn_index" title="Dunn index">Dunn index</a></li> <li><a href="/wiki/Hopkins_statistic" title="Hopkins statistic">Hopkins statistic</a></li> <li><a href="/wiki/Jaccard_index" title="Jaccard index">Jaccard index</a></li> <li><a href="/wiki/Rand_index" title="Rand index">Rand index</a></li> <li><a href="/wiki/Similarity_measure" title="Similarity measure">Similarity measure</a></li> <li><a href="/wiki/Simple_matching_coefficient" title="Simple matching coefficient">SMC</a></li> <li><a href="/wiki/SimHash" title="SimHash">SimHash</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ranking_(information_retrieval)" title="Ranking (information retrieval)">Ranking</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean_reciprocal_rank" title="Mean reciprocal rank">MRR</a></li> <li><a href="/wiki/NDCG" class="mw-redirect" title="NDCG">NDCG</a></li> <li><a href="/wiki/Average_precision" class="mw-redirect" title="Average precision">AP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_Vision" class="mw-redirect" title="Computer Vision">Computer Vision</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/PSNR" class="mw-redirect" title="PSNR">PSNR</a></li> <li><a href="/wiki/SSIM" class="mw-redirect" title="SSIM">SSIM</a></li> <li><a href="/wiki/Intersection_over_union" class="mw-redirect" title="Intersection over union">IoU</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Natural_language_processing" title="Natural language processing">NLP</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Perplexity" title="Perplexity">Perplexity</a></li> <li><a href="/wiki/BLEU" title="BLEU">BLEU</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Deep Learning Related Metrics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Inception_score" title="Inception score">Inception score</a></li> <li><a href="/wiki/Fr%C3%A9chet_inception_distance" title="Fréchet inception distance">FID</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Recommender_system" title="Recommender system">Recommender system</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coverage_probability" title="Coverage probability">Coverage</a></li> <li><a href="/w/index.php?title=Intra-list_Similarity&action=edit&redlink=1" class="new" title="Intra-list Similarity (page does not exist)">Intra-list Similarity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Similarity</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cosine_similarity" title="Cosine similarity">Cosine similarity</a></li> <li><a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson correlation coefficient</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li></ul> 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