SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Explained variation</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Explained variation</span></span>

</h1>

</header>

<p>In <a href="Statistics" title="Statistics">statistics</a>, <b>explained variation</b> measures the proportion to which a mathematical model accounts for the variation (<a href="Dispersion_(statistics)" class="mw-redirect" title="Dispersion (statistics)">dispersion</a>) of a given data set. Often, variation is quantified as <a href="Variance" title="Variance">variance</a>; then, the more specific term <b>explained variance</b> can be used.

</p><p>The complementary part of the total variation is called <b><a href="Fraction_of_variance_unexplained" title="Fraction of variance unexplained">unexplained</a></b> or <b><a href="Residual_(statistics)" class="mw-redirect" title="Residual (statistics)">residual</a> variation</b>; likewise, when discussing variance as such, this is referred to as <b>unexplained</b> or <b>residual variance</b>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition_in_terms_of_information_gain">Definition in terms of information gain</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Information_gain_by_better_modelling">Information gain by better modelling</h3></div>

<p>Following Kent (1983),<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> we use the Fraser information (Fraser 1965)<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>

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(\theta )=\int {\textrm {d}}r\,g(r)\,\ln f(r;\theta )}</annotation>

</semantics>

</math></span><img src="./3e78f4206af60c0b19e28e016eeff136022309fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.384ex; height:5.676ex;" alt="{\displaystyle F(\theta )=\int {\textrm {d}}r\,g(r)\,\ln f(r;\theta )}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(r)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(r)}</annotation>

</semantics>

</math></span><img src="./e09e9ec780782afb0b2ae8c172811dba1e4eb63c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.974ex; height:2.843ex;" alt="{\displaystyle g(r)}" loading="lazy"></span> is the probability density of a random variable <span class="mwe-math-element mwe-math-element-inline"><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\,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\,}</annotation>

</semantics>

</math></span><img src="./293563891196765d2d51e0dd54e1ae1000ba9def.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle R\,}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><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(r;\theta )\,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(r;\theta )\,}</annotation>

</semantics>

</math></span><img src="./da7de60ec06aef3ca8d612c1f0ba46f279d509ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.648ex; height:2.843ex;" alt="{\displaystyle f(r;\theta )\,}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \in \Theta _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta \in \Theta _{i}}</annotation>

</semantics>

</math></span><img src="./0004cffd0d3f215700b1fb4e3f45ff9fd749c9df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.539ex; height:2.509ex;" alt="{\displaystyle \theta \in \Theta _{i}}" loading="lazy"></span> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=0,1\,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=0,1\,}</annotation>

</semantics>

</math></span><img src="./c8a3560aaba7f48274f871d97940b2ae6bd92f77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.647ex; height:2.509ex;" alt="{\displaystyle i=0,1\,}" loading="lazy"></span>) are two families of parametric models. Model family 0 is the simpler one, with a restricted parameter space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{0}\subset \Theta _{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta _{0}\subset \Theta _{1}}</annotation>

</semantics>

</math></span><img src="./5a0c09228d02e85d7bd4a3f8eb963f7758adfb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.823ex; height:2.509ex;" alt="{\displaystyle \Theta _{0}\subset \Theta _{1}}" loading="lazy"></span>.

</p><p>Parameters are determined by <a href="Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimation</a>,

</p>

<msub>

</mrow>

</msub>

<msub>

<mi>argmax</mi>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{i}=\operatorname {argmax} _{\theta \in \Theta _{i}}F(\theta ).}</annotation>

</semantics>

</math></span><img src="./8181bd342cc9044d6bdbfd3fcd33eb123ccd1f6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.228ex; height:3.176ex;" alt="{\displaystyle \theta _{i}=\operatorname {argmax} _{\theta \in \Theta _{i}}F(\theta ).}" loading="lazy"></span></dd></dl>

<p>The information gain of model 1 over model 0 is written as

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (\theta _{1}:\theta _{0})=2[F(\theta _{1})-F(\theta _{0})]\,}</annotation>

</semantics>

</math></span><img src="./47409f4a2a6b23bb5ac88a2e7e38d560bde7005c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.66ex; height:2.843ex;" alt="{\displaystyle \Gamma (\theta _{1}:\theta _{0})=2[F(\theta _{1})-F(\theta _{0})]\,}" loading="lazy"></span></dd></dl>

<p>where a factor of 2 is included for convenience. Γ is always nonnegative; it measures the extent to which the best model of family 1 is better than the best model of family 0 in explaining <i>g</i>(<i>r</i>).

</p>

<div class="mw-heading mw-heading3"><h3 id="Information_gain_by_a_conditional_model">Information gain by a conditional model</h3></div>

<p>Assume a two-dimensional random variable <span class="mwe-math-element mwe-math-element-inline"><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=(X,Y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R=(X,Y)}</annotation>

</semantics>

</math></span><img src="./32d0a2344eb82acd357aa9f05e5ccca10aa3e51d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.459ex; height:2.843ex;" alt="{\displaystyle R=(X,Y)}" loading="lazy"></span> where <i>X</i> shall be considered as an explanatory variable, and <i>Y</i> as a dependent variable. Models of family 1 "explain" <i>Y</i> in terms of <i>X</i>,

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(y\mid x;\theta )}</annotation>

</semantics>

</math></span><img src="./40c74b29d98fd131afb31586f1660a92fc671480.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.635ex; height:2.843ex;" alt="{\displaystyle f(y\mid x;\theta )}" loading="lazy"></span>,</dd></dl>

<p>whereas in family 0, <i>X</i> and <i>Y</i> are assumed to be independent. We define the randomness of <i>Y</i> by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(Y)=\exp[-2F(\theta _{0})]}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(Y)=\exp[-2F(\theta _{0})]}</annotation>

</semantics>

</math></span><img src="./ba2ac9841409c1a2e3d07e5c78eaea70a1db5ce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.117ex; height:2.843ex;" alt="{\displaystyle D(Y)=\exp[-2F(\theta _{0})]}" loading="lazy"></span>, and the randomness of <i>Y</i>, given <i>X</i>, by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(Y\mid X)=\exp[-2F(\theta _{1})]}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(Y\mid X)=\exp[-2F(\theta _{1})]}</annotation>

</semantics>

</math></span><img src="./e45e18272ce1ac6393cf68b4376267b8f352628c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.034ex; height:2.843ex;" alt="{\displaystyle D(Y\mid X)=\exp[-2F(\theta _{1})]}" loading="lazy"></span>. Then,

</p>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{C}^{2}=1-D(Y\mid X)/D(Y)}</annotation>

</semantics>

</math></span><img src="./49b4717435e2878e533bf051e5ccacc788ef5e89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.878ex; height:3.176ex;" alt="{\displaystyle \rho _{C}^{2}=1-D(Y\mid X)/D(Y)}" loading="lazy"></span></dd></dl>

<p>can be interpreted as proportion of the data dispersion which is "explained" by <i>X</i>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Special_cases_and_generalized_usage">Special cases and generalized usage</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Linear_regression">Linear regression</h3></div>

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Fraction_of_variance_unexplained" title="Fraction of variance unexplained">Fraction of variance unexplained</a></div>

<p>The fraction of variance unexplained is an established concept in the context of <a href="Linear_regression" title="Linear regression">linear regression</a>. The usual definition of the <a href="Coefficient_of_determination" title="Coefficient of determination">coefficient of determination</a> is based on the fundamental concept of explained variance.

</p>

<div class="mw-heading mw-heading3"><h3 id="Correlation_coefficient_as_measure_of_explained_variance">Correlation coefficient as measure of explained variance</h3></div>

<p>Let <i>X</i> be a random vector, and <i>Y</i> a random variable that is modeled by a normal distribution with centre <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\Psi ^{\textrm {T}}X}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =\Psi ^{\textrm {T}}X}</annotation>

</semantics>

</math></span><img src="./17a3c64d733bc31bf0271d273d5174418d5cc1f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.707ex; height:3.176ex;" alt="{\displaystyle \mu =\Psi ^{\textrm {T}}X}" loading="lazy"></span>. In this case, the above-derived proportion of explained variation <span class="mwe-math-element mwe-math-element-inline"><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 _{C}^{2}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{C}^{2}}</annotation>

</semantics>

</math></span><img src="./ef6219ab707109c298a5659435387f43443767af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.683ex; height:3.176ex;" alt="{\displaystyle \rho _{C}^{2}}" loading="lazy"></span> equals the squared <a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">correlation coefficient</a> <span class="mwe-math-element mwe-math-element-inline"><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}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{2}}</annotation>

</semantics>

</math></span><img src="./5ce07e278be3e058a6303de8359f8b4a4288264a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{2}}" loading="lazy"></span>.

</p><p>Note the strong model assumptions: the centre of the <i>Y</i> distribution must be a linear function of <i>X</i>, and for any given <i>x</i>, the <i>Y</i> distribution must be normal. In other situations, it is generally not justified to interpret <span class="mwe-math-element mwe-math-element-inline"><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}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{2}}</annotation>

</semantics>

</math></span><img src="./5ce07e278be3e058a6303de8359f8b4a4288264a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{2}}" loading="lazy"></span> as proportion of explained variance.

</p>

<div class="mw-heading mw-heading3"><h3 id="In_principal_component_analysis">In principal component analysis</h3></div>

<p>Explained variance is routinely used in <a href="Principal_component_analysis" title="Principal component analysis">principal component analysis</a>. The relation to the Fraser–Kent information gain remains to be clarified.

</p>

<div class="mw-heading mw-heading2"><h2 id="Criticism">Criticism</h2></div>

<p>As the fraction of "explained variance" equals the squared correlation coefficient <span class="mwe-math-element mwe-math-element-inline"><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}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{2}}</annotation>

</semantics>

</math></span><img src="./5ce07e278be3e058a6303de8359f8b4a4288264a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{2}}" loading="lazy"></span>, it shares all the disadvantages of the latter: it reflects not only the quality of the regression, but also the distribution of the independent (conditioning) variables.

</p><p>In the words of one critic: "Thus <span class="mwe-math-element mwe-math-element-inline"><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}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{2}}</annotation>

</semantics>

</math></span><img src="./5ce07e278be3e058a6303de8359f8b4a4288264a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{2}}" loading="lazy"></span> gives the 'percentage of variance explained' by the regression, an expression that, for most social scientists, is of doubtful meaning but great rhetorical value. If this number is large, the regression gives a good fit, and there is little point in searching for additional variables. Other regression equations on different data sets are said to be less satisfactory or less powerful if their <span class="mwe-math-element mwe-math-element-inline"><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}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{2}}</annotation>

</semantics>

</math></span><img src="./5ce07e278be3e058a6303de8359f8b4a4288264a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{2}}" loading="lazy"></span> is lower. Nothing about <span class="mwe-math-element mwe-math-element-inline"><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}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{2}}</annotation>

</semantics>

</math></span><img src="./5ce07e278be3e058a6303de8359f8b4a4288264a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{2}}" loading="lazy"></span> supports these claims".<sup id="cite_ref-Achen_1982_3-0" class="reference"><a href="#cite_note-Achen_1982-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 58">: 58 </span></sup> And, after constructing an example where <span class="mwe-math-element mwe-math-element-inline"><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}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{2}}</annotation>

</semantics>

</math></span><img src="./5ce07e278be3e058a6303de8359f8b4a4288264a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{2}}" loading="lazy"></span> is enhanced just by jointly considering data from two different populations: "'Explained variance' explains nothing."<sup id="cite_ref-Achen_1982_3-1" class="reference"><a href="#cite_note-Achen_1982-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 183">: 183 </span></sup>

</p>

<ul><li><a href="Analysis_of_variance" title="Analysis of variance">Analysis of variance</a></li>

<li><a href="Variance_reduction" title="Variance reduction">Variance reduction</a></li>

<li><a href="Variance-based_sensitivity_analysis" title="Variance-based sensitivity analysis">Variance-based sensitivity analysis</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFKent1983" class="citation journal cs1">Kent, J. T. (1983). "Information gain and a general measure of correlation". <i><a href="Biometrika" title="Biometrika">Biometrika</a></i>. <b>70</b> (1): <span class="nowrap">163–</span>173. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F70.1.163">10.1093/biomet/70.1.163</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2335954">2335954</a>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFFraser1965" class="citation journal cs1">Fraser, D. A. S. (1965). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177700061">"On Information in Statistics"</a>. <i>Ann. Math. Statist</i>. <b>36</b> (3): <span class="nowrap">890–</span>896. <a href="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/10.1214%2Faoms%2F1177700061">10.1214/aoms/1177700061</a></span>.</cite></span>

</li>

<li id="cite_note-Achen_1982-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Achen_1982_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Achen_1982_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFAchen1982" class="citation book cs1">Achen, C. H. (1982). <i>Interpreting and Using Regression</i>. Beverly Hills: Sage. pp. <span class="nowrap">58–</span>59. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8039-1915-8</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFAchen1990" class="citation journal cs1">Achen, C. H. (1990). "'What Does "Explained Variance" Explain?: Reply". <i>Political Analysis</i>. <b>2</b> (1): <span class="nowrap">173–</span>184. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fpan%2F2.1.173">10.1093/pan/2.1.173</a>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080413144223/http://darwin.cwru.edu/~witte/statistics/explained_variance.htm">Explained and Unexplained Variance on a graph</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-05-09" href="https://en.wikipedia.org/wiki/?title=Explained_variation&oldid=1222957648">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>