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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Random effects model</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Random_coefficient_model" class="mw-redirect" title="Random coefficient model">Random coefficient model</a>.</div>
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</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></th></tr><tr><th class="sidebar-heading">
Models</th></tr><tr><td class="sidebar-content">
<ul><li><a href="Linear_regression" title="Linear regression">Linear regression</a></li>
<li><a href="Simple_linear_regression" title="Simple linear regression">Simple regression</a></li>
<li><a href="Polynomial_regression" title="Polynomial regression">Polynomial regression</a></li>
<li><a href="General_linear_model" title="General linear model">General linear model</a></li></ul></td>
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<ul><li><a href="Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></li>
<li><a href="Vector_generalized_linear_model" title="Vector generalized linear model">Vector generalized linear model</a></li>
<li><a href="Discrete_choice" title="Discrete choice">Discrete choice</a></li>
<li><a href="Binomial_regression" title="Binomial regression">Binomial regression</a></li>
<li><a href="Binary_regression" title="Binary regression">Binary regression</a></li>
<li><a href="Logistic_regression" title="Logistic regression">Logistic regression</a></li>
<li><a href="Multinomial_logistic_regression" title="Multinomial logistic regression">Multinomial logistic regression</a></li>
<li><a href="Mixed_logit" title="Mixed logit">Mixed logit</a></li>
<li><a href="Probit_model" title="Probit model">Probit</a></li>
<li><a href="Multinomial_probit" title="Multinomial probit">Multinomial probit</a></li>
<li><a href="Ordered_logit" title="Ordered logit">Ordered logit</a></li>
<li><a href="Ordered_probit" class="mw-redirect" title="Ordered probit">Ordered probit</a></li>
<li><a href="Poisson_regression" title="Poisson regression">Poisson</a></li></ul></td>
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<ul><li><a href="Multilevel_model" title="Multilevel model">Multilevel model</a></li>
<li><a href="Fixed_effects_model" title="Fixed effects model">Fixed effects</a></li>
<li><a href="Mixed_model" title="Mixed model">Linear mixed-effects model</a></li>
<li><a href="Nonlinear_mixed-effects_model" title="Nonlinear mixed-effects model">Nonlinear mixed-effects model</a></li></ul></td>
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<ul><li><a href="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>
<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>
<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>
<li><a href="Robust_regression" title="Robust regression">Robust</a></li>
<li><a href="Quantile_regression" title="Quantile regression">Quantile</a></li>
<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>
<li><a href="Principal_component_regression" title="Principal component regression">Principal components</a></li>
<li><a href="Least-angle_regression" title="Least-angle regression">Least angle</a></li>
<li><a href="Local_regression" title="Local regression">Local</a></li>
<li><a href="Segmented_regression" title="Segmented regression">Segmented</a></li></ul></td>
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<ul><li><a href="Errors-in-variables_models" class="mw-redirect" title="Errors-in-variables models">Errors-in-variables</a></li></ul></td>
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Estimation</th></tr><tr><td class="sidebar-content">
<ul><li><a href="Least_squares" title="Least squares">Least squares</a></li>
<li><a href="Linear_least_squares" title="Linear least squares">Linear</a></li>
<li><a href="Non-linear_least_squares" title="Non-linear least squares">Non-linear</a></li></ul></td>
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<ul><li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary</a></li>
<li><a href="Weighted_least_squares" title="Weighted least squares">Weighted</a></li>
<li><a href="Generalized_least_squares" title="Generalized least squares">Generalized</a></li>
<li><a href="Generalized_estimating_equation" title="Generalized estimating equation">Generalized estimating equation</a></li></ul></td>
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<ul><li><a href="Partial_least_squares_regression" title="Partial least squares regression">Partial</a></li>
<li><a href="Total_least_squares" title="Total least squares">Total</a></li>
<li><a href="Non-negative_least_squares" title="Non-negative least squares">Non-negative</a></li>
<li><a href="Tikhonov_regularization" class="mw-redirect" title="Tikhonov regularization">Ridge regression</a></li>
<li><a href="Regularized_least_squares" title="Regularized least squares">Regularized</a></li></ul></td>
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<ul><li><a href="Least_absolute_deviations" title="Least absolute deviations">Least absolute deviations</a></li>
<li><a href="Iteratively_reweighted_least_squares" title="Iteratively reweighted least squares">Iteratively reweighted</a></li>
<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian</a></li>
<li><a href="Bayesian_multivariate_linear_regression" title="Bayesian multivariate linear regression">Bayesian multivariate</a></li>
<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li></ul></td>
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Background</th></tr><tr><td class="sidebar-content">
<ul><li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>
<li><a href="Mean_and_predicted_response" class="mw-redirect" title="Mean and predicted response">Mean and predicted response</a></li>
<li><a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>
<li><a href="Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></li>
<li><a href="Studentized_residual" title="Studentized residual">Studentized residual</a></li>
<li><a href="Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a></li></ul></td>
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<p>In <a href="Econometrics" title="Econometrics">econometrics</a>, a <b>random effects model</b>, also called a <b>variance components model</b>, is a <a href="Statistical_model" title="Statistical model">statistical model</a> where the model effects are <a href="Random_variable" title="Random variable">random variables</a>. It is a kind of <a href="Hierarchical_linear_model" class="mw-redirect" title="Hierarchical linear model">hierarchical linear model</a>, which assumes that the data being analysed are drawn from a hierarchy of different populations whose differences relate to that hierarchy. A random effects model is a special case of a <a href="Mixed_model" title="Mixed model">mixed model</a>.
</p><p>Contrast this to the <a href="Biostatistics" title="Biostatistics">biostatistics</a> definitions,<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><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 id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> as biostatisticians use "fixed" and "random" effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, <a href="Latent_variables" class="mw-redirect" title="Latent variables">latent variables</a>).
</p>
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<div class="mw-heading mw-heading2"><h2 id="Qualitative_description">Qualitative description</h2></div>
<p>Random effect models assist in controlling for <a href="Unobserved_heterogeneity" class="mw-redirect" title="Unobserved heterogeneity">unobserved heterogeneity</a> when the heterogeneity is constant over time and not correlated with independent variables. This constant can be removed from longitudinal data through differencing, since taking a first difference will remove any time invariant components of the model.<sup id="cite_ref-:0_6-0" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
</p><p>Two common assumptions can be made about the individual specific effect: the random effects assumption and the fixed effects assumption. The random effects assumption is that the individual unobserved heterogeneity is uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables.<sup id="cite_ref-:0_6-1" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
</p><p>If the random effects assumption holds, the random effects estimator is more <a href="Efficiency_(statistics)" title="Efficiency (statistics)">efficient</a> than the fixed effects model.
</p>
<div class="mw-heading mw-heading2"><h2 id="Simple_example">Simple example</h2></div>
<p>Suppose <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 m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
</semantics>
</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> large elementary schools are chosen randomly from among thousands in a large country. Suppose also that <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> pupils of the same age are chosen randomly at each selected school. Their scores on a standard aptitude test are ascertained. Let <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 Y_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y_{ij}}</annotation>
</semantics>
</math></span><img src="./6035187f7e2cee387277f07091bb7827e8e66818.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.828ex; height:2.843ex;" alt="{\displaystyle Y_{ij}}" loading="lazy"></span> be the score of the <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 j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j}</annotation>
</semantics>
</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span>-th pupil at the <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>-th school.
</p><p>A simple way to model this variable is
</p>
<dl><dd><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 Y_{ij}=\mu +U_{i}+W_{ij},\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
<mo>+</mo>
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>W</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y_{ij}=\mu +U_{i}+W_{ij},\,}</annotation>
</semantics>
</math></span><img src="./c1a1811fdee21567598587dc06b558988a77e137.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.101ex; height:2.843ex;" alt="{\displaystyle Y_{ij}=\mu +U_{i}+W_{ij},\,}" 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 \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> is the average test score for the entire population.
</p><p>In this model <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 U_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation>
</semantics>
</math></span><img src="./2b21a6f475b0e68475c6019abe1fed0b415e0e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}" loading="lazy"></span> is the school-specific <b>random effect</b>: it measures the difference between the average score at school <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> and the average score in the entire country. The term <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 W_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>W</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W_{ij}}</annotation>
</semantics>
</math></span><img src="./29c09e9d719bb634d8ca5a6172b0562b945bf325.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.671ex; height:2.843ex;" alt="{\displaystyle W_{ij}}" loading="lazy"></span> is the individual-specific random effect, i.e., it's the deviation of the <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 j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j}</annotation>
</semantics>
</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span>-th pupil's score from the average for the <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>-th school.
</p><p>The model can be augmented by including additional explanatory variables, which would capture differences in scores among different groups. For example:
</p>
<dl><dd><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 Y_{ij}=\mu +\beta _{1}\mathrm {Sex} _{ij}+\beta _{2}\mathrm {ParentsEduc} _{ij}+U_{i}+W_{ij},\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
<mo>+</mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">S</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">P</mi>
<mi mathvariant="normal">a</mi>
<mi mathvariant="normal">r</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">n</mi>
<mi mathvariant="normal">t</mi>
<mi mathvariant="normal">s</mi>
<mi mathvariant="normal">E</mi>
<mi mathvariant="normal">d</mi>
<mi mathvariant="normal">u</mi>
<mi mathvariant="normal">c</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>W</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y_{ij}=\mu +\beta _{1}\mathrm {Sex} _{ij}+\beta _{2}\mathrm {ParentsEduc} _{ij}+U_{i}+W_{ij},\,}</annotation>
</semantics>
</math></span><img src="./17cf1bdb21aba3031c4459141e0f76d80a51a5c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:50.031ex; height:2.843ex;" alt="{\displaystyle Y_{ij}=\mu +\beta _{1}\mathrm {Sex} _{ij}+\beta _{2}\mathrm {ParentsEduc} _{ij}+U_{i}+W_{ij},\,}" 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 \mathrm {Sex} _{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">S</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {Sex} _{ij}}</annotation>
</semantics>
</math></span><img src="./c478f3bad0104372ecb0292527c9553b79cb5ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.03ex; height:2.843ex;" alt="{\displaystyle \mathrm {Sex} _{ij}}" loading="lazy"></span> is a binary <a href="Dummy_variable_(statistics)" title="Dummy variable (statistics)">dummy variable</a> 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 \mathrm {ParentsEduc} _{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">P</mi>
<mi mathvariant="normal">a</mi>
<mi mathvariant="normal">r</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">n</mi>
<mi mathvariant="normal">t</mi>
<mi mathvariant="normal">s</mi>
<mi mathvariant="normal">E</mi>
<mi mathvariant="normal">d</mi>
<mi mathvariant="normal">u</mi>
<mi mathvariant="normal">c</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {ParentsEduc} _{ij}}</annotation>
</semantics>
</math></span><img src="./1a9b63a65b4964f2f445cee4236b562f4f898b84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.48ex; height:2.843ex;" alt="{\displaystyle \mathrm {ParentsEduc} _{ij}}" loading="lazy"></span>records, say, the average education level of a child's parents. This is a <a href="Mixed_model" title="Mixed model">mixed model</a>, not a purely random effects model, as it introduces <a href="Fixed_effects_model" title="Fixed effects model">fixed-effects</a> terms for Sex and Parents' Education.
</p>
<div class="mw-heading mw-heading3"><h3 id="Variance_components">Variance components</h3></div>
<p>The variance of <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 Y_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y_{ij}}</annotation>
</semantics>
</math></span><img src="./6035187f7e2cee387277f07091bb7827e8e66818.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.828ex; height:2.843ex;" alt="{\displaystyle Y_{ij}}" loading="lazy"></span> is the sum of the variances <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 \tau ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>τ<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau ^{2}}</annotation>
</semantics>
</math></span><img src="./86a31603f8cb4eba1905d7c1c468146553dfa40c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.312ex; height:2.676ex;" alt="{\displaystyle \tau ^{2}}" 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 \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="./53a5c55e536acf250c1d3e0f754be5692b843ef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}" loading="lazy"></span> of <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 U_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation>
</semantics>
</math></span><img src="./2b21a6f475b0e68475c6019abe1fed0b415e0e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}" 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 W_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>W</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W_{ij}}</annotation>
</semantics>
</math></span><img src="./29c09e9d719bb634d8ca5a6172b0562b945bf325.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.671ex; height:2.843ex;" alt="{\displaystyle W_{ij}}" loading="lazy"></span> respectively.
</p><p>Let
</p>
<dl><dd><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 {\overline {Y}}_{i\bullet }={\frac {1}{n}}\sum _{j=1}^{n}Y_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>Y</mi>
<mo accent="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>∙<!-- ∙ --></mo>
</mrow>
</msub>
<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>j</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>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\overline {Y}}_{i\bullet }={\frac {1}{n}}\sum _{j=1}^{n}Y_{ij}}</annotation>
</semantics>
</math></span><img src="./c5cde5df1435e98a98d8b4148b1197513fab9415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.071ex; height:7.176ex;" alt="{\displaystyle {\overline {Y}}_{i\bullet }={\frac {1}{n}}\sum _{j=1}^{n}Y_{ij}}" loading="lazy"></span></dd></dl>
<p>be the average, not of all scores at the <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>-th school, but of those at the <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>-th school that are included in the <a href="Random_sample" class="mw-redirect" title="Random sample">random sample</a>. Let
</p>
<dl><dd><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 {\overline {Y}}_{\bullet \bullet }={\frac {1}{mn}}\sum _{i=1}^{m}\sum _{j=1}^{n}Y_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>Y</mi>
<mo accent="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>∙<!-- ∙ --></mo>
<mo>∙<!-- ∙ --></mo>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>m</mi>
<mi>n</mi>
</mrow>
</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>m</mi>
</mrow>
</munderover>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</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>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\overline {Y}}_{\bullet \bullet }={\frac {1}{mn}}\sum _{i=1}^{m}\sum _{j=1}^{n}Y_{ij}}</annotation>
</semantics>
</math></span><img src="./fc137b4b82732e2fef4e391d3dcc5f6c40299a4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.108ex; height:7.176ex;" alt="{\displaystyle {\overline {Y}}_{\bullet \bullet }={\frac {1}{mn}}\sum _{i=1}^{m}\sum _{j=1}^{n}Y_{ij}}" loading="lazy"></span></dd></dl>
<p>be the <a href="Grand_average" class="mw-redirect" title="Grand average">grand average</a>.
</p><p>Let
</p>
<dl><dd><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 SSW=\sum _{i=1}^{m}\sum _{j=1}^{n}(Y_{ij}-{\overline {Y}}_{i\bullet })^{2}\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle SSW=\sum _{i=1}^{m}\sum _{j=1}^{n}(Y_{ij}-{\overline {Y}}_{i\bullet })^{2}\,}</annotation>
</semantics>
</math></span><img src="./9bfb69d0ffaf16d39d7e4fe4f5640303783e74aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.333ex; height:7.176ex;" alt="{\displaystyle SSW=\sum _{i=1}^{m}\sum _{j=1}^{n}(Y_{ij}-{\overline {Y}}_{i\bullet })^{2}\,}" loading="lazy"></span></dd></dl>
<dl><dd><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 SSB=n\sum _{i=1}^{m}({\overline {Y}}_{i\bullet }-{\overline {Y}}_{\bullet \bullet })^{2}\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mi>S</mi>
<mi>B</mi>
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<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle SSB=n\sum _{i=1}^{m}({\overline {Y}}_{i\bullet }-{\overline {Y}}_{\bullet \bullet })^{2}\,}</annotation>
</semantics>
</math></span><img src="./58c323ed353738d019e7bfce212f92cafffdfb90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.913ex; height:6.843ex;" alt="{\displaystyle SSB=n\sum _{i=1}^{m}({\overline {Y}}_{i\bullet }-{\overline {Y}}_{\bullet \bullet })^{2}\,}" loading="lazy"></span></dd></dl>
<p>be respectively the sum of squares due to differences <i>within</i> groups and the sum of squares due to difference <i>between</i> groups. Then it can be shown that
</p>
<dl><dd><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 {\frac {1}{m(n-1)}}E(SSW)=\sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>m</mi>
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<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{m(n-1)}}E(SSW)=\sigma ^{2}}</annotation>
</semantics>
</math></span><img src="./ffc9494e92a1e216cbe1b9c03c6e36caf544dfc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.585ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{m(n-1)}}E(SSW)=\sigma ^{2}}" loading="lazy"></span></dd></dl>
<p>and
</p>
<dl><dd><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 {\frac {1}{(m-1)n}}E(SSB)={\frac {\sigma ^{2}}{n}}+\tau ^{2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mo stretchy="false">(</mo>
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<mi>n</mi>
</mrow>
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</mrow>
<mi>E</mi>
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<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>n</mi>
</mfrac>
</mrow>
<mo>+</mo>
<msup>
<mi>τ<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(m-1)n}}E(SSB)={\frac {\sigma ^{2}}{n}}+\tau ^{2}.}</annotation>
</semantics>
</math></span><img src="./79ac9d5bc29ba74e84d33d64929ed9ae601ecf64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.549ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{(m-1)n}}E(SSB)={\frac {\sigma ^{2}}{n}}+\tau ^{2}.}" loading="lazy"></span></dd></dl>
<p>These "<a href="Expected_mean_square" class="mw-redirect" title="Expected mean square">expected mean squares</a>" can be used as the basis for <a href="Estimation" title="Estimation">estimation</a> of the "variance components" <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 \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="./53a5c55e536acf250c1d3e0f754be5692b843ef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}" loading="lazy"></span> and <i><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 \tau ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>τ<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau ^{2}}</annotation>
</semantics>
</math></span><img src="./86a31603f8cb4eba1905d7c1c468146553dfa40c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.312ex; height:2.676ex;" alt="{\displaystyle \tau ^{2}}" loading="lazy"></span>.</i>
</p><p>The <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 \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="./53a5c55e536acf250c1d3e0f754be5692b843ef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}" loading="lazy"></span> parameter is also called the <a href="Intraclass_correlation#Modern_ICC_definitions:_simpler_formula_but_positive_bias" title="Intraclass correlation">intraclass correlation coefficient</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Marginal_likelihood">Marginal likelihood</h2></div>
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<p>For random effects models the <a href="Marginal_likelihood" title="Marginal likelihood">marginal likelihoods</a> are important.<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>
<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>
<p>Random effects models used in practice include the <a href="B%C3%BChlmann_model" title="Bühlmann model">Bühlmann model</a> of insurance contracts and the <a href="Fay-Herriot_model" class="mw-redirect" title="Fay-Herriot model">Fay-Herriot model</a> used for <a href="Small_area_estimation" title="Small area estimation">small area estimation</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="B%C3%BChlmann_model" title="Bühlmann model">Bühlmann model</a></li>
<li><a href="Hierarchical_linear_modeling" class="mw-redirect" title="Hierarchical linear modeling">Hierarchical linear modeling</a></li>
<li><a href="Fixed_effects" class="mw-redirect" title="Fixed effects">Fixed effects</a></li>
<li><a href="MINQUE" title="MINQUE">MINQUE</a></li>
<li><a href="Covariance_estimation" class="mw-redirect" title="Covariance estimation">Covariance estimation</a></li>
<li><a href="Conditional_variance" title="Conditional variance">Conditional variance</a></li>
<li><a href="Panel_analysis" title="Panel analysis">Panel analysis</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>
<ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">
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/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFBaltagi2008" class="citation book cs1">Baltagi, Badi H. (2008). <i>Econometric Analysis of Panel Data</i> (4th ed.). New York, NY: Wiley. pp. <span class="nowrap">17–</span>22. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-51886-1</bdi>.</cite></li>
<li><cite id="CITEREFHsiao2003" class="citation book cs1">Hsiao, Cheng (2003). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/analysispaneldat00chsi"><i>Analysis of Panel Data</i></a></span> (2nd ed.). New York, NY: Cambridge University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/analysispaneldat00chsi/page/n90">73</a>–92. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-52271-4</bdi>.</cite></li>
<li><cite id="CITEREFWooldridge2002" class="citation book cs1">Wooldridge, Jeffrey M. (2002). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/econometricanaly0000wool"><i>Econometric Analysis of Cross Section and Panel Data</i></a></span>. Cambridge, MA: MIT Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/econometricanaly0000wool/page/257">257–265</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-23219-7</bdi>.</cite></li>
<li><cite id="CITEREFGomes2022" class="citation journal cs1">Gomes, Dylan G.E. (20 January 2022). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8784019">"Should I use fixed effects or random effects when I have fewer than five levels of a grouping factor in a mixed-effects model?"</a>. <i>PeerJ</i>. <b>10</b>: e12794. <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/10.7717%2Fpeerj.12794">10.7717/peerj.12794</a></span>. <a href="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/PMC8784019">8784019</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/35116198">35116198</a>.</cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite id="CITEREFDiggleHeagertyLiangZeger2002" class="citation book cs1">Diggle, Peter J.; Heagerty, Patrick; Liang, Kung-Yee; Zeger, Scott L. (2002). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/analysislongitud00digg_730"><i>Analysis of Longitudinal Data</i></a></span> (2nd ed.). Oxford University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/analysislongitud00digg_730/page/n96">169</a>–171. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-852484-6</bdi>.</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="CITEREFFitzmauriceLairdWare2004" class="citation book cs1">Fitzmaurice, Garrett M.; Laird, Nan M.; Ware, James H. (2004). <i>Applied Longitudinal Analysis</i>. Hoboken: John Wiley & Sons. pp. <span class="nowrap">326–</span>328. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-21487-6</bdi>.</cite></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFLairdWare1982" class="citation journal cs1">Laird, Nan M.; Ware, James H. (1982). "Random-Effects Models for Longitudinal Data". <i><a href="Biometrics_(journal)" title="Biometrics (journal)">Biometrics</a></i>. <b>38</b> (4): <span class="nowrap">963–</span>974. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2529876">10.2307/2529876</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/2529876">2529876</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/7168798">7168798</a>.</cite></span>
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<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="CITEREFGardinerLuoRoman2009" class="citation journal cs1">Gardiner, Joseph C.; Luo, Zhehui; Roman, Lee Anne (2009). "Fixed effects, random effects and GEE: What are the differences?". <i><a href="Statistics_in_Medicine_(journal)" title="Statistics in Medicine (journal)">Statistics in Medicine</a></i>. <b>28</b> (2): <span class="nowrap">221–</span>239. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fsim.3478">10.1002/sim.3478</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19012297">19012297</a>.</cite></span>
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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFGomes2022" class="citation journal cs1">Gomes, Dylan G.E. (20 January 2022). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8784019">"Should I use fixed effects or random effects when I have fewer than five levels of a grouping factor in a mixed-effects model?"</a>. <i>PeerJ</i>. <b>10</b>: e12794. <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/10.7717%2Fpeerj.12794">10.7717/peerj.12794</a></span>. <a href="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/PMC8784019">8784019</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/35116198">35116198</a>.</cite></span>
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<li id="cite_note-:0-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFWooldridge2010" class="citation book cs1">Wooldridge, Jeffrey (2010). <i>Econometric analysis of cross section and panel data</i> (2nd ed.). Cambridge, Mass.: MIT Press. p. 252. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780262232586</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/627701062">627701062</a>.</cite></span>
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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Hedeker, D., Gibbons, R. D. (2006). Longitudinal Data Analysis. Deutschland: Wiley. Page 163 <a rel="nofollow" class="external free" href="https://books.google.com/books?id=f9p9iIgzQSQC&pg=PA163">https://books.google.com/books?id=f9p9iIgzQSQC&pg=PA163</a></span>
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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="http://teaching.sociology.ul.ie/DCW/confront/node45.html">Fixed and random effects models</a></li>
<li><a rel="nofollow" class="external text" href="http://www.pitt.edu/~SUPER1/lecture/lec1171/012.htm">How to Conduct a Meta-Analysis: Fixed and Random Effect Models</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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