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id="toc-Estimation-sublist" class="vector-toc-list"> <li id="toc-Choice_of_random_effects_structure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Choice_of_random_effects_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Choice of random effects structure</span> </div> </a> <ul id="toc-Choice_of_random_effects_structure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Software" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Software"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Software</span> </div> </a> <ul id="toc-Software-sublist" class="vector-toc-list"> </ul> </li> </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">5</span> <span>See also</span> </div> </a> <ul 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class="mw-page-title-main">Mixed model</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gemischtes_Modell" title="Gemischtes Modell – German" lang="de" hreflang="de" data-title="Gemischtes Modell" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Mod%C3%A8le_mixte" title="Modèle mixte – French" lang="fr" hreflang="fr" data-title="Modèle mixte" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%B7%B7%E5%90%88%E3%83%A2%E3%83%87%E3%83%AB" title="混合モデル – Japanese" lang="ja" hreflang="ja" data-title="混合モデル" 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/Blandede_modeller" title="Blandede modeller – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Blandede modeller" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Modelo_misto" title="Modelo misto – Portuguese" lang="pt" hreflang="pt" data-title="Modelo misto" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%BC%D1%96%D1%88%D0%B0%D0%BD%D0%B0_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C" title="Змішана модель – Ukrainian" lang="uk" hreflang="uk" 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</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="/wiki/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="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple regression</a></li> <li><a href="/wiki/Polynomial_regression" title="Polynomial regression">Polynomial regression</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></li> <li><a href="/wiki/Vector_generalized_linear_model" title="Vector generalized linear model">Vector generalized linear model</a></li> <li><a href="/wiki/Discrete_choice" title="Discrete choice">Discrete choice</a></li> <li><a href="/wiki/Binomial_regression" title="Binomial regression">Binomial regression</a></li> <li><a href="/wiki/Binary_regression" title="Binary regression">Binary regression</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a></li> <li><a href="/wiki/Multinomial_logistic_regression" title="Multinomial logistic regression">Multinomial logistic regression</a></li> <li><a href="/wiki/Mixed_logit" title="Mixed logit">Mixed logit</a></li> <li><a href="/wiki/Probit_model" title="Probit model">Probit</a></li> <li><a href="/wiki/Multinomial_probit" title="Multinomial probit">Multinomial probit</a></li> <li><a href="/wiki/Ordered_logit" title="Ordered logit">Ordered logit</a></li> <li><a href="/wiki/Ordered_probit" class="mw-redirect" title="Ordered probit">Ordered probit</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Poisson</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Multilevel_model" title="Multilevel model">Multilevel model</a></li> <li><a href="/wiki/Fixed_effects_model" title="Fixed effects model">Fixed effects</a></li> <li><a href="/wiki/Random_effects_model" title="Random effects model">Random effects</a></li> <li><a class="mw-selflink selflink">Linear mixed-effects model</a></li> <li><a href="/wiki/Nonlinear_mixed-effects_model" title="Nonlinear mixed-effects model">Nonlinear mixed-effects model</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Quantile_regression" title="Quantile regression">Quantile</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Principal_component_regression" title="Principal component regression">Principal components</a></li> <li><a href="/wiki/Least-angle_regression" title="Least-angle regression">Least angle</a></li> <li><a href="/wiki/Local_regression" title="Local regression">Local</a></li> <li><a href="/wiki/Segmented_regression" title="Segmented regression">Segmented</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Errors-in-variables_models" class="mw-redirect" title="Errors-in-variables models">Errors-in-variables</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Estimation</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Least_squares" title="Least squares">Least squares</a></li> <li><a href="/wiki/Linear_least_squares" title="Linear least squares">Linear</a></li> <li><a href="/wiki/Non-linear_least_squares" title="Non-linear least squares">Non-linear</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary</a></li> <li><a href="/wiki/Weighted_least_squares" title="Weighted least squares">Weighted</a></li> <li><a href="/wiki/Generalized_least_squares" title="Generalized least squares">Generalized</a></li> <li><a href="/wiki/Generalized_estimating_equation" title="Generalized estimating equation">Generalized estimating equation</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Partial_least_squares_regression" title="Partial least squares regression">Partial</a></li> <li><a href="/wiki/Total_least_squares" title="Total least squares">Total</a></li> <li><a href="/wiki/Non-negative_least_squares" title="Non-negative least squares">Non-negative</a></li> <li><a href="/wiki/Tikhonov_regularization" class="mw-redirect" title="Tikhonov regularization">Ridge regression</a></li> <li><a href="/wiki/Regularized_least_squares" title="Regularized least squares">Regularized</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Least_absolute_deviations" title="Least absolute deviations">Least absolute deviations</a></li> <li><a href="/wiki/Iteratively_reweighted_least_squares" title="Iteratively reweighted least squares">Iteratively reweighted</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian</a></li> <li><a href="/wiki/Bayesian_multivariate_linear_regression" title="Bayesian multivariate linear regression">Bayesian multivariate</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Background</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mean_and_predicted_response" class="mw-redirect" title="Mean and predicted response">Mean and predicted response</a></li> <li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></li> <li><a href="/wiki/Studentized_residual" title="Studentized residual">Studentized residual</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a></li></ul></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Regression_bar" title="Template:Regression bar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Regression_bar" title="Template talk:Regression bar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Regression_bar" title="Special:EditPage/Template:Regression bar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>A <b>mixed model</b>, <b>mixed-effects model</b> or <b>mixed error-component model</b> is a <a href="/wiki/Statistical_model" title="Statistical model">statistical model</a> containing both <a href="/wiki/Fixed_effect" class="mw-redirect" title="Fixed effect">fixed effects</a> and <a href="/wiki/Random_effect" class="mw-redirect" title="Random effect">random effects</a>.<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-Gomes2022_2-0" class="reference"><a href="#cite_note-Gomes2022-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> These models are useful in a wide variety of disciplines in the physical, biological and social sciences. They are particularly useful in settings where <a href="/wiki/Repeated_measures_design" title="Repeated measures design">repeated measurements</a> are made on the same <a href="/wiki/Statistical_unit" title="Statistical unit">statistical units</a> (see also <a href="/wiki/Longitudinal_study" title="Longitudinal study">longitudinal study</a>), or where measurements are made on clusters of related statistical units.<sup id="cite_ref-Gomes2022_2-1" class="reference"><a href="#cite_note-Gomes2022-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Mixed models are often preferred over traditional <a href="/wiki/Analysis_of_variance" title="Analysis of variance">analysis of variance</a> regression models because they don't rely on the independent observations assumption. Further, they have their flexibility in dealing with missing values and uneven spacing of repeated measurements.<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> The Mixed model analysis allows measurements to be explicitly modeled in a wider variety of <a href="/wiki/Correlation" title="Correlation">correlation</a> and <a href="/wiki/Variance" title="Variance">variance</a>-<a href="/wiki/Covariance" title="Covariance">covariance</a> <a href="/wiki/Variance" title="Variance">avoiding biased estimations</a> structures. </p><p>This page will discuss mainly <b>linear mixed-effects models</b> rather than <a href="/wiki/Generalized_linear_mixed_model" title="Generalized linear mixed model">generalized linear mixed models</a> or <a href="/wiki/Nonlinear_mixed-effects_model" title="Nonlinear mixed-effects model">nonlinear mixed-effects models</a>.<sup id="cite_ref-:0_4-0" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Qualitative_Description">Qualitative Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixed_model&action=edit&section=1" title="Edit section: Qualitative Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Linear mixed models (LMMs) are <a href="/wiki/Statistical_models" class="mw-redirect" title="Statistical models">statistical models</a> that incorporate <a href="/wiki/Fixed_effects" class="mw-redirect" title="Fixed effects">fixed</a> and <a href="/wiki/Random_effects" class="mw-redirect" title="Random effects">random effects</a> to accurately represent non-independent data structures. LMM is an alternative to <a href="/wiki/Analysis_of_variance" title="Analysis of variance">analysis of variance</a>. Often, ANOVA assumes the <a href="/wiki/Independence" title="Independence">independence</a> of observations within each group, however, this assumption may not hold in non-independent data, such as multilevel/<a href="/wiki/Hierarchical" class="mw-redirect" title="Hierarchical">hierarchical</a>, <a href="/wiki/Longitudinal_study" title="Longitudinal study">longitudinal</a>, or <a href="/wiki/Correlated" class="mw-redirect" title="Correlated">correlated</a> datasets. </p><p>Non-independent sets are ones in which the variability between outcomes is due to correlations within groups or between groups. Mixed models properly account for <a href="/wiki/Nesting_(computing)" title="Nesting (computing)">nest</a> structures/hierarchical data structures where observations are influenced by their nested associations. For example, when studying education methods involving multiple schools, there are multiple levels of variables to consider. The individual level/lower level comprises individual students or teachers within the school. The observations obtained from this student/teacher is nested within their school. For example, Student A is a unit within the School A. The next higher level is the school. At the higher level, the school contains multiple individual students and teachers. The school level influences the observations obtained from the students and teachers. For Example, School A and School B are the higher levels each with its set of Student A and Student B respectively. This represents a hierarchical data scheme. A solution to modeling hierarchical data is using linear mixed models. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Heiarchial_Data_Strucutre_Education.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Heiarchial_Data_Strucutre_Education.jpg/376px-Heiarchial_Data_Strucutre_Education.jpg" decoding="async" width="376" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Heiarchial_Data_Strucutre_Education.jpg/564px-Heiarchial_Data_Strucutre_Education.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Heiarchial_Data_Strucutre_Education.jpg/752px-Heiarchial_Data_Strucutre_Education.jpg 2x" data-file-width="1327" data-file-height="778" /></a><figcaption>Representation of how data, related to education system, is non-independent and structured in nested/hierarchical levels.</figcaption></figure> <p>LMMs allow us to understand the important effects between and within levels while incorporating the corrections for standard errors for non-independence embedded in the data structure.<sup id="cite_ref-:0_4-1" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:4_5-0" class="reference"><a href="#cite_note-:4-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In experimental fields such as social psychology, psycholinguistics, cognitive psychology (and neuroscience), where studies often involve multiple grouping variables, failing to account for random effects can lead to inflated Type I error rates and unreliable conclusions.<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><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> For instance, when analyzing data from experiments that involve both samples of participants and samples of stimuli (e.g., images, scenarios, etc.), ignoring variation in either of these grouping variables (e.g., by averaging over stimuli) can result in misleading conclusions. In such cases, researchers can instead treat both participant and stimulus as random effects with LMMs, and in doing so, can correctly account for the variation in their data across multiple grouping variables. Similarly, when analyzing data from comparative longitudinal surveys, failing to include random effects at all relevant levels—such as country and country-year—can significantly distort the results.<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> </p> <div class="mw-heading mw-heading3"><h3 id="The_Fixed_Effect">The Fixed Effect</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixed_model&action=edit&section=2" title="Edit section: The Fixed Effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fixed effects encapsulate the tendencies/trends that are consistent at the levels of primary interest. These effects are considered fixed because they are non-random and assumed to be constant for the population being studied.<sup id="cite_ref-:4_5-1" class="reference"><a href="#cite_note-:4-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> For example, when studying education a fixed effect could represent overall school level effects that are consistent across all schools. </p><p>While the hierarchy of the data set is typically obvious, the specific fixed effects that affect the average responses for all subjects must be specified. Some fixed effect coefficients are sufficient without corresponding random effects where as other fixed coefficients only represent an average where the individual units are random. These may be determined by incorporating random <a href="/wiki/Y-intercept" title="Y-intercept">intercepts</a> and <a href="/wiki/Slopes" class="mw-redirect" title="Slopes">slopes</a>.<sup id="cite_ref-:1_9-0" class="reference"><a href="#cite_note-:1-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_10-0" class="reference"><a href="#cite_note-:2-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:3_11-0" class="reference"><a href="#cite_note-:3-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>In most situations, several related models are considered and the model that best represents a universal model is adopted. </p> <div class="mw-heading mw-heading3"><h3 id="The_Random_Effect,_ε"><span id="The_Random_Effect.2C_.CE.B5"></span>The Random Effect, ε</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixed_model&action=edit&section=3" title="Edit section: The Random Effect, ε"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A key component of the mixed model is the incorporation of random effects with the fixed effect. Fixed effects are often fitted to represent the underlying model. In Linear mixed models, the true <a href="/wiki/Linear_regression" title="Linear regression">regression</a> of the population is linear, β. The fixed data is fitted at the highest level. Random effects introduce <a href="/wiki/Statistical_variability" class="mw-redirect" title="Statistical variability">statistical variability</a> at different levels of the data hierarchy. These account for the unmeasured sources of variance that affect certain groups in the data. For example, the differences between student 1 and student 2 in the same class, or the differences between class 1 and class 2 in the same school.  <sup id="cite_ref-:1_9-1" class="reference"><a href="#cite_note-:1-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_10-1" class="reference"><a href="#cite_note-:2-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:3_11-1" class="reference"><a href="#cite_note-:3-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History_and_current_status">History and current status</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixed_model&action=edit&section=4" title="Edit section: History and current status"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Bias_described_using_LMM.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Bias_described_using_LMM.jpg/235px-Bias_described_using_LMM.jpg" decoding="async" width="235" height="606" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c7/Bias_described_using_LMM.jpg 1.5x" data-file-width="242" data-file-height="623" /></a><figcaption>Representation of biased vs. unbiased data and the differences between fitted estimates using least squares regression (LSR) and a linear mixed model (LMM).</figcaption></figure> <p><a href="/wiki/Ronald_Fisher" title="Ronald Fisher">Ronald Fisher</a> introduced <a href="/wiki/Random_effects_model" title="Random effects model">random effects models</a> to study the correlations of trait values between relatives.<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> In the 1950s, <a href="/wiki/Charles_Roy_Henderson" title="Charles Roy Henderson">Charles Roy Henderson</a> provided <a href="/wiki/Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">best linear unbiased estimates</a> of <a href="/wiki/Fixed_effects_estimator" class="mw-redirect" title="Fixed effects estimator">fixed effects</a> and <a href="/wiki/Best_linear_unbiased_prediction" title="Best linear unbiased prediction">best linear unbiased predictions</a> of random effects.<sup id="cite_ref-GKR1991_13-0" class="reference"><a href="#cite_note-GKR1991-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><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><sup id="cite_ref-LDVV1989_15-0" class="reference"><a href="#cite_note-LDVV1989-15"><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> Subsequently, mixed modeling has become a major area of statistical research, including work on computation of maximum likelihood estimates, non-linear mixed effects models, missing data in mixed effects models, and <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian</a> estimation of mixed effects models. Mixed models are applied in many disciplines where multiple correlated measurements are made on each unit of interest. They are prominently used in research involving human and animal subjects in fields ranging from genetics to marketing, and have also been used in baseball <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> and industrial statistics.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The mixed linear model association has improved the prevention of false positive associations. Populations are deeply interconnected and the relatedness structure of population dynamics is extremely difficult to model without the use of mixed models. Linear mixed models may not, however, be the only solution. LMM's have a constant-<a href="/wiki/Residual_(statistics)" class="mw-redirect" title="Residual (statistics)">residual</a> <a href="/wiki/Variance" title="Variance">variance</a> assumption that is sometimes violated when accounting for deeply associated <a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">continuous</a> and <a href="/wiki/Binary_function" title="Binary function">binary</a> traits.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixed_model&action=edit&section=5" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Matrix_notation#Notation" class="mw-redirect" title="Matrix notation">matrix notation</a> a linear mixed model can be represented 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 {\boldsymbol {y}}=X{\boldsymbol {\beta }}+Z{\boldsymbol {u}}+{\boldsymbol {\epsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">y</mi> </mrow> <mo>=</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mo>+</mo> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϵ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {y}}=X{\boldsymbol {\beta }}+Z{\boldsymbol {u}}+{\boldsymbol {\epsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e00ba9a8e13237ca374d55bb3070aebd12a5b8e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.051ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {y}}=X{\boldsymbol {\beta }}+Z{\boldsymbol {u}}+{\boldsymbol {\epsilon }}}"></span></dd></dl> <p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3ae71d44145d721c4b15d442e03005e5ea9850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.371ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {y}}}"></span> is a known vector of observations, with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E({\boldsymbol {y}})=X{\boldsymbol {\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">y</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E({\boldsymbol {y}})=X{\boldsymbol {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e80868378849f4243dd83136dbcd55f5bae55a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.569ex; height:2.843ex;" alt="{\displaystyle E({\boldsymbol {y}})=X{\boldsymbol {\beta }}}"></span>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702cafc420cc00c54896f6d125112820956aaf6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}"></span> is an unknown vector of fixed effects;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d60e374e33b2c1d75888c0e8759f9e770e718f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {u}}}"></span> is an unknown vector of random effects, with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E({\boldsymbol {u}})={\boldsymbol {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E({\boldsymbol {u}})={\boldsymbol {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc9b89cf3bf8af31f308b852232ea1e164e06ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.603ex; height:2.843ex;" alt="{\displaystyle E({\boldsymbol {u}})={\boldsymbol {0}}}"></span> and <a href="/wiki/Covariance_matrix" title="Covariance matrix">variance–covariance matrix</a> <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 \operatorname {var} ({\boldsymbol {u}})=G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {var} ({\boldsymbol {u}})=G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92e7197d495b985e3b4908e414ecb60ef3d58d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.619ex; height:2.843ex;" alt="{\displaystyle \operatorname {var} ({\boldsymbol {u}})=G}"></span>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\epsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϵ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\epsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be3ff7747cde11092061b1cf99a03eb19bfeba31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.123ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\epsilon }}}"></span> is an unknown vector of random errors, with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E({\boldsymbol {\epsilon }})={\boldsymbol {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E({\boldsymbol {\epsilon }})={\boldsymbol {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6276da282871b9e361aa0a794c61d2fc059b77f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.143ex; height:2.843ex;" alt="{\displaystyle E({\boldsymbol {\epsilon }})={\boldsymbol {0}}}"></span> and variance <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 \operatorname {var} ({\boldsymbol {\epsilon }})=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {var} ({\boldsymbol {\epsilon }})=R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3172e3fd6b8283eae893b11d6b945d8d37f089a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.096ex; height:2.843ex;" alt="{\displaystyle \operatorname {var} ({\boldsymbol {\epsilon }})=R}"></span>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> is the known <a href="/wiki/Design_matrix" title="Design matrix">design matrix</a> for the fixed effects relating the observations <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 {\boldsymbol {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3ae71d44145d721c4b15d442e03005e5ea9850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.371ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {y}}}"></span> 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 {\boldsymbol {\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702cafc420cc00c54896f6d125112820956aaf6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}"></span>, respectively</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> is the known <a href="/wiki/Design_matrix" title="Design matrix">design matrix</a> for the random effects relating the observations <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 {\boldsymbol {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3ae71d44145d721c4b15d442e03005e5ea9850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.371ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {y}}}"></span> 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 {\boldsymbol {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d60e374e33b2c1d75888c0e8759f9e770e718f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {u}}}"></span>, respectively.</li></ul> <p>For example, if each observation can belong to any zero or more of <span class="texhtml mvar" style="font-style:italic;">k</span> categories then <span class="texhtml mvar" style="font-style:italic;">Z</span>, which has one row per observation, can be chosen to have <span class="texhtml mvar" style="font-style:italic;">k</span> columns, where a value of <span class="texhtml">1</span> for a matrix element of <span class="texhtml mvar" style="font-style:italic;">Z</span> indicates that an observation is known to belong to a category and a value of <span class="texhtml">0</span> indicates that an observation is known to not belong to a category. The inferred value of <span class="texhtml mvar" style="font-style:italic;">u</span> for a category is then a category-specific <a href="/wiki/Y-intercept" title="Y-intercept">intercept</a>. If <span class="texhtml mvar" style="font-style:italic;">Z</span> has additional columns, where the non-zero values are instead the value of an independent variable for an observation, then the corresponding inferred value of <span class="texhtml mvar" style="font-style:italic;">u</span> is a category-specific <a href="/wiki/Slope" title="Slope">slope</a> for that independent variable. The prior distribution for the category intercepts and slopes is described by the covariance matrix <span class="texhtml mvar" style="font-style:italic;">G</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Estimation">Estimation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixed_model&action=edit&section=6" title="Edit section: Estimation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The joint density 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 {\boldsymbol {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3ae71d44145d721c4b15d442e03005e5ea9850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.371ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {y}}}"></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 {\boldsymbol {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d60e374e33b2c1d75888c0e8759f9e770e718f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {u}}}"></span> can be written 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 f({\boldsymbol {y}},{\boldsymbol {u}})=f({\boldsymbol {y}}|{\boldsymbol {u}})\,f({\boldsymbol {u}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">y</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f({\boldsymbol {y}},{\boldsymbol {u}})=f({\boldsymbol {y}}|{\boldsymbol {u}})\,f({\boldsymbol {u}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/176b5a405d4fe1aeda9d9c6b740bec143dd152a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.921ex; height:2.843ex;" alt="{\displaystyle f({\boldsymbol {y}},{\boldsymbol {u}})=f({\boldsymbol {y}}|{\boldsymbol {u}})\,f({\boldsymbol {u}})}"></span>. Assuming normality, <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 {\boldsymbol {u}}\sim {\mathcal {N}}({\boldsymbol {0}},G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}\sim {\mathcal {N}}({\boldsymbol {0}},G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15295eb4e351bc0572f8094b532af1e3827034f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.963ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {u}}\sim {\mathcal {N}}({\boldsymbol {0}},G)}"></span>, <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 {\boldsymbol {\epsilon }}\sim {\mathcal {N}}({\boldsymbol {0}},R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\epsilon }}\sim {\mathcal {N}}({\boldsymbol {0}},R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf2c338eff29cee9d70136af2032106812d304b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.44ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {\epsilon }}\sim {\mathcal {N}}({\boldsymbol {0}},R)}"></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 \mathrm {Cov} ({\boldsymbol {u}},{\boldsymbol {\epsilon }})={\boldsymbol {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">v</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Cov} ({\boldsymbol {u}},{\boldsymbol {\epsilon }})={\boldsymbol {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2387a151a2bb4c4f2a9e5cbf2f11b1d661388cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.052ex; height:2.843ex;" alt="{\displaystyle \mathrm {Cov} ({\boldsymbol {u}},{\boldsymbol {\epsilon }})={\boldsymbol {0}}}"></span>, and maximizing the joint density over <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 {\boldsymbol {\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702cafc420cc00c54896f6d125112820956aaf6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}"></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 {\boldsymbol {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d60e374e33b2c1d75888c0e8759f9e770e718f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {u}}}"></span>, gives Henderson's "mixed model equations" (MME) for linear mixed models:<sup id="cite_ref-GKR1991_13-1" class="reference"><a href="#cite_note-GKR1991-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-LDVV1989_15-1" class="reference"><a href="#cite_note-LDVV1989-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <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 {\begin{pmatrix}X'R^{-1}X&X'R^{-1}Z\\Z'R^{-1}X&Z'R^{-1}Z+G^{-1}\end{pmatrix}}{\begin{pmatrix}{\hat {\boldsymbol {\beta }}}\\{\hat {\boldsymbol {u}}}\end{pmatrix}}={\begin{pmatrix}X'R^{-1}{\boldsymbol {y}}\\Z'R^{-1}{\boldsymbol {y}}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>X</mi> <mo>′</mo> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>X</mi> </mtd> <mtd> <msup> <mi>X</mi> <mo>′</mo> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>Z</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>Z</mi> <mo>′</mo> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>X</mi> </mtd> <mtd> <msup> <mi>Z</mi> <mo>′</mo> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>Z</mi> <mo>+</mo> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>X</mi> <mo>′</mo> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">y</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>Z</mi> <mo>′</mo> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">y</mi> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}X'R^{-1}X&X'R^{-1}Z\\Z'R^{-1}X&Z'R^{-1}Z+G^{-1}\end{pmatrix}}{\begin{pmatrix}{\hat {\boldsymbol {\beta }}}\\{\hat {\boldsymbol {u}}}\end{pmatrix}}={\begin{pmatrix}X'R^{-1}{\boldsymbol {y}}\\Z'R^{-1}{\boldsymbol {y}}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95a98287183c43db7b3ee3f0b9ab930fd957ca84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.6ex; height:6.509ex;" alt="{\displaystyle {\begin{pmatrix}X'R^{-1}X&X'R^{-1}Z\\Z'R^{-1}X&Z'R^{-1}Z+G^{-1}\end{pmatrix}}{\begin{pmatrix}{\hat {\boldsymbol {\beta }}}\\{\hat {\boldsymbol {u}}}\end{pmatrix}}={\begin{pmatrix}X'R^{-1}{\boldsymbol {y}}\\Z'R^{-1}{\boldsymbol {y}}\end{pmatrix}}}"></span></dd></dl> <p>where for example <span class="texhtml mvar" style="font-style:italic;">X′</span> is the <a href="/wiki/Matrix_transpose" class="mw-redirect" title="Matrix transpose">matrix transpose</a> of <span class="texhtml mvar" style="font-style:italic;">X</span> and <span class="texhtml"><i>R</i><sup>−1</sup></span> is the <a href="/wiki/Matrix_inverse" class="mw-redirect" title="Matrix inverse">matrix inverse</a> of <span class="texhtml mvar" style="font-style:italic;">R</span>. </p><p>The solutions to the MME, <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 \textstyle {\hat {\boldsymbol {\beta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\boldsymbol {\beta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b85d0e04df1e0176fc55514ac6763ebf8e1b20a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.571ex; height:3.176ex;" alt="{\displaystyle \textstyle {\hat {\boldsymbol {\beta }}}}"></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 \textstyle {\hat {\boldsymbol {u}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\boldsymbol {u}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a68c211cb74c368dbc3795aab48a47a6dd7cb7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.343ex;" alt="{\displaystyle \textstyle {\hat {\boldsymbol {u}}}}"></span> are best linear unbiased estimates and predictors for <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 {\boldsymbol {\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702cafc420cc00c54896f6d125112820956aaf6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}"></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 {\boldsymbol {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d60e374e33b2c1d75888c0e8759f9e770e718f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {u}}}"></span>, respectively. This is a consequence of the <a href="/wiki/Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a> when the <a href="/wiki/Conditional_variance" title="Conditional variance">conditional variance</a> of the outcome is not scalable to the identity matrix. When the conditional variance is known, then the inverse variance weighted least squares estimate is best linear unbiased estimates. However, the conditional variance is rarely, if ever, known. So it is desirable to jointly estimate the variance and weighted parameter estimates when solving MMEs. </p> <div class="mw-heading mw-heading3"><h3 id="Choice_of_random_effects_structure">Choice of random effects structure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixed_model&action=edit&section=7" title="Edit section: Choice of random effects structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One choice that analysts face with mixed models is which random effects (i.e., grouping variables, random intercepts, and random slopes) to include. One prominent recommendation in the context of confirmatory hypothesis testing<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> is to adopt a "maximal" random effects structure, including all possible random effects justified by the experimental design, as a means to control Type I error rates. </p> <div class="mw-heading mw-heading3"><h3 id="Software">Software</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixed_model&action=edit&section=8" title="Edit section: Software"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One method used to fit such mixed models is that of the <a href="/wiki/Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">expectation–maximization algorithm</a> (EM) where the variance components are treated as unobserved <a href="/wiki/Nuisance_parameter" title="Nuisance parameter">nuisance parameters</a> in the joint likelihood.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Currently, this is the method implemented in statistical software such as <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> (statsmodels package) and <a href="/wiki/SAS_(software)" title="SAS (software)">SAS</a> (proc mixed), and as initial step only in <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>'s nlme package lme(). The solution to the mixed model equations is a <a href="/wiki/Maximum_likelihood_estimate" class="mw-redirect" title="Maximum likelihood estimate">maximum likelihood estimate</a> when the distribution of the errors is normal.<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><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Mixedandfixedeffects.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Mixedandfixedeffects.jpg/346px-Mixedandfixedeffects.jpg" decoding="async" width="346" height="237" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Mixedandfixedeffects.jpg/519px-Mixedandfixedeffects.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Mixedandfixedeffects.jpg/692px-Mixedandfixedeffects.jpg 2x" data-file-width="1915" data-file-height="1310" /></a><figcaption>Fixed, mixed, and random effects influence linear regression models.</figcaption></figure> <p>There are several other methods to fit mixed models, including using a mixed effect model (MEM) initially, and then Newton-Raphson (used by <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a> package nlme<sup id="cite_ref-pinheiro_bates2006_25-0" class="reference"><a href="#cite_note-pinheiro_bates2006-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>'s lme()), penalized least squares to get a profiled log likelihood only depending on the (low-dimensional) variance-covariance parameters 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 {\boldsymbol {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d60e374e33b2c1d75888c0e8759f9e770e718f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {u}}}"></span>, i.e., its cov matrix <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 {\boldsymbol {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">G</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef237517a5b4d24e57a1e81481f18ec67256b99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.061ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {G}}}"></span>, and then modern direct optimization for that reduced objective function (used by <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>'s lme4<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> package lmer() and the <a href="/wiki/Julia_(programming_language)" title="Julia (programming language)">Julia</a> package MixedModels.jl) and direct optimization of the likelihood (used by e.g. <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>'s glmmTMB). Notably, while the canonical form proposed by Henderson is useful for theory, many popular software packages use a different formulation for numerical computation in order to take advantage of sparse matrix methods (e.g. lme4 and MixedModels.jl). </p><p>In the context of Bayesian methods, the brms package provides a user-friendly interface for fitting mixed models in R using Stan, allowing for the incorporation of prior distributions and the estimation of posterior distributions.<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-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> In python, Bambi provides a similarly streamlined approach for fitting mixed effects models using PyMC.<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=Mixed_model&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Nonlinear_mixed-effects_model" title="Nonlinear mixed-effects model">Nonlinear mixed-effects model</a></li> <li><a href="/wiki/Fixed_effects_model" title="Fixed effects model">Fixed effects model</a></li> <li><a href="/wiki/Generalized_linear_mixed_model" title="Generalized linear mixed model">Generalized linear mixed model</a></li> <li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Mixed-design_analysis_of_variance" title="Mixed-design analysis of variance">Mixed-design analysis of variance</a></li> <li><a href="/wiki/Multilevel_model" title="Multilevel model">Multilevel model</a></li> <li><a href="/wiki/Random_effects_model" title="Random effects model">Random effects model</a></li> <li><a href="/wiki/Repeated_measures_design" title="Repeated measures design">Repeated measures design</a></li> <li><a href="/wiki/Empirical_Bayes_method" title="Empirical Bayes method">Empirical Bayes method</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixed_model&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output 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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 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(2022-01-11), <a rel="nofollow" class="external text" href="https://arxiv.org/abs/2012.10754"><i>Bambi: A simple interface for fitting Bayesian linear models in Python</i></a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.48550%2FarXiv.2012.10754">10.48550/arXiv.2012.10754</a><span class="reference-accessdate">, retrieved <span class="nowrap">2025-01-11</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bambi%3A+A+simple+interface+for+fitting+Bayesian+linear+models+in+Python&rft.date=2022-01-11&rft_id=info%3Adoi%2F10.48550%2FarXiv.2012.10754&rft.aulast=Capretto&rft.aufirst=Tom%C3%A1s&rft.au=Piho%2C+Camen&rft.au=Kumar%2C+Ravin&rft.au=Westfall%2C+Jacob&rft.au=Yarkoni%2C+Tal&rft.au=Martin%2C+Osvaldo+A.&rft_id=https%3A%2F%2Farxiv.org%2Fabs%2F2012.10754&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMixed+model" class="Z3988"></span></span> </li> </ol></div></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=Mixed_model&action=edit&section=11" 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="CITEREFGałeckiBurzykowski2013" class="citation book cs1">Gałecki, Andrzej; Burzykowski, Tomasz (2013). <i>Linear Mixed-Effects Models Using R: A Step-by-Step Approach</i>. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-3900-4" title="Special:BookSources/978-1-4614-3900-4"><bdi>978-1-4614-3900-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Mixed-Effects+Models+Using+R%3A+A+Step-by-Step+Approach&rft.place=New+York&rft.pub=Springer&rft.date=2013&rft.isbn=978-1-4614-3900-4&rft.aulast=Ga%C5%82ecki&rft.aufirst=Andrzej&rft.au=Burzykowski%2C+Tomasz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMixed+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMillikenJohnson1992" class="citation book cs1">Milliken, G. A.; Johnson, D. E. (1992). <i>Analysis of Messy Data: Vol. I. Designed Experiments</i>. New York: Chapman & Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+of+Messy+Data%3A+Vol.+I.+Designed+Experiments&rft.place=New+York&rft.pub=Chapman+%26+Hall&rft.date=1992&rft.aulast=Milliken&rft.aufirst=G.+A.&rft.au=Johnson%2C+D.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMixed+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWestWelchGalecki2007" class="citation book cs1">West, B. T.; Welch, K. B.; Galecki, A. T. (2007). <i>Linear Mixed Models: A Practical Guide Using Statistical Software</i>. New York: Chapman & Hall/CRC.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Mixed+Models%3A+A+Practical+Guide+Using+Statistical+Software&rft.place=New+York&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=2007&rft.aulast=West&rft.aufirst=B.+T.&rft.au=Welch%2C+K.+B.&rft.au=Galecki%2C+A.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMixed+model" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Mixed_model&oldid=1275088582">https://en.wikipedia.org/w/index.php?title=Mixed_model&oldid=1275088582</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Regression_models" title="Category:Regression models">Regression models</a></li><li><a href="/wiki/Category:Analysis_of_variance" title="Category:Analysis of variance">Analysis of variance</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:CS1:_long_volume_value" title="Category:CS1: long volume value">CS1: long volume value</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 11 February 2025, at 00:11<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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