ⓘ Mixed model
A mixed model is a statistical model containing both fixed effects and random effects. These models are useful in a wide variety of disciplines in the physical, biological and social sciences. They are particularly useful in settings where repeated measurements are made on the same statistical units, or where measurements are made on clusters of related statistical units. Because of their advantage in dealing with missing values, mixed effects models are often preferred over more traditional approaches such as repeated measures ANOVA.
1. History and current status
Ronald Fisher introduced random effects models to study the correlations of trait values between relatives. In the 1950s, Charles Roy Henderson provided best linear unbiased estimates BLUE of fixed effects and best linear unbiased predictions BLUP of random effects. Subsequently, mixed modeling has become a major area of statistical research, including work on computation of maximum likelihood estimates, nonlinear mixed effects models, missing data in mixed effects models, and Bayesian 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 and industrial statistics.
2. Definition
In matrix notation a mixed model can be represented as
y = X β + Z u + ϵ {\displaystyle {\boldsymbol {y}}=X{\boldsymbol {\beta }}+Z{\boldsymbol {u}}+{\boldsymbol {\epsilon }}}where
 ϵ {\displaystyle {\boldsymbol {\epsilon }}} is an unknown vector of random errors, with mean E ϵ = 0 {\displaystyle E{\boldsymbol {\epsilon }}={\boldsymbol {0}}} and variance var ϵ = R {\displaystyle \operatorname {var} {\boldsymbol {\epsilon }}=R} ;
 y {\displaystyle {\boldsymbol {y}}} is a known vector of observations, with mean E y = X β {\displaystyle E{\boldsymbol {y}}=X{\boldsymbol {\beta }}} ;
 X {\displaystyle X} and Z {\displaystyle Z} are known design matrices relating the observations y {\displaystyle {\boldsymbol {y}}} to β {\displaystyle {\boldsymbol {\beta }}} and u {\displaystyle {\boldsymbol {u}}}, respectively.
 β {\displaystyle {\boldsymbol {\beta }}} is an unknown vector of fixed effects;
 u {\displaystyle {\boldsymbol {u}}} is an unknown vector of random effects, with mean E u = 0 {\displaystyle E{\boldsymbol {u}}={\boldsymbol {0}}} and variance–covariance matrix var u = G {\displaystyle \operatorname {var} {\boldsymbol {u}}=G} ;
3. Estimation
The joint density of y {\displaystyle {\boldsymbol {y}}} and u {\displaystyle {\boldsymbol {u}}} can be written as: f y, u = f y  u f u {\displaystyle f{\boldsymbol {y}},{\boldsymbol {u}}=f{\boldsymbol {y}}{\boldsymbol {u}}\,f{\boldsymbol {u}}}. Assuming normality, u ∼ N 0, G {\displaystyle {\boldsymbol {u}}\sim {\mathcal {N}}{\boldsymbol {0}},G}, ϵ ∼ N 0, R {\displaystyle {\boldsymbol {\epsilon }}\sim {\mathcal {N}}{\boldsymbol {0}},R} and C o v u, ϵ = 0 {\displaystyle \mathrm {Cov} {\boldsymbol {u}},{\boldsymbol {\epsilon }}={\boldsymbol {0}}}, and maximizing the joint density over β {\displaystyle {\boldsymbol {\beta }}} and u {\displaystyle {\boldsymbol {u}}}, gives Hendersons "mixed model equations" MME:
X ′ R − 1 X ′ R − 1 Z ′ R − 1 X Z ′ R − 1 Z + G − 1 β ^ u ^ = X ′ R − 1 y Z ′ R − 1 y {\displaystyle {\begin{pmatrix}XR^{1}X&XR^{1}Z\\ZR^{1}X&ZR^{1}Z+G^{1}\end{pmatrix}}{\begin{pmatrix}{\hat {\boldsymbol {\beta }}}\\{\hat {\boldsymbol {u}}}\end{pmatrix}}={\begin{pmatrix}XR^{1}{\boldsymbol {y}}\\ZR^{1}{\boldsymbol {y}}\end{pmatrix}}}The solutions to the MME, β ^ {\displaystyle \textstyle {\hat {\boldsymbol {\beta }}}} and u ^ {\displaystyle \textstyle {\hat {\boldsymbol {u}}}} are best linear unbiased estimates BLUE and predictors BLUP for β {\displaystyle {\boldsymbol {\beta }}} and u {\displaystyle {\boldsymbol {u}}}, respectively. This is a consequence of the Gauss–Markov theorem when the conditional variance 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 BLUE. 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.
One method used to fit such mixed models is that of the EM algorithm where the variance components are treated as unobserved nuisance parameters in the joint likelihood. Currently, this is the implemented method for the major statistical software packages R lme in the nlme package, or lmer in the lme4 package, Python statsmodels package, Julia MixedModels.jl package, and SAS proc mixed. The solution to the mixed model equations is a maximum likelihood estimate when the distribution of the errors is normal.
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