# ⓘ 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 ..

## ⓘ 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, non-linear 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.

• grammatical items. Yaron Matras distinguishes between three types of models for mixed language: language maintenance and language shift, unique and predetermined
• multilevel models and as mixed model In general, fitting GLMMs is more computationally complex and intensive than fitting GEEs. Generalized additive models GAMs
• equations and the well - mixed weakly stratified layer known as the ocean mixed layer near the ocean surface. However, level coordinate models have problems when
• Mixed government or a mixed constitution is a form of government that combines elements of democracy, aristocracy and monarchy, ostensibly making impossible
• Robbie s girlfriend is played by Top London Model Kirsty Rose Heslewood. An accompanying music video for Mixed Signals filmed in Shoreditch, East London
• section. A division was composed of two or three mixed brigades. The mixed brigade was based on a model that would replace the columns columnas and militias
• A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data and similar data
• Box models are simplified versions of complex systems, reducing them to boxes or reservoirs linked by fluxes. The boxes are assumed to be mixed homogeneously
• ownership with some caveats within a mixed economy. Each of the Nordic countries has its own economic and social models sometimes with significant differences
• final process, which is a form of Rayleigh Taylor instability. Early models of the mixed layer such as those of Mellor and Durbin included the final two processes
• removing the loincloth from a male model in a mixed classroom. Similarly, Victorian modesty sometimes required the female model to pose nude with her face draped

#### Users also searched:

fixed effects model, generalized linear mixed models, panel regression with fixed effects, random effects regression model, re regression,

...
 ...
...