Mixed-Effects Model

ANALYSIS OF VARIANCE (ANOVA) models can be used to study how a dependent variable Y depends on one or several factors. A factor here is defined as a categorical independent variable—in other words, an explanatory variable with a nominal LEVEL OF MEASUREMENT. Factors can have fixed or random effects. For a factor with fixed effects, the parameters expressing the effect of the categories of each factor are fixed (i.e., nonrandom) numbers. For a factor with random effects, the effects of its categories are modeled as random variables. Fixed effects are appropriate when the statistical inference aims at finding conclusions that hold for precisely the categories of the factor that are present in the data set at hand. Random effects are appropriate when the categories of the factor are regarded as a random sample from some population, and the statistical inference aims at finding conclusions that hold for this population. If the model contains only fixed or only random effects, we speak of FIXED-EFFECTS MODELS or RANDOM-EFFECTS MODELS, respectively. A model containing fixed as well as random effects is called a mixed-effects model.

An example of a mixed-effects ANOVA model could be a study about work satisfaction of employees in several job categories in several organizations. If the researcher is interested in a limited number of specific job categories and in a population of organizations, as well as, inter alia, the amount of variability between the organizations in this population, then it is natural to use a mixed model, with fixed effects for the job categories and random effects for the organizations.

Random effects, as defined above and in the entry on random-effects models, can be regarded as random main effects of the levels of a given factor. Random interaction effects can be defined similarly. In the example from the previous paragraph, a random job category-by-organization interaction effect also could be included.

Mixed-effects models are also used for the analysis of data collected according to SPLIT-PLOT DESIGNS or REPEATED-MEASURES DESIGNS. In such designs, individual subjects (or other units) are investigated under several conditions. Suppose that the conditions form one factor, called B; the individual subjects are also regarded as a factor, called S. The mixed-effects model for this design has a fixed effect for B, a random effect for S, and a random B × S interaction effect.

The Linear Mixed Model

A generalization is a random coefficient of a numerical explanatory variable. This leads to the generalization of the GENERAL LINEAR MODEL, which incorporates fixed as well as random coefficients, commonly called the linear mixed model or random coefficient regression model. In matrix form, this can be written as

where Y is the dependent variable, X is the design matrix for the fixed effects, β is the vector of fixed regression coefficients, Z is the design matrix for the random coefficients, U is the vector of random coefficients, and E is the vector of random residuals. The standard assumption is that the vectors U and E are independent and have multivariate normal distributions with zero mean vectors and with covariance matrices G and R, respectively. This assumption implies that Y has the multivariate normal distribution with mean Xβ and covariance matrix ZGZ′ + R. Depending on the study design, a further structure will usually be imposed on the matrices G and R. For example, if there are K factors with random effects and the residuals are purely random, then the model can be specialized to

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