Random-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. In a random-effects ANOVA model (also called a Type II ANOVA model), the effects of the categories of each factor are modeled as random variables; one could say that this is a model with random parameters. More specifically, when a given factor has [Page 916]values 1, 2,…, K in the data set, and for case i, the value of this factor is indicated by k(i), then a random effect of this factor for case i is defined by the term Uk(i) in the linear model for Yi, where it is assumed that U1,…, UK are independent and identically distributed RANDOM VARIABLES. Usually, a normal distribution is assumed for these random variables. Such a model is appropriate if the categories of this factor are regarded as a random sample from some population and if the statistical inference aims at drawing conclusions that hold for this population. This contrasts with FIXED EFFECTS MODELS, which are described in another entry. Random-effects ANOVA models also are called VARIANCE COMPONENTS MODELS.

Two examples of studies where random-effects models can be appropriate are (a) educational studies about educational achievement of pupils where teachers are an important factor, and the researcher is primarily interested in drawing conclusions that apply to the population of teachers rather than specifically to the teachers included in the present study; and (b) studies in the framework of GENERALIZABILITY THEORY and INTERRATER RELIABILITY, where ratings on individuals are obtained from each of several observers, and one of the interests is in the contribution of the observers (drawn randomly from some population of observers) to the variability of the measurement obtained by averaging the ratings of one individual over the observers.

Random effects occur similarly in extensions to other, more complicated models, such as GENERALIZED LINEAR MODELS or nonlinear regression models. Their definition, then, is the same: The effects of the categories of each factor are modeled as random variables.

The term random-effect models is reserved for models in which each factor has random effects. Models containing some random as well as some fixed effects are called MIXED-EFFECT MODELS, and these are used more frequently than pure random effect models. The random effects as defined above can be regarded as random main effects of the levels of a given factor. A generalization is a random coefficient of a numerical explanatory variable (see RANDOM-COEFFICIENT MODELS). For hierarchically nested factors, models including random coefficients as well as fixed effects are also called HIERARCHICAL LINEAR MODELS. These models are the basis of most procedures of MULTILEVEL ANALYSIS.