Fixed versus Random Effects

The terms fixed and random are commonly used in the regression modeling literature and pertain [Page 509]to whether particular coefficients in a model are treated as fixed or random values. A statistical model is classified as a fixed effects model if all independent variables are regarded as fixed, a random effects model if all independent variables are regarded as random, and a mixed effects model if the independent variables constitute a mix of fixed and random effects. Analytic methods vary depending on the model. The approach selected depends on the nature of the available data and the study objectives.

A fixed variable is one that is assumed to be measured without error. The values of the fixed variable from one study are assumed to be the same as the values in any attempted replication of the study; that is, they are the only levels of a factor that are of interest (hence the term fixed). Gender and marital status are examples of fixed variables because they have a small fixed number of categories (levels). There is no larger population of gender categories that the levels male and female are sampled from. Fixed effects regression and analysis of variance (ANOVA) refer to assumptions about the independent variable and the error distribution. The independent variables are assumed to be fixed, and the generalization of results applies to similar values of the independent variable in the population or in other studies.

A random variable is one whose levels are assumed to be a random sample from a larger population of levels for that variable. Subjects, hospitals, physicians, schools, and litters are examples of random factors since investigators usually want to make inferences beyond the particular values of the independent variable that were captured to a larger population. Designation of variables as fixed or random is not always straightforward. Some basic questions an investigator should ask are the following: (a) Is it reasonable to assume that the levels of an independent variable were randomly sampled from some population? (b) Is the goal to make inferences to a population from which the levels of the variable were selected or from the particular levels on hand? Treatments or drug doses from a clinical trial are usually considered fixed variables since they represent all levels of interest for a study; however, they can be considered as random if their levels are a subset of the possible values one wants to generalize to.

Random effects models are referred to as variance component models, hierarchical linear models, multilevel regression models, nested models, generalized linear mixed models, and random coefficient or mixed models (using both fixed and random effects). These models can be considered as extensions of linear models and have gained popularity with advances in computing and software availability.