Mixed Effects Models

Abstract

Mixed effects models, or simply mixed models, are widely used in practice. These models are characterized by the involvement of the so-called random effects. To understand the basic elements of a mixed model, let us first recall a linear regression model, which can be expressed as y = Xβ + ε, where y is a vector of observations, X is a matrix of known covariates, β is a vector of unknown regression coefficients, and ε is a vector of (unobservable random) errors. In this model, the regression coefficients are considered fixed. However, there are cases in which it makes sense to assume that some of these coefficients are random. These cases typically occur when the observations are correlated. For example, in medical studies, observations are often collected from the same individuals over time.