Modeling Heterogeneous Variance-Covariance Components in Two-Level Models

Applications of multilevel models to continuous outcomes nearly always assume constant residual variance and constant random effects variances and covariances. However, modeling heterogeneity of variance can prove a useful indicator of model misspecification, and in some educational and behavioral studies, it may even be of direct substantive interest. The purpose of this article is to review, describe, and illustrate a set of recent extensions to two-level models that allow the residual and random effects variance-covariance components to be specified as functions of predictors. These predictors can then be entered with random coefficients to allow the Level-1 heteroscedastic relationships to vary across Level-2 units. We demonstrate by simulation that ignoring Level-2 variability in residual variances leads the Level-1 variance function regression coefficients to be estimated with spurious precision. We discuss software options for fitting these extensions, and we illustrate them by reanalyzing the classic High School and Beyond data and two-level school effects models presented by Raudenbush and Bryk.