Modeling Heterogeneity in Relationships Between Initial Status and Rates of Change: Treating Latent Variable Regression Coefficients as Random Coefficients in a Three-Level Hierarchical Model

In studies of change in education and numerous other fields, interest often centers on how differences in the status of individuals at the start of a period of substantive interest relate to differences in subsequent change. In this article, the authors present a fully Bayesian approach to estimating three-level Hierarchical Models in which latent variable regression (LVR) coefficients capturing the relationship between initial status and rates of change within each of J schools (Bw _j , j = 1, …, J) are treated as varying across schools. Specifically, the authors treat within-group LVR coefficients as random coefficients in three-level models. Through analyses of data from the Longitudinal Study of American Youth, the authors show how modeling differences in Bw_j as a function of school characteristics can broaden the kinds of questions they can address in school effects research. They also illustrate the possibility of conducting sensitivity analyses using t distributional assumptions at each level of such models (termed latent variable regression in a three-level hierarchical model [LVR-HM3s]), and present results from a small-scale simulation study that help provide some guidance concerning the specification of priors for variance components in LVR-HM3s. They outline extensions of LVR-HM3s to settings in which growth is nonlinear, and discuss the use of LVR-HM3s in other types of research including multisite evaluation studies in which time-series data are collected during a preintervention period, and cross-sectional studies in which within-cluster LVR slopes are treated as varying across clusters.