Modeling Heterogeneity in Relationships Between Initial Status and Rates of Change: Latent Variable Regression 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 time period of substantive interest relate to differences in subsequent change. In this report we present a fully Bayesian approach to estimating three-level hierarchical models in which latent variable regression coefficients capturing the relationship between initial status and rates of change within each of J schools (Bwj, j = 1, …, J) are treated as varying across schools. Through analyses of data from the Longitudinal Study of American Youth, we show how modeling differences in Bwj as a function of school characteristics can broaden the kinds of questions we can address in school effects research. We illustrate the possibility of conducting sensitivity analyses employing t distributional assumptions at each level of such models (termed LVR-HM3s) and present results from a simulation study that focuses on the coverage properties of marginal posterior intervals for fixed effects in LVR-HM3s. We 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 pre-intervention period, and cross-sectional studies in which within-cluster latent variable regression slopes are treated as varying across clusters.