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

Abstract

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 threelevel 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 ¼ 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.

Estimating a Three-Level Latent Variable Regression Model With Cross-Classified Multiple Membership Data

“…Using fully Bayesian estimation can provide more precise and robust estimates for parameters in a more complex model, such as with LVR models, as long as prior distributions are appropriately chosen. Choi and Seltzer (2010) found that uniform priors, which are analogous to inverse-Pareto distributions for the level-2 LVR variance components, resulted in less bias than using default inverse-gamma distributions. These results assist future researchers when selecting the best prior distributions to use for a threelevel LVR analysis.…”

“…Therefore, more hypotheses about growth trajectories can be tested by using the more flexible LVR framework. Choi and Seltzer (2010) presented a fully Bayesian approach to estimate a threelevel LVR model with purely clustered data, and conducted a small simulation study to evaluate the differences in the choice of prior distributions for the level-2 random effects' variance components. Using fully Bayesian estimation can provide more precise and robust estimates for parameters in a more complex model, such as with LVR models, as long as prior distributions are appropriately chosen.…”

“…This hypothesis cannot be directly tested using a conventional GCM although this interaction effect can be tested using an LVR model that incorporates gender as a predictor of the coefficient representing the influence of initial status on growth rate (Seltzer et al, 2003). Alternatively, there might be an interaction effect in that differences in growth rates between educational intervention and control groups may vary as a function of individuals' initial status values (Choi & Seltzer, 2010;Muthén & Curran, 1997). In addition, the LVR coefficient representing the prediction of growth rate by initial status could be modeled as varying across organizational units, as well as a function of organizational characteristics in the model (Choi & Seltzer, 2010).…”

“…Alternatively, there might be an interaction effect in that differences in growth rates between educational intervention and control groups may vary as a function of individuals' initial status values (Choi & Seltzer, 2010;Muthén & Curran, 1997). In addition, the LVR coefficient representing the prediction of growth rate by initial status could be modeled as varying across organizational units, as well as a function of organizational characteristics in the model (Choi & Seltzer, 2010).…”

Technical Efficiency in Education from Value-Added Measures: An Application to Spanish Primary Schools

“…2 The effect of the differential performance of each student in the first assessment (November, 2005) compared to the mean for the population on the schools' growth rate (linear and quadratic) has been controlled. The reason for entering this predictor in the model is that several studies have related the differences in the students' initial performances to the different rates of change (Choi & Seltzer, 2005, 2010Klein & Muthén, 2006;Castro, Ruiz, & López, 2009). When calculating technical efficiency, the estimates of value added in mathematics and in reading comprehension calculated in the previous step are included as outputs of the education system.…”