Addressing Questions Concerning Equity in Longitudinal Studies of School Effectiveness and Accountability: Modeling Heterogeneity in Relationships Between Initial Status and Rates of Change

Attending to the relationship between where individuals start (e.g., their initial status and how rapidly they progress (e.g., their rates of change) can help draw attention to possible concerns regarding the distribution of student achievement within schools in longitudinal studies of school effectiveness. Focusing on the relationship between initial status and rates of change, we address questions concerning equity: Why is it that student achievement is distributed in a more equitable fashion in some schools than in other schools? To what extent might this be due to school characteristics, school policies, or school practices? Do schools that start off with high mean achievement have a more positive relationship or negative relationship?

To address questions of this kind, we present a latent variable regression modeling strategy that incorporates latent variable regression into a three-level hierarchical modeling framework (LVR-HM3). To illustrate key ideas and the distinctive features of the LVR-HM3, we fit a series of LVR-HM3s to the data from the Longitudinal Study of American Youth (LSAY) using the Gibbs sampler. We also present results from sensitivity analyses that involve employing tdistributional assumptions in LVR-HM3s, and we examine the convergence of the Gibbs sampler using different formulations and parameterization of the LVR-HM3. In a final section, we discuss some implications and possibilities that arise in longitudinal multisite intervention studies using LVR-HM3s.