Studying change in student achievement is of central importance in numerous areas of educational research, including efforts to monitor school performance, investigations of the effects of educational interventions over time, and school effects studies focusing on how differences in school policies and practices relate to differences in student progress. In this article, we argue that in studying patterns of change, it is often important to consider the relationship between where students start (i.e., their initial status) and how rapidly they progress (i.e., their rates of change). Drawing on recent advances in growth modeling methodology, we illustrate the potential value of such an approach in the context of monitoring school performance. In particular, we highlight the ways in which attending to initial status in analyses of student progress can help draw attention to possible concerns regarding the distribution of achievement within schools. To convey the logic of our approach and illustrate various analysis possibilities, we fit a series of growth models to the time series data for students in several schools in the Longitudinal Study of American Youth (LSAY) sample. In a final section, we discuss some of the possibilities that arise in employing a modeling approach of this kind in evaluating educational programs and in conducting school effects research.

This review introduces classical item response theory (IRT) models as well as more contemporary extensions to the case of multilevel, multidimensional, and mixtures of discrete and continuous latent variables through the lens of discrete multivariate analysis. A general modeling framework is discussed, and the applications of this framework in diverse contexts are presented, including large-scale educational surveys, randomized efficacy studies, and diagnostic measurement. Other topics covered include parameter estimation and model fit evaluation. Both classical (numerical integration based) and more modern (stochastic) parameter estimation approaches are discussed. Similarly, limited information goodness-of-fit testing and posterior predictive model checking are reviewed and contrasted. The review concludes with a discussion of some emerging strands in IRT research such as response time modeling, crossed random effects models, and non-standard models for response processes.

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.

The No Child Left Behind Act (NCLB, 2002) establishes ambitious goals for increasing student learning and attaining equity in the distribution of student performance. Schools must assure that all students, including all significant subgroups, show adequate yearly progress (AYP) toward the goal of 100% proficiency by the year 2014. In this paper, we illustrate an alternative way of evaluating AYP that both emphasizes individual student growth over time and focuses on the distribution of student growth between performance subgroups. We do so through analyses of a longitudinal data set from an urban school district in the state of Washington. We also examine what these patterns tell us about schools that have been designated as meeting their AYP targets and those that have not. This alternative way of measuring AYP helps bring to light potentially important aspects of school performance that might be masked if we limit our focus to classifying schools based only on current AYP criteria. In particular, we are able to identify some schools meeting Washington state's AYP criteria in which above‐average students are making substantial progress but below‐average students making little to no progress. In contrast, other schools making AYP have below‐average students making adequate progress but above‐average students showing little gains. These contrasts raise questions about the meaning of “adequate” progress and to whom the notion of progress refers. We believe that closely examining the distribution of student progress may provide an important supplementary or alternative measure of AYP.

Much work on sensitivity analysis for hierarchical models (HMs) has focused on level-2 outliers (e.g., in multisite evaluations, a site at which an intervention was unusually successful). However, efforts to draw sound conclusions concerning parameters of interest in HMs also require that we attend to extreme level-1 units (e.g., a person in the treatment group at a particular site whose post-test score [y ij ] is unusually small vis-á-vis the other members of that person's group). One goal of this article is to examine the ways in which level-1 outliers can impact the estimation of fixed effects and random effects in HMs. A second goal is to outline and illustrate the use of Markov Chain Monte Carlo algorithms for conducting sensitivity analyses under t level-1 assumptions, including algorithms for settings in which the degrees of freedom at level 1 (v 1 ) is treated as an unknown parameter.