A MultivAriAte Growth Curve Model for three-level dAtA

One of the most vexing challenges that has faced the behavioral sciences over the past century has been how to optimally measure, summarize, and predict individual variability in stability and change over time. It has long been known that a multitude of advantages are associated with the collection and analysis of repeated measures data; indeed, longitudinal data have become nearly requisite in many disciplines within the behavioral sciences. The challenge of how to best empirically capture individual change cuts across every aspect of the empirical research endeavor, including study design, psychometric measurement, subject sampling, data analysis, and substantive interpretation. Although many textbooks have been devoted to each of these research dimensions, here we have the much more modest goal of exploring just one specific type of longitudinal data analytic method: the multivariate growth model. Given our love of jargon in the social sciences, our field has coined a rather large number of terms to describe patterns of intraindividual change over time. These terms include (but are not limited to) growth, curve, trajectory, and path, among many others. Whether the term is growth models, growth trajectories, growth curves, latent trajectories, developmental curves, latent curves, time paths, or latent developmental growth curve time path trajectories of growth,1 all tend to refer to the same thing. Namely, repeated measures are collected on a sample of individuals followed over time, and models are designed to capture both the mean and variance components associated with patterns of stability and change over time. There are two broad types of growth models: the structural equation model (SEM) and the multilevel linear model (MLM). Whereas the SEM approaches the repeated measures as observed indicators of an underlying latent-growth process (e.g., Bollen & Curran, 2006; McArdle, 1988; Meredith & Tisak, 1990), the MLM approaches these data as the hierarchical structuring of repeated measures nested within the individual (e.g., Bryk & Raudenbush, 1987; Raudenbush, 2001; Singer & Willett, 2003). A great deal of prior research has explored the similarities and dissimilarities of these two approaches, and the lines that demarcate the SEM and MLM are becoming increasingly blurred with the passing of each year (e.g., Bauer, 2003; Curran, 2003; Mehta & Neale, 2005; Newsom, 2002; Willett & Sayer, 1994). Suffice it to say that both methods are powerful and flexible approaches to the analysis of longitudinal data, the optimal choice of which depends strictly on the characteristics of the substantive question and the experimental design at hand (Raudenbush, 2001). That said, here we focus exclusively on the growth model as estimated within the framework of the MLM, which stems directly from the substantive question on which we are currently working. As we