Latent Curve Models: A Structural Equation Perspective (review)

The collection and widespread dissemination of panel data is one of the most important social science innovations of the past 40 years. During this period, social statisticians have struggled to develop models and methods that make the most of these data. Over the past 10 years or so, these efforts have coalesced around a group of methods that are becoming very widely used across sociology, psychology, epidemiology, political science and economics. "Latent curve models" is one name for these methods. Other names include growth models, multilevel models, hierarchical linear models, random coefficient models and linear or generalized mixed models for longitudinal data. These are all quite similar. They involve a dependent variable that typically is measured on a continuous scale, at multiple time points for each individual in the study. "Individuals" are often people in a panel data study, but could be cities, states or other units for which repeated measures are collected on some dependent variable. Independent variables measured for these individuals may be either constant over time, typically measured at the beginning of the study, or time-varying, taking on different values at each data collection time point.

These models are most easily understood as they are presented in hierarchical linear modeling. Each individual has a level-1 equation, in which the dependent variable is regressed against time (as well as any time-varying predictors that may be present). This produces a "growth curve" or trajectory of the dependent variable, defined by the y-intercept and slope coefficients for each individual. In the level-2 equations, these coefficients (the y-intercept and slope) are then allowed to depend upon unchanging characteristics of the individual. So it's a bit like "regressing regression coefficients," in which the y-intercept (beginning value) and slope (growth rate) of the dependent variable for each individual are allowed to vary as a function of the unchanging characteristics (for example, gender, SES, race) of the individual.

One hindrance to understanding has been the very different ways that social statisticians have presented these models and the software for estimating them. Over-simplifying a bit, there are three schools. One, the hierarchical, or multilevel modeling approach, exemplified by Raudenbush and Bryk (2002), and originally developed by sociologists and education researchers, presents the models in the level-1, level-2 format discussed above, and employs HLM software for estimation. A second approach, developed by statisticians from biosciences, epidemiology, medicine, and related fields, has been named mixed models for longitudinal data, typically presented in textbooks such as Verbeke and Molenberghs (2000), and estimated via SAS routines (described, for example, in Littell et al (2006)). The third approach, exemplified by this volume, presents these models within the structural equation modeling (SEM) tradition, and adapts standard SEM software, such as LISREL or Amos, or in advanced applications, Muthen and Muthen's (2000) MPlus software, for estimation.

Because Bollen and Curran are significant contributors to the development of SEM methodology, it is appropriate that they should present what is likely to be the standard text for latent curve models from a SEM perspective. The volume has eight chapters, and would best be approached after courses in both regression analysis and SEM. In a sociology department, it could be taught to second-year graduate students, following a SEM course.

The introductory chapter motivates the volume by discussing data on crime rates over four time points for each of 359 communities in the state of New York. It also provides a limited history of the development of these models and methods. The second chapter focuses on the first level, or trajectory equation, for each unit of observation. The approach is nicely motivated by focusing on the 359 separate trajectories that might be estimated by OLS regression for each of the communities. In this "case by case" approach, the authors show histograms of the 359 Y-intercepts and slopes resulting from separate regressions for each community. They also show how a SEM path analysis diagram for...