Chapter 1: Latent Variable Hybrids: Overview of Old and New Models

The conference that this book builds upon contained many different special topics within the general area of modeling with categorical latent variables, also referred to as mixture modeling. The many different models addressed at that conference and within this volume may overwhelm a newcomer to the field. In fact, however, there are really only a small number of variations on a common theme. This chapter aims to distinguish the different themes, show how they relate to each other, and give some key references for further study. Some new mixture models are also proposed.

Table 1.1 gives a summary of different types of latent variable models. An overview discussion of the models of Table 1.1 was presented in Muthén (2002). The entries of the table are types of models, with the rows dividing the models into cross-sectional and longitudinal and the columns dividing models into traditional models with continuous latent variables, models with categorical latent variables, and newer hybrids using both types of latent variables. The upper left cell includes conventional psychometric models such as factor analysis (FA) and structural equation models (SEMs). The bottom left cell contains the generalization to longitudinal settings, where the continuous latent variables appear in the form of random effects describing individual differences in development over time. The categorical latent variable column includes cross-sectional models such as latent class analysis (LCA), which in longitudinal settings generalizes to latent transition analysis (LTA). LTA is a longitudinal model in the class of auto-regressive models (also including “hidden Markov” models), where the status at one time point influences the status at the next time point. Another LCA-related model is latent class growth analysis (LCGA), where the outcomes are influenced by growth factors analogous to conventional random effects growth modeling. The current chapter gives an overview that emphasizes the last column of hybrid models, with the typical examples of factor mixture analysis (FMA) and growth mixture modeling (GMM). As will be discussed, these models present useful generalizations of the models in the other columns, allowing for both classification of subjects in the form of latent classes and determination of continuous latent scores within these classes. All analyses to be discussed can be carried out using maximum-likelihood estimation in the Mplus program (Muthén & Muthén, 1998–2007).