Marginal modeling with latent variables

Abstract

So far in this book, all analyses have been conducted at the manifest, observed data level with the implicit assumption that the observed data resulted from perfectly valid and reliable measurements of the intended concepts. However, observed data are often only indirect measures of what researchers want to measure and are often contaminated by measurement errors. Latent variable models provide very powerful tools for discovering many kinds of systematic and unsystematic measurement errors in the observed data, and offer many ways to correct for them. For categorical data— where the measurement errors are usually denoted as misclassifications — the most appropriate and flexible latent variable model is the latent class model (Lazarsfeld & Henry, 1968; Goodman, 1974a; Goodman, 1974b; Haberman, 1979; Clogg, 1981b; Hagenaars & McCutcheon, 2002). Combining latent class models with marginal restrictions makes it, in principle, possible to conduct the marginal analyses at the latent level. Important pioneering work regarding maximum likelihood estimation for marginal latent class models has been conducted by Becker and Yang (1998). Because not all readers may be familiar with latent class models, the basic latent class model will be introduced in the next section, mainly in the form of a loglinear model with latent variables. Thereafter, in Section 6.2, a simple marginal homogeneity model for the joint distribution of two latent variables will be presented. Further, it will be shown that application of the marginal-modeling approach advocated in this book makes it possible to handle latent class models that lie outside the loglinear framework (Section 6.3). Next, two examples of marginal categorical SEMs with latent variables will be presented in Section 6.4. The explanation of the appropriate ML estimation procedures for latent marginal modeling is presented in the final Section 6.5.