ABSTRACT
Latent variable models are widely used in Social Sciences for measuring constructs (latent variables) such as ability , belief, attitude , behavior, welfare, satisfaction, and knowledge. Those unobserved constructs are measured through a number of observed indicators (items). Those indicators may be binary categorical (yes/no), ordinal categorical (strongly disagree, disagree, agree, strongly agree), nominal (political party, religion), or metric (weight, height, income, expenditures). Latent variable models have two main objectives. The first objective is t o reduce the dimensionality of multivariate observed data by trying to identify a small number of latent dimensions that can explain the relationships among the observed items. The second objective is to score population members on those identified latent dimensions based on what they have responded to the observed items. Those latent scores can sometimes be used in regression models as dependent or independent variables. However, latent variable models have also been recently extended to allow effects of explanatory variables directly on the latent variables and/or on the indicator variables (Moustaki, 2003; Sammel, Ryan, & Legier, 1997). One can distinguish two m a in groups of estimation methods for latent variable models. The first group of methods is based on Monte Carlo Markov Chain Bayes estimation (MCMC) . The second group includes methods based either on Newton-Raphson iteration or on the E - M algorithm. MCMC methods have recently become popular in the area of latent variable modelling mainly because they allow estimation of complex models (see e.g., Albert , 1992; Albert & Chib , 1993; Patz & Junker, 1999a, 1999b; Dunson, 2000). In the psychometric, educational and medical literature models have been developed for binary, nominal, ordinal , and metric manifest indicators see, for example, Bartholomew and Knott (1999), Bock and Aitkin (1981), Bock, Gibbons, and Muraki (1988), Muraki and Carlson (1995), and Samejima (1969). Moustaki (1996) and Sammel et al. (1997) also looked at models with mixed metric and binary observed indicators and recently Moustaki (2000a) and Moustaki and Knott (2000a) presented a general exponential family framework for fitting latent variable models t o any type of observed items or to data including several different types of items. The models discussed in those papers are estimated by marginal maximum likelihood using an E - M algorithm.
