A latent trait and a latent class model for mixed observed variables

Latent variable models are widely used in social sciences in which interest is centred on entities such as attitudes, beliefs or abilities for which there exist no direct measuring instruments. Latent modelling tries to extract these entities, here described as latent (unobserved) variables, from measurements on related manifest (observed) variables. Methodology already exists for fitting a latent variable model to manifest data that is either categorical (latent trait and latent class analysis) or continuous (factor analysis and latent profile analysis). In this paper a latent trait and a latent class model are presented for analysing the relationships among a set of mixed manifest variables using one or more latent variables. The set of manifest variables contains metric (continuous or discrete) and binary items. For the latent trait model the latent variables are assumed to follow a multivariate standard normal distribution. Our method gives maximum likelihood estimates of the model parameters and standard errors of the estimates by analysing the data as they are without using any underlying variables. The mixed latent trait and latent class models are fitted using an EM algorithm. To illustrate the use of the mixed model three data sets have been analysed. Two of the data sets contain five memory questions, the first on Thatcher's resignation and the second on the Hillsborough football disaster; these five questions were included in British Market Research Bureau International August 1993 face-to-face omnibus survey. The third data set is from the 1991 British Social Attitudes Survey; the questions which have been analysed are from the environment section.