Latent variables, measurement error and methods for analysing longitudinal binary and ordinal data

We explore structural equations with latent variables for modelling between-individual variability and measurement error in the analysis of longitudinal binary and ordinal data. The structural equation formulation provides insight into the assumptions and differences in interpretation of methods that are popular for longitudinal data analysis. Introducing the concept of continuous latent variables makes it clear that marginal and cluster-specific models differ because their predicted variables are scaled to different standard deviations, and that adjustment for measurement error in the outcome involves a change in scale as well. We apply both structural equation modelling and common longitudinal modelling approaches to data from a study of sleep disorders. In the process, we compare results from marginal modelling using an SAS GEE routine (Karim and Zeger, 1988), Qu's GAUSS program (Qu, 1992) for generalized mixed models using GEE, the MIXOR package for cluster-specific mixed effects models (Hedeker and Gibbons, 1994), and LISCOMP for structural models (Muthén, 1988).