Latent-variable models for longitudinal data with bivariate ordinal outcomes.

We use the concept of latent variables to derive the joint distribution of bivariate ordinal outcomes, and then extend the model to allow for longitudinal data. Specifically, we relate the observed ordinal outcomes using threshold values to a bivariate latent variable, which is then modelled as a linear mixed model. Random effects terms are used to tie all together repeated observations from the same subject. The cross-sectional association between the two outcomes is modelled through the correlation coefficient of the bivariate latent variable, conditional on random effects. Assuming conditional independence given random effects, the marginal likelihood, under the missing data at random assumption, is approximated using an adaptive Gaussian quadrature for numerical integration. The model provides fixed effects parameters that are subject-specific, but retain the population-averaged interpretation when properly scaled. This is particularly well suited for the situation in which population comparisons and individual level contrasts are of equal importance. Data from a psychiatric trial, the Fluvoxamine (an antidepressant drug) study, are used to illustrate the methodology.