A latent class model for bivariate binary responses from twins

We propose a latent class model for bivariate binary responses from monozygotic and dizygotic twin pairs. We study the comorbidity of two diseases, where each disease and their association are presumed to be genetically related via latent predispositions. Each twin belongs to one of four latent classes, conceptualized as a 2X2 table and parameterized by the probabilities for the two genetic predispositions and the dependence ratio between them. Both members of a monozygotic pair belong to the same latent class, but a dizygotic pair falls in one of the 4X4 cells from the cross-classification of the four individual classes. The association structure in the 4X4 table is also parameterized by dependence ratios. All parameters of the model are probabilities or ratios of probabilities having clear-cut interpretations, e.g. prevalence and penetrance. We propose a measure of heritability of the complex consisting of the four latent classes which contrasts the concordance of monozygotic and dizygotic twins. We reanalyse two data sets and compare our model with previously used models which are based on other measures of association.