A joint normal-binary (probit) model

In biomedical research, often hierarchical binary and continuous responses need to be jointly modelled. In joint generalised linear mixed models, this can be done with correlated random effects, which allows examining the association structure between the various responses and the evolution of this association over time. In addition, the effect of covariates on all outcomes can be assessed simultaneously. Still, investigating this association is often limited to examining the correlations between the responses on an underlying scale. In addition, the interpretation of this hierarchical model is conditional on the subject-specific random effects. This paper extends this approach and shows how manifest correlations can be computed, that is, the associations between the observed responses. Further, a marginal model is formulated, in which the interpretation is no longer conditional on the random effects. In addition, prediction intervals are derived of one subvector of responses conditional on the other. These methods are applied in a case study of the lung function and allergic bronchopulmonary aspergillosis in patients with cystic fibrosis.