A joint normal-binary (probit) model for high-dimensional longitudinal data

In many biomedical studies multiple responses are collected over time, which results in highdimensional longitudinal data. It is often of interest to model the continuous and binary responses jointly, which can be done with joint generalized mixed models in which the association is modelled through random effects. Investigating the association between the responses is often limited to scrutinizing the correlations between the latent random effects. In this article, this approach is extended by deriving closed-form formulas for the manifest correlations (and corresponding standard errors), which reflects the correlation between the observed responses as observed. In addition, the marginal joint model is constructed, from which predictions of subvectors of one response conditional on subvectors of other response(s) and potentially a subvector of the history of the response can be derived. Corresponding prediction and confidence intervals are constructed. Two case studies are discussed, in which further pseudo-likelihood methodology is applied to reduce the computational complexity.