Multivariate Generalised Linear Mixed Models With Graphical Latent Covariance Structure

This paper introduces a method for studying the correlation structure of a range of responses modelled by a multivariate generalised linear mixed model (MGLMM). The methodology requires the existence of clusters of observations and that each of the several responses studied is modelled using a generalised linear mixed models (GLMM) containing random components representing the clusters. We construct a MGLMM by assuming that the distribution of each of the random components representing the clusters is the marginal distribution of a (sufficiently regular) multivariate elliptically contoured distribution. We use an undirected graphical model to represent the correlation structure of the random components representing the clusters of observations for each response. This representation allows us to draw conclusions regarding unknown underlying determining factors related to the clusters of observations. Using a combination of an undirected graph and a directed acyclic graph (DAG), we jointly represent the correlation structure of the responses and the related random components. Applying the theory of graphical models allows us to describe and draw conclusions on the correlation and, in some cases, the dependence between responses of different statistical nature (\eg following different distributions, different linear predictors and link functions). We present some simulation studies illustrating the proposed methodology.