A multivariate generalized linear mixed model for joint modelling of clustered outcomes in the exponential family

Clustered and repeated measures data are very common in biomedical applications, for example when one or more variables are measured on each patient at a number of hospital visits, or when a number of questions are asked at a series of interviews. The Generalized Linear Mixed Model (GLMM) can be used for fully parametric subject-specific inference for clustered or repeated measures responses in the exponential family. In this paper a multivariate generalization is proposed to deal with situations when multiple outcome variables in the exponential family are present. Separate GLMM's are assumed for each response variable and then the responses are combined in a single model by imposing a joint multivariate normal distribution for the variable-specific random effects. This allows maximum-likelihood estimation approaches such as Gauss-Hermitian quadrature and Monte Carlo EM algorithm to be extended from the univariate to the multivariate case. Two data sets are used for illustration. The outcome variables are assumed to be conditionally independent given the random effects which is a restrictive assumption in some cases. Score tests for checking this assumption are proposed and alternative models are considered for one of the data examples.