Conditional Inference for Multivariate Generalised Linear Mixed Models

Abstract: We propose a method for inference in generalised linear mixed models (GLMMs) and several extensions of these models. First, we extend the GLMM by allowing the distribution of the random components to be non-Gaussian, that is, assuming an absolutely continuous distribution with respect to the Lebesgue measure that is symmetric around zero, unimodal and with finite moments up to fourth-order. Second, we allow the conditional distribution to follow a dispersion model instead of exponential dispersion models. Finally, we extend these models to a multivariate framework where multiple responses are combined by imposing a multivariate absolute continuous distribution on the random components representing common clusters of observations in all the marginal models. Maximum likelihood inference in these models involves evaluating an integral that often cannot be computed in closed form. We suggest an inference method that predicts values of random components and does not involve the integration of conditional likelihood quantities. The multivariate GLMMs that we studied can be constructed with marginal GLMMs of different statistical nature, and at the same time, represent complex dependence structure providing a rather flexible tool for applications.