Missing responses in generalised linear mixed models when the missing data mechanism is nonignorable

We propose a method for estimating parameters in the generalised linear mixed model with nonignorable missing response data and with nonmonotone patterns of missing data in the response variable. We develop a Monte Carlo EM algorithm for estimating the parameters in the model via the Gibbs sampler. For the normal random effects model, we derive a novel analytical form for the E- and M-steps, which is facilitated by integrating out the random effects. This form leads to a computationally feasible and extremely efficient Monte Carlo EM algorithm for computing maximum likelihood estimates and standard errors. In addition, we propose a very general joint multinomial model for the missing data indicators, which can be specified via a sequence of one-dimensional conditional distributions. This multinomial model allows for an arbitrary correlation structure between the missing data indicators, and has the potential of reducing the number of nuisance parameters. Real datasets from the International Breast Cancer Study Group and an environmental study involving dyspnoea in cotton workers are presented to illustrate the proposed methods.