Fitting exponential family mixed models

The generalized linear model (McCullagh and Nelder, 1972) and the semiparametric multiplicative hazard model (Cox, 1972) have significantly influenced the way in which statistical modelling is taught and practiced. Common for the two model families is the assumption that conditionally on covariate information (including time) the observations are independent. The obvious difficulty in identifying and measuring all relevant covariates has pushed for methods that can jointly handle both mean and dependence structures. The early 1990s saw a myriad of approaches for dealing with multivariate generalized linear models. More recently, the hazard models have been extended to multivariate settings. Here we review (i) penalized likelihood, (ii) Monte Carlo EM, and (iii) Bayesian Markov chain Monte Carlo methods for fitting the generalized linear mixed models and the frailty models, and we discuss the rationale for choosing between the methods. The similarities of the toolboxes for these two multivariate model families open up for a new level of generality both in teaching and applied research. Two examples are used for illustration, involving censored failure time responses and Poisson responses, respectively.