Flexible modelling of random effects in linear mixed models—A Bayesian approach

Flexible modelling of random effects in linear mixed models has attracted some attention recently. In this paper, we propose the use of finite Gaussian mixtures as in Verbeke and Lesaffre [A linear mixed model with heterogeneity in the random-effects population, J. Amu. Statist. Assoc. 91, 217–221]. We adopt a fully Bayesian hierarchical framework that allows simultaneous estimation of the number of mixture components together with other model parameters. The technique employed is the Reversible Jump MCMC algorithm (Richardson and Green [On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion). J. Roy. Statist. Soc. Ser. B 59, 731–792]). This approach has the advantage of producing a direct comparison of different mixture models through posterior probabilities from a single run of the MCMC algorithm. Moreover, the Bayesian setting allows us to integrate over different mixture models to obtain a more robust density estimate of the random effects. We focus on linear mixed models with a random intercept and a random slope. Numerical results on simulated data sets and a real data set are provided to demonstrate the usefulness of the proposed method.