Flexible parametric models for random-effects distributions

It is commonly assumed that random effects in hierarchical models follow a normal distribution. This can be extremely restrictive in practice. We explore the use of more flexible alternatives for this assumption, namely the t distribution, and skew extensions to the normal and t distributions, implemented using Markov Chain Monte Carlo methods. Models are compared in terms of parameter estimates, deviance information criteria, and predictive distributions. These methods are applied to examples in meta-analysis and health-professional variation, where the distribution of the random effects is of direct interest. The results highlight the importance of allowing for potential skewing and heavy tails in random-effects distributions, especially when estimating a predictive distribution. We describe the extension of these random-effects models to the bivariate case, with application to a meta-analysis examining the relationship between treatment effect and baseline response. We conclude that inferences regarding the random effects can crucially depend on the assumptions made and recommend using a distribution, such as those suggested here, which is more flexible than the normal.