Flexible parametric models for random‐effects distributions

Abstract

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. Copyright © 2007 John Wiley & Sons, Ltd.