Specification of random effects in multilevel models: a review

The analysis of highly structured data requires models with unobserved components (random effects) able to adequately account for the patterns of variances and correlations. The specification of the unobserved components is a key and challenging task. In this paper, we first review the literature about the consequences of misspecifying the distribution of the random effects and the related diagnostic tools; we then outline the main alternatives and generalizations, also considering some issues arising in Bayesian inference. The relevance of suitably structuring the unobserved components is illustrated by means of an application exploiting a model with heteroscedastic random effects.