A Note on the Indeterminacy of the Random-Effects Distribution in Hierarchical Models

Hierarchical models are regularly used for the analysis of longitudinal or clustered data. The estimators and inferential procedures are frequently derived from maximum likelihood theory, which assumes that the underlying probability model is correctly specified. Recent research shows that the results obtained from these models are not always robust against violations of the distributional assumptions for the random effects. As a consequence, diagnostic tools to detect such misspecification on the one hand, and alternative models allowing for more flexible random-effects distributions on the other hand, have been proposed in the literature. However, the intangible nature of random effects makes the evaluation of their distributional assumptions conceptually difficult. In the present work we show that such an evaluation requires additional assumptions regarding the conditional model (describing the outcomes, given the random effects), without which the distribution of the random effects is indeterminate. Consequently, the evaluation of the distributional assumptions of the random effects will necessarily demand a careful evaluation of the conditional model and it may require the use of extra scientific elements that go beyond the data.