Skew random effects in multilevel binomial models

Abstract

Compared to modelling observable data, it is more difficult to choose a suitable distribution to describe latent variables since no prior knowledge or observable information can be used and only normal or nonparametric distributions are mainly applied to random effects for generalized linear mixed models (GLMMs) in the literature. To enhance the modelling toolkit, this article investigates a class of parametric skew elliptical random effects in multilevel binomial regression models using a Bayesian approach; the class includes skew normal, skew Students’ t-distributions and others. Skewness mechanism is considered through multiplying skewness parameter Δ by standardized folded elliptical random variables, and the posterior sampling is realized by working on a binary skewness indicator (BSI) instead of continuous Δ for parameter identifiability. Simulation study shows that the original continuous skewness parameter Δ and the posterior mean of BSI may have dichotomous signs to describe the directional (right/left) skewness; thus we address the importance of assuming specific random effects distribution and interpreting the skewness carefully. The methodology is exemplified through reanalyzing a teratogenic activity study of two niacin analogs published in the biological literature, and sampling-based model comparison shows that the parametric skew normal random effects model works largely better than nonparametric Dirichlet process mixture models for this data set.