Statistical Modelling 8 (2008), 221–241

Skew random effects in multilevel binomial models: an alternative to nonparametric approach

Jungfeng Liu
Biometrics Division, The Cancer Institute of New Jersey,
195 Little Albany Street,
New Brunswick, NJ 08901
and
Department of Biostatistics, School of Public Health,
University of Medicine and Dentistry of New Jersey,
Piscataway, NJ 08854
U.S.A.
eMail: liu16@umdnj.edu

Dipak K. Dey
Department of Statistics,
University of Connecticut,
Storrs
U.S.A.

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.

Keywords:

Binomial; binary skewness indicator (BSI); generalized linear mixed model (GLMM); multilevel; random effects; sign switching; skewness

Downloads:

Example data and software code in zipped archive.

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