Bayesian Multivariate Distributional Regression With Skewed Responses and Skewed Random Effects

The normal and the <i>t</i> distribution are classical tools for building random effects regression models where both can be used for the specification of either the conditional response distribution or the random effects distribution. However, the underlying assumption of symmetry can be questionable in many applications. We, therefore, propose regression models where the skew-normal and skew-<i>t</i> distribution are considered for both the response and the random effects specification and embed these models in the framework of distributional regression such that regression predictors can be specified for all distributional parameters. The distributional regression framework also allows us to consider multivariate versions of the skew-normal and the skew-<i>t</i> distribution. For Bayesian inference, we adapt iteratively weighted least-square proposals within Markov chain Monte Carlo simulations such that they can also facilitate the inclusion of nonnormal random effects specifications. Model choice is based on the Watanabe–Akaike information criterion, in particular, to differentiate between skew and nonskew distributional specifications in a number of simulation studies. Finally, to illustrate their practical applicability, the developed models are applied to a study on cholesterol levels originating from the Framingham Heart Study and a dataset from the Demographic and Health Surveys on undernutrition among children in Nigeria. Supplementary material for this article is available online.