Parameter Interpretation in Skewed Logistic Regression with Random Intercept

This paper aims at providing the prior and posterior interpretations for the parameters in the logistic regression model with random or cluster-level intercept when univariate and multivariate classes of skew normal distributions are assumed to model the random effects behavior. We obtain the prior distributions for the odds ratio and their medians under skew normality for the random effects. Original results related to linear combinations of skew-normal distributions are obtained as a by-product and, in the univariate case, a new class of log-skew-normal distribution is introduced. Robust results are obtained whenever a class of multivariate skew-normal distribution is assumed. We also evaluate the effect of the misspecification of the random effects distributions in the odds ratio estimation. We consider both simulated and the Teratogenic activity experiment datasets. The latter was previously analysed in the literature. We concluded that the misspecification of the random effects distribution yields poor odds ratios estimates and that the median odds ratio is not necessarily the best measure of heterogeneity among the clusters as suggested in the literature.