Modelos logísticos com classes estendidas de distribuições normais para os efeitos aleatórios

Abstract

This work aims at extending the mixed logistic regression model (MLRM) by considering classes of univariate and multivariate skew-normal distributions for the random effects. The Bayesian paradigm is used. Random effects are introduced into the model to represent unmeasured covariates or as a way of modeling data correlation or overdispersion and, usually, it is assumed that the random effects are normally distributed. As for the logistic regression, the odds ratio is used for the parameters interpretation. However, under the MLRM, for some cases, the odds ratio depends on the random effectsthus it is a random quantity under both Bayesian and classical approaches. From the classical point of view, Larsen et al. (2000) find the distribution of the odds ratio under normality for the random effects. They suggest using the median odds ratio (MOR) to interpret the parameters. According to Larsen et al. (2000), the MOR quantifies appropriately theheterogeneity among the different groups. One of the main contributions of this work is to extend such results whenever both univariate and multivariate skew-normal distributions are considered to model the uncertainty about the random effects. As a byproduct, some resultsrelated to linear combination of skew-normal random variables are obtained. Considering simulated data sets, a study is performed in order to evaluate the effect of the misspecification of the distributions of the random effects in the estimations of the fixed and random effects as well as the odds ratio. In general the model which assumes the univariate skew-normal for the random effects leads to the improvement of the estimates for the random effects and to the detection of the asymmetry in their distribution if it exists. Similar inferences are obtained if normal and the multivariate skew-normal are assumed. The proposed models are considered to analyze the data set reported in Liu e Dey (2008).