A new class of discrete models for the analysis of zero-modified count data

Abstract

In this work, a new class of discrete models for the analysis of zero-modified count data has been introduced. The proposed class is composed of hurdle versions of the Poisson-Lindley, Poisson-Shanker, and Poisson-Sujatha baseline distributions, which are uniparametric Poisson mixtures that can accommodate different levels of overdispersion. Unlike the traditional formulation of zero-modified distributions, the primary assumption under hurdle models is that the positive observations are entirely represented by zero-truncated distributions. In the sense of extending the applicability of the theoretical models, it has also been developed a fixed-effects regression framework, in which the probability of zero-valued observations being generated as well as the average number of positive observations per individual could be modeled in the presence of covariates. Besides, an even more flexible structure allowing the inclusion of both fixed and random-effects in the linear predictors of the hurdle models has also been developed. In the derived mixed-effects structure, it has been considered the use of scalar random-effects to quantify the within-subjects heterogeneity arising from clustering or repeated measurements. In this work, all inferential procedures were conducted under a fully Bayesian perspective. Different prior distributions have been considered (e.g., Jeffreys' and g-prior), and the task of generating pseudo-random values from a posterior distribution without closed-form has been performed by one out of the three following algorithms (depending on the structure of each model): Rejection Sampling, Random-walk Metropolis, and Adaptive Metropolis. Intensive Monte Carlo simulation studies were performed in order to evaluate the performance of the adopted Bayesian methodologies. The usefulness of the proposed zero-modified models was illustrated by using several real datasets presenting different structures and sources of variation. Beyond parameter estimation, it has been performed sensitivity analyses to identify influent points, and, in order to evaluate the fitted models, it has been computed the Bayesian p-values, the randomized quantile residuals, among other measures. Finally, when compared with well-established distributions for the analysis of count data, the competitiveness of the proposed models has been proved in all provided examples.