Modelos de regressão misto para resposta limitada usando distribuições do tipo Johson-SB

Abstract

In this work, new properties, estimation methods, residual analysis and extensions are developed for regression models in the (0,1) interval considering the Johnson-SB type distributions (JSB). The extensions are zero-and-one inflated models and mixed regression models. New mixed-effects models for bounded longitudinal data in the interval $(0,1)$ based on the $JSB$ distributions are presented. The penalized likelihood estimators are obtained by maximizing the penalized likelihood and are computed by the Rigby and Stasinopoulos (RS) algorithm. From the Bayesian perspective, the No-U-Turn-Sampler (NUTS) is used to sample from the posterior distribution. Residual analysis is performed considering randomized quantile residuals. Simulation studies considering robustness to outliers from the distributions and extensions of the models to support 0 and 1 observations are presented. Three real data sets motivate the use of the new models. The first dataset contains the proportion of individuals vulnerable to poverty of the 645 municipalities from São Paulo state in Brazil and with no covariate. The second dataset incorporates the proportion of votes obtained by a political party in five Brazilian presidential elections, every four years, from 1994 to 2010, from the 75 municipalities from Sergipe state in Brazil. The third dataset comes from the public health area in Brazilian states. It contains the mortality rates from bronchial and lung cancer from the 27 Brazilian states over the last 30 years. The aim is to identify if factors like sex, age, and the Municipal Human Development Index of the state can influence the mortality rate. The $JSB$ mixed regression models and the Beta mixed model were applied. The $JSB$ mixed models display better values than the Beta mixed model for the model comparison criteria. The results and the residual analysis reveal that the $JSB$ models are an alternative to the Beta model.