A quantile parametric mixed regression model for bounded response variables

Abstract

Bounded response variables are common in many applications where the responses are percentages, proportions, or rates. New regression models have been proposed recently to model the relationship among one or more covariates and the conditional mean of a response variable based on the beta distribution or a mixture of beta distributions. However, when we are interested in knowing how covariates impact different levels of the response variable, quantile regression models play an important role. A new quantile parametric mixed regression model for bounded response variables is presented by considering the distribution introduced by [27]. A Bayesian approach is adopted for inference using Markov Chain Monte Carlo (MCMC) methods. Model comparison criteria are also discussed. The inferential methods can be easily programmed and then easily used for data modeling. Results from a simulation study are reported showing the good performance of the proposed inferential methods. Furthermore, results from data analyses using regression models with fixed and mixed effects are given. Specifically, we show that the quantile parametric model proposed here is an alternative and complementary modeling tool for bounded response variables such as the poverty index in Brazilian municipalities, which is linked to the Gini coefficient and the human development index.

A new class of regression model for a bounded response with application in the study of the incidence rate of colorectal cancer

“…Note also that it is possible to consider another parameterization for the location parameter based on other quantile, different from the median considered here, and then a quantile parametric regression model for bounded response can be formulated as alternative for Bayes et al 6 Finally, although we use own R code for fit the models presented here, these models can be implemented easily, for example in Stan code. One example for the Johnson SB Regression model is shown in Appendix 1 and can be easily generalized by substituting the expression for the log density of the normal distribution for the corresponding power normal distribution and using other transformation in substitution of the logistic transformation or quantile of the Logistic distribution and finally considering alternatives links for the location parameter.…”

“…Thus, alternative regression models to the beta regression model have been proposed over the last few years. For example, Qiu et al 5 proposes a regression model based on the simplex distribution, and Bayes et al 6 introduced the parametric quantile regression model based on the Kumaraswamy distribution. Lemonte and Baza´n 7 proposed such a regression model based on a more general class of distributions which include as special case the Johnson S B distribution.…”

A parametric quantile regression approach for modelling zero‐or‐one inflated double bounded data

“…There are advantages to using quantile regression, such as the robustness to outliers, and can be more intuitive than the mean, especially for skewed distributions. According to Bayes, Bazán, and De Castro (2017), the main advantage of the quantile regression is its flexibility for modeling data with heterogeneous conditional distributions. Furthermore, in contrast to the mean regression model, quantile regression can provide an overall assessment of the covariate effects at different quantiles.…”

Parametric mode regression for bounded responses

“…Also considering Bayesian regression analysis for bounded data, Migliorati, Di Brisco, and Ongaro (2018) proposed a flexible beta distribution for the response given covariates based on a special mixture of two beta distributions to balance between flexibility and tractability. Unlike all the above regression models which focus on inferring the conditional mean of a bounded response, Bayes, Bazán, and De Castro (2017) developed quantile regression models for bounded responses built upon on beta distributions. Barrientos, Jara, and Quintana (2017) took on a fully nonparametric Bayesian approach to model the covariates‐dependent distribution of a bounded response.…”