Multivariate quasi-beta regression models for continuous bounded data

Abstract

We propose a multivariate regression model to deal with multiple continuous bounded data. The proposed model is based on second-moment assumptions, only. We adopted the quasi-score and Pearson estimating functions for estimation of the regression and dispersion parameters, respectively. Thus, the proposed approach does not require a multivariate probability distribution for the variable response vector. The multivariate quasi-beta regression model can easily handle multiple continuous bounded outcomes taking into account the correlation between the response variables. Furthermore, the model allows us to analyze continuous bounded data on the interval [0, 1], including zeros and/or ones. Simulation studies were conducted to investigate the behavior of the NORmal To Anything (NORTA) algorithm and to check the properties of the estimating function estimators to deal with multiple correlated response variables generated from marginal beta distributions. The model was motivated by a data set concerning the body fat percentage, which was measured at five regions of the body and represent the response variables. We analyze each response variable separately and compare it with the fit of the multivariate proposed model. The multivariate quasi-beta regression model provides better fit than its univariate counterparts, as well as allows us to measure the correlation between response variables. Finally, we adapted diagnostic tools to the proposed model. In the supplementary material, we provide the data set and R code.

Multivariate generalized linear mixed models for continuous bounded outcomes: Analyzing the body fat percentage data

“…As future work, we suggest: (i) specifying the proposed model by using the simplex distribution; (ii) extending the MGLMM to deal with multiple continuous bounded variables inflated with zeroes and/or ones; (iii) modeling the precision parameter as a function of a set of covariates; (iv) proposing methods for residuals analysis and diagnostic; (v) comparing the proposed model with other approaches for the analysis of multiple continuous bounded data, such as the multivariate regression models based on copula functions 30 and second-moment assumptions 31 ; and (vi) introducing multivariate hypothesis test 56,57 to deal with multiple continuous bounded variables in the MGLMM framework.…”

“…The motivating example analyzed in this article uses data of the body fat percentage of subjects assisted at the Endocrinology and Metabology Service of the Clinic's Hospital of the Paraná Federal University, Curitiba, Brazil. The data set contains 298 observations and was previously analyzed by Petterle et al 31 by using the multivariate quasi-beta regression model. The data set is composed by two continuous and two categorical covariates, which correspond to: agesubject age (in years, ranging from 18 to 87 years), BMI (in kg/m 2 , ranging from 18.5 to 29.9 kg/m 2 ), gender-gender of subjects (F: female; M: male) and IPAQ-level of physical activity practiced routinely (S: sedentary; IA: insufficiently active; A: active).…”

“…The goals of the data analysis are: investigate the effect of the age, BMI, gender, and physical activity level (IPAQ) in the BFP (arms, legs, trunk, android, and gynecoid) and evaluate the correlation between the response variables, taking into account the effect of covariates in the model. For additional description of the BFP data set, see Petterle et al 31 3 Probability distributions on the interval (0, 1)…”

“…Using copula functions, Cepeda-Cuervo et al 29 presented the bivariate beta regression model, while Souza and Moura 30 extended this model to the multivariate case (with fixed effects) and proposed another multivariate beta model with correlated random effects. Based on second-moment assumptions, Petterle et al 31 proposed the multivariate quasi-beta regression models to deal with multiple continuous bounded variables on the interval [0, 1]. However, to the best of our knowledge, no model was proposed to deal with multiple continuous bounded variables using the MGLMM framework.…”

Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications

“…This application considers a real data set first reported in [41], which was also analyzed in [7]. This data set contains 298 observations about the body fat proportion of patients in a public hospital located in Curitiba, Paraná, Brazil.…”