Multivariate Beta Regression with Application in Small Area Estimation

Abstract

Multivariate beta regression models for jointly modelling two or more variables whose values belong in the (0,1) interval, such as indexes, rates or proportions, are proposed for making small area predictions. The multivariate model can help the estimation process by borrowing strength between units and obtaining more precise estimates, especially for small samples. Each response variable is assumed to have a beta distribution so the models could accommodate multivariate asymmetric data. Copula functions are used to construct the joint distribution of the dependent variables; all the marginal distributions are fixed as beta. A hierarchical beta regression model is additionally proposed with correlated random effects. We present an illustration of the proposed approach by estimating two indexes of educational attainment at school level in a Brazilian state. Our predictions are compared with separate univariate beta regressions. The inference process was conducted using a full Bayesian approach.

Small area estimation of proportions under area-level compositional mixed models

“…Under the regression-based setup, Ferrante and Trivisano (2010) proposed a multivariate SAE approach for count data based on the multivariate Poisson-lognormal distribution an derived hierarchical Bayes predictors. Souza and Moura (2016) and Fabrizi et al (2016) deal with multivariate Beta regression models in SAE. Saei and Chambers (2003), Molina et al (2007) and López-Vizcaíno et al (2013 have applied multinomial logit mixed models to category counts for estimating domain totals of labour status categories.…”

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 R Package tipsae: Tools for Mapping Proportions and Indicators on the Unit Interval

“…We recall the works by Marhuenda, Molina, and Morales (2013); Marhuenda, Morales, and del Carmen Pardo (2014), Morales, Pagliarella, and Salvatore (2015), and Esteban, Morales, Pérez, and Santamaría (2012); Esteban, Lombardía, López-Vizcaíno, Morales, and Pérez (2020). The second strand focus on classical beta regression, both in the univariate case (Liu, Lahiri, and Kalton 2007;Bauder et al 2015;Giovinazzi and Cocchi 2021) and in the multivariate ones (Fabrizi et al 2011;Souza and Moura 2016), considering also zero and/or one inflated extensions (Wieczorek, Nugent, and Hawala 2012;Trivisano 2016, 2020). Lastly, a beta mixture approach in SAE has been proposed by De Nicolò, Ferrante, and Pacei (2022b).…”