Statistics and Its Interface

Volume 14 (2021)

Number 2

Spatial regression models for bounded response variables with evaluation of the degree of dependence

Pages: 95 – 107



Sandra E. Flores (Instituto de Matemática e Estatística, Universidade de São Paulo, SP, Brazil)

Marcos O. Prates (Departamento de Estatística, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil)

Jorge L. Bazán (Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, SP, Brazil)

Heleno B. Bolfarine (Instituto de Matemática e Estatística, Universidade de São Paulo, SP, Brazil)


Bounded response variables such as percentages, proportions, or rates are common in applications involving social and educational datasets, including rates of poverty or rates of achievement by municipalities, counties or provinces. New regression models have been proposed in recent years by considering distributions such as the Beta, Simplex and Kumaraswamy models for this type of data. However, to this type of dataset, it is common to observe the spatial dependence of units. For instance, municipalities or counties are organized into states. For this case, the supposition of independence among observations in the same state removes relevant relations between neighboring provinces. In this paper, we present a model of spatially bounded distribution regression with a Bayesian estimation approach where spatial relations are modeled by a spatial random variable with a particular dependence structure, such as the intrinsic conditional autoregressive model or the Leroux definition. Additionally, the Bayesian inferential method and model comparison criteria are discussed. Simulation studies and an application in reading comprehension spatial data are used to illustrate the performance of the proposed model and the estimation method adopted.


bounded distribution, Bayesian inference, proportions, spatial models

This study was financed in part by the “Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil” (CAPES) - Finance Code 001. The second author acknowledges financial support from CNPq grants 436948/2018-4 and 307547/2018-4 and FAPEMIG grant PPM-00532-16. The third author was partially supported by FAPESP-Brazil 2017/15452-5.

Received 2 May 2019

Accepted 29 April 2020

Published 22 December 2020