Bayesian Analysis

A New Regression Model for Bounded Responses

Sonia Migliorati, Agnese Maria Di Brisco, and Andrea Ongaro

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Aim of this contribution is to propose a new regression model for continuous variables bounded to the unit interval (e.g. proportions) based on the flexible beta (FB) distribution. The latter is a special mixture of two betas, which greatly extends the shapes of the beta distribution mainly in terms of asymmetry, bimodality and heavy tail behaviour. Its special mixture structure ensures good theoretical properties, such as strong identifiability and likelihood boundedness, quite uncommon for mixture models. Moreover, it makes the model computationally very tractable also within the Bayesian framework here adopted.

At the same time, the FB regression model displays easiness of interpretation as well as remarkable fitting capacity for a variety of data patterns, including unimodal and bimodal ones, heavy tails and presence of outliers. Indeed, simulation studies and applications to real datasets show a general better performance of the FB regression model with respect to competing ones, namely the beta (Ferrari and Cribari-Neto, 2004) and the beta rectangular (Bayes et al., 2012), in terms of precision of estimates, goodness of fit and posterior predictive intervals.

Article information

Bayesian Anal. (2017), 28 pages.

First available in Project Euclid: 25 October 2017

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proportions beta regression flexible beta mixture models MCMC outliers heavy tails

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Migliorati, Sonia; Di Brisco, Agnese Maria; Ongaro, Andrea. A New Regression Model for Bounded Responses. Bayesian Anal., advance publication, 25 October 2017. doi:10.1214/17-BA1079.

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Supplemental materials

  • Supplementary Material for A New Regression Model for Bounded Responses. The online supplementary material contains proofs of the Propositions 1 and 2, sensible recommendations about the choice of priors for ϕ, a list of the thinning intervals adopted as to guarantee Raftery–Lewis diagnostics in most cases around 1. Furthermore, it includes a detailed description of the Simulation scenarios of Section 5.1, further results for the regression model for the mean with two explanatory variables for the reading accuracy dataset (Section 6.1) and a detailed analysis of residuals for sport data of Section 6.2.