Regression modelling with the tilted beta distribution: A Bayesian approach

Abstract

Beta regression models are commonly used in the case of a dependent variable y that exists on the range (0,1). However, when y can additionally take on the values of zero and/or one, limitations of the beta distribution and beta regression models become apparent. One recent approach is to use an inflated beta regression model which has discrete point-valued components. In this article, we introduce a new class of regression models for y ∈ [0, 1] that is fully continuous. This allows the entirety of y to be treated as a continuum instead of discontinuously, which appears to be a new development for the literature. We use a Bayesian approach for estimation. We also illustrate the impact of different choices of prior distributions on empirical findings and perform a simulation study examining model fit.

Tilted beta regression and beta-binomial regression models: Mean and variance modeling

“…The tilted distribution was proposed by Hahn and López Martín (2015), and the following alternative definition was proposed by Cepeda-Cuervo and Cifuentes-Amado (2017): a random variable Y follows a tilted distribution with parameter ν if its density function is given by: c (y | υ) = 2υ − 2 (2υ − 1) y I (0,1) (y), 0 ≤ ν ≤ 1.…”

“…This section presents a new parameterization of the tilted beta distribution proposed by Hahn and López Martín (2015), in terms of the mean (µ b ) and the variance (σ 2 ) parameters of the beta distribution and the mean of the tilted distribution µ t . This new parameterization of the tilted beta distribution is obtained from the convex combination of the µ t -tilted distribution proposed by Cepeda-Cuervo and Cifuentes-Amado (2017) and the µ b σ 2 -beta distribution proposed by Cepeda-Cuervo (2023).…”

“…The beta rectangular distribution consists of a convex combination between the beta distribution and the uniform distribution U(0, 1). Subsequently, Hahn and López Martín (2015) proposed the tilted beta distribution, that consists of a mixture of the beta distribution and the tilted distribution, which has as particular cases the beta rectangular distribution and the beta distribution. In this paper, from the mean and variance beta distribution, a new parameterization of the tilted beta distribution is proposed, where mean and variance beta distributions and beta rectangular distributions were applied to improve the above proposal by including the interpretation advantages of the mean and variance beta distributions.…”

A Model for Bimodal Rates and Proportions

“…Variations of the beta model can be found in Ferrari and Cribari-Neto (2004), Ospina and Ferrari (2008), Bayes et al (2012), Hahn (2021), among others. However, all the models cited above are not suitable for capturing bimodality.…”

“…• The proposed model allows the boundary values to lie on a smooth unified continuum along with the rest of the open interval (0, 1), as opposed existing as one or two discontinuities, i.e., it does not require boundary values to be either discarded or else treated separately (Hahn, 2021). Thus, one of the main motivation of this paper is to contribute with another attractive regression model for modeling of double bounded response variables.…”