Bayesian analysis for a Class of Beta Mixed Models

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There has been an increased interest in the class of Generalized Linear Mixed Models (GLMM) in the last 10 years. One possible reason for such popularity is that GLMM combine Generalized Linear Models (GLM) citep{Nelder1972} with Gaussian random effects, adding flexibility to the models and accommodating complex data structures such as hierarchical, repeated measures, longitudinal, among others which typically induce extra variability and/or dependence.

GLMMs can also be viewed as a natural extension of Mixed Linear Models citep{Pinheiro:2000}, allowing flexible distributions to response variables. Common choices are Gaussian for continuous data, Poisson and Negative Binomial for count data and Binomial for binary data. These three situations include the majority of applications within this class of models. Some examples can be found in citep{Breslow:1993} and citep{Molenberghs:2005}.

Despite that flexibility, exist situations where the response variable is continuous but, bounded such as rates, percentages, indexes and proportions. In these situations the traditional GLMM based on the Gaussian distribution, is not adequate, since bounded is ignored. An approach that has been used to model this kind of data are based on the beta distribution. The beta distribution is very flexible with density function that can display quite different shapes, including left or right skews, symmetric, J-shape, and inverted J-shape citep{Da-Silva2011}.

Regression models for independent and identically distributed beta variable proposed by cite{Paolino2001}, cite{Kieschnick2003} and cite{Ferrari2004}. The basic assumption is that the response follow a beta law whose expected value is related to a linear predictor through a link func...

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...ent results which help to choose prior distributions.

The main goal this paper is therefore to present Bayesian inference for beta mixed models using INLA.
We discuss the choice of prior distributions and measures of model comparisons.
Results obtained from INLA are compared to those obtained using an MCMC algorithm and likelihood analysis. The model is illustrated with the analysis a real dataset, previously analyzed by citet{Bonat2013}.
Additional care is given to choice of prior distributions for precision parameter of the beta law.

The structure this paper is the follows. In Section 2, we define the Bayesian beta mixed model, Section 3 we describe the Integrated Nested Laplace Approximation (INLA). In Section 4 the model is introduced for the motivating example and the results of the analyses are presented. We close with concluding remarks in Section 5.

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