Document Type : Original Paper
Authors
Department of Statistics, Tarbiat Modares University, Tehran, Islamic Republic of Iran
Abstract
In modeling the variables related to each other, regression models are usually used assuming that the response variable is Normal. But in problems dealing with data such as the rate or ratio of an event distributed in the (0,1) interval, these models may provide out-of-range predictions for the response variable. In addition rate or ratio data are oftem asymmetrically distributed, and the use of symmetric distributions leads to invalid results. In such cases, the Beta regression model is used, in which the distribution of the response variable is in the Beta family. Bayesian analysis of these models generally requires the calculation of multiple integrals. The use of MCMC algorithms sometimes encounters long computation times and divergence. This work presents approximate methods for obtaining posterior distributions for Bayesian analysis of Beta regression models. Then the Integrated Nested Laplace Approximation will be offered for getting the posterior distributions in the Bayesian analysis of these models. Moreover, these models' application is illustrated on a real data set.
Keywords
- Bonat WH and Ribeiro Jr PJ. Bayesian analysis for a class of beta mixed Chilean J Stat. 2014; 6(1): 3–13.
- Bonat WH, Ribeiro Jr PJ and Zeviani WM Likelihood analysis for a class of beta mixed models. J Appl Stat. 2015; 42(2): 252-256.
- Ferrari SLP, and Cribari-Neto F. Beta Regression for Modelling Rates and Proportions. J Appl Stat. 2004; 31: 799-815.
- Smithson M and Verkuilen J. A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychol Methods. 2006 Mar; 11: 54-71.
- Verkuilen J and Smithson M. Mixed and mixture regression models for continuous bounded responses using the Beta distribution. J Educ Behav Stat. 2012; 37: 82-113.
- Cepeda E and Gammerman D. Bayesian modeling of variance heterogeneity in normal regression models. Brazil J Probab Stat. 2001; 14: 207-221.
- Cepeda-Cuervo E. Modeling Variability in Generalized Linear Models, Unpublished Ph.D. Thesis. Mathematics Institute, Universidade Federal do Rio de Janeiro. 2001; https://repositorio.unal.edu.co/handle/unal/11839.
- Cepeda-Cuervo E, Jaimes D, Marin M and Rojas J. Bayesian beta regression with Bayesianbetareg R-package. Computational Statistics. 2016; 31: 165-187.
|
|
- Rue H and. Martino S. Approximate bayesian inference for hierarchical gaussian markov random fields models. J Stat Plan Infer. 2007; 137: 3177-3192.
- Rue H and Martino S, Chopin N. Approximate bayesian inference for latent gaussian models using integrated nested laplace approximations (with discussion). J Royal Stat Soc. 2009; 71: 319-392.
- Martino S and Rue H. Case studies in bayesian computation using INLA, In: Mantovan P., Secchi P. (eds) complex data modeling and computationally intensive statistical methods. Contributions to Statistics. 2010; Milano: Springer. p. 99-114.
- Tierney L and Kadane J. Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc. 1986; 81: 82-86.
- Rue H and Held I. Gaussian Markov Random Field: Theory and Application, vol 104 of Monographs on Statistics and Applied Probability, Chapman and Hall, London; 2005.
- Pollice A, Jona Lasinio G, Rossi R, Amato M, Kneib T and Lang S. Bayesian measurement error correction in structured additive distributional regression with an application to the analysis of sensor data on soil–plant variability. Stochast Environ Res Risk Assess. 2019; 33: 747-763.
)