Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

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

  • Received : 2023.02.19
  • Accepted : 2023.12.16
  • Published : 2024.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

This paper proposes new parameterizations of the tilted beta binomial distribution, obtained from the combination of the binomial distribution and the tilted beta distribution, where the beta component of the mixture is parameterized as a function of their mean and variance. These new parameterized distributions include as particular cases the beta rectangular binomial and the beta binomial distributions. After that, we propose new linear regression models to deal with overdispersed binomial datasets. These new models are defined from the proposed new parameterization of the tilted beta binomial distribution, and assume regression structures for the mean and variance parameters. These new linear regression models are fitted by applying Bayesian methods and using the OpenBUGS software. The proposed regression models are fitted to a school absenteeism dataset and to the seeds germination rate according to the type seed and root.

Keywords

References

  1. Cepeda-Cuervo E (2001). Modelagem da Variabilidade em Modelos Lineares Generalizados. (Unpublished Ph.D. thesis), Mathematics Institute, Universidade Federal do Rio de Janeiro, Available from: http://www.docentes.unal.edu.co/ecepedac/docs/TESISFINAL.pdf
  2. Cepeda-Cuervo E (2015). Beta regression models: Joint mean and variance modeling, Journal of Statistical Theory and Practice, 9, 134-145.
  3. Cepeda-Cuervo E (2023). µσ2-beta and µσ2-beta binomial regression models, Revista Colombiana de Estad'istica, 46, 63-79.
  4. Cepeda-Cuervo E and Cifuentes-Amado MV (2017). Double generalized beta-binomial and negative binomial regression models, Revista Colombiana de Estad'istica, 40, 141-163.
  5. Cepeda-Cuervo E and Cifuentes-Amado MV (2020). Tilted beta binomial linear regression model: A bayesian approach, Journal of Mathematics and Statistics, 16, 1-8.
  6. Collet D (1991). Modeling Binary Data, Chapman Hall, London.
  7. Cox D (1983). Some remarks on overdispersion, Biometrika, 70, 269-274.
  8. Crowder MJ (1978). Beta-binomial ANOVA for proportions, Applied Statistics, 27, 34-37.
  9. Efron B (1986). Double exponential families and their use in generalized linear regression, Journal of the American Statistical Association, 81, 709-721.
  10. Ferrari S and Cribari-Neto F (2004). Beta Regression for Modelling Rates and Proportions, Journal of Applied Statistics, 31, 799-815.
  11. Garcia Perez J, Lopez MM, Garcia Garcia C, and MA SG (2016). Project management under uncertainty beyond beta: The generalized bicubic distribution, Operations Research Perspectives, 67-76.
  12. Perez JG, Martin MD, M. L., Garc'ia, C. G., and Granero MAS (2016). Project management under un-certainty beyond beta: The generalized bicubic distribution, Operations Research Perspectives, 3, 67-76.
  13. Hahn E (2008). Mixture densities for project management activity times: A robust approach to pert, European Journal of Operational Research, 188, 450-459.
  14. Hahn E (2022). The tilted beta-binomial distribution in overdispersed data: Maximum likelihood and bayesian estimation, Journal of Statistical Theory and Practice, 16, 43.
  15. Hahn E and Lopez Martin M (2015). Robust project management with the tilted beta distribution, SORT, 39, 253-272.
  16. Hinde J and Demetrio C (1998). Overdispersion: Models and estimation, Computational Statistics & Data Analysis, 27, 151-170.
  17. Jorgensen B (1997). Proper dispersion models, Brazilian Journal of Probability and Statistics, 11, 89-128.
  18. Quine S (1975). Achievement orientation of aboriginal and white australian adolescents (Ph.d. thesis), Australian National University, Australia.
  19. Simas AB, Barreto-Souza W, and Rocha AV (2010). Improved estimators for a general class of beta regression models, Computational Statistics and Data Analysis, 54, 348-366.
  20. Spiegelhalter DJ, Thomas A, Best N, and Lunn D (2003). WinBUGS Version 1.4., MRC Biostatistics Unit, Cambridge, UK.
  21. Udoumoh EF, Ebong DW, and Iwok IA (2017). Simulation of project completion time with Burr XII activity distribution, Asian Research Journal of Mathematics, 6, 1-14.
  22. Williams D (1982). Extra-binomial variation in logistic linear models, Journal of the Royal Statistical Society: Series C (Applied Statistics), 31, 144-148.