Document Type: Final File

Authors

1 Department of Statistics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Islamic Republic of Iran

Abstract

In some applications, the response variable assumes values in the unit interval. The standard linear regression model is not appropriate for modelling this type of data because the normality assumption is not met. Alternatively, the beta regression model has been introduced to analyze such observations. A beta distribution represents a flexible density family on (0, 1) interval that covers symmetric and skewed families. In this paper, a beta generalized linear mixed model with spatial random effect is proposed emphasizing on small values of the spatial range parameter and small sample sizes. Then some models with both fixed and varying precision parameter and different combinations of priors and sample sizes are discussed. Next, the Bayesian estimation of the model parameters is evaluated in an intensive simulation study. Selected priors improved the Bayesian estimation of the parameters, especially for small sample sizes and small values of range parameter. Finally, an application of the proposed model on data provided by Household Income and Expenditure Survey (HIES) of Tehran city is presented.

Keywords

  1. Ferrari S., Cribari-Neto F. Beta Regression for Modelling Rates and Proportions. J. Appl. Stat. 31: 799-815 (2004).
  2. Ferrari S.L., Pinheiro E.C. Improved Likelihood Inference in Beta Regression. J. Stat. Comput. Simul. 81: 431-443 (2011).
  3. Cepeda-Cuervo  E., Gamerman D. Bayesian Methodology for Modelling Parameters in the two Parameter Exponential Family. Rev. Estad. 57: 93-105 (2005).
  4. Smithson M., Verkuilen J. A Better Lemon Squeezer? Maximum-Likelihood Regression with Beta Distributed Dependent Variables. Psychol. Methods. 11: 54-71 (2006).
  5. Branscum A.J., Johnson W.O., Thurmond M.C. Bayesian Beta Regression: Applications to Household Expenditure Data and Genetic Distance Between Foot-and-Mouth Disease Viruses. Aust. N. Z. J. Stat. 49: 287-301 (2007).
  6. Simas A.B., Barreto-Souza W., Rocha A.V. Improved Estimators for a General Class of Beta Regression Models. Comput. Stat. Data Anal. 54: 348-366 (2010).
  7. Zimprich D. Modelling Change in Skewed Variables using Mixed Beta Regression Models. Res Hum Dev. 7: 9-26 (2010).
  8. Figueroa-Zúñiga J.I., Arellano-Valle R.B., Ferrari S.L. Mixed Beta Regression: A Bayesian Perspective. Comput. Stat. Data Anal. 61: 137-147 (2013).
  9. Verkuilen J., Smithson M. Mixed and Mixture Regression Models for Continuous Bounded Responses using the Beta Distribution. J. Educ. Behav. Stat. 37: 82-113 (2012).

10. Ferreira G., Figueroa-Zúñiga J.I., de Castro M. Partially Linear Beta regression Model with Autoregressive Errors. TEST24: 752-775 (2015).

11. Cepeda-Cuervo E., Urdinola B.P., Rodriguez D. Double Generalized Spatial Econometric Models. Commun. Stat. Simul. Comput. 41:  671-685 (2012).

12. Cepeda-Cuervo, E., Nunez-Anton V. Spatial Double Generalized Beta Regression Models Extensions and Application to Study Quality of Education in Colombia. J. Educ. Behav. Stat. 38:604-628 (2013).

13. Gholizadeh K., Mohammadzadeh M., Ghayyomi Z. Spatial Analysis of Structured Additive Regression and Modelling of Crime Data in Tehran City Using Integrated Nested Laplace Approximation. Journal of Statistical Society.7: 103-124 (2013).

14. Fustos R. Modelo Lineal Generalizado Espacial Con Variable Respuesta Beta. Engineer's Degree Dissertation. Department of Statistics. University of Concepcion, Chile. (2013).

15. Lagos-Alvarez B.M., Fustos-Toribio R., Figueroa-Zúñiga J., and Mateu, J. Geostatistical Mixed Beta Regression: A Bayesian Approach. SERRA. 31: 571-584 (2016).

16. Kalhori L., Mohammadzadeh M. Spatial Beta Regression Model with Random Effect. J. SRI. 13: 214-230 (2016).

17. Brooks S.P., Gelman A. General Methods for Monitoring Convergence of Iterative Simulations. J. Comput. Graph. Stat. 7: 434-455 (1998).

18. Heidelberger P., Welch P.D. A Spectral Method for Confidence Interval Generation and Run Length Control in Simulations. Commun. ACM. 24: 233-245 (1981).

19.

Appendix I:

Full conditional distributions are as follow

 

                                                                       

 

 

                                           

                                          

 

 

                       

 

{

 

Diggle P.J., Tawn J., Moyeed R. Model-Based Geostatistics.  J. Royal. Stat. Soc: Ser. C Appl. Stat. 47: 299-350 (1998).

20. Mark S. Handcock M.L.S. A Bayesian Analysis of Kriging. Technometrics. 35: 403-410 (1993).

21. Stein M.L. Interpolation of Spatial Data: Some Theory for Kriging, Springer Science & Business Media, New York. (2012).

22. Stein M.L. Interpolation of Spatial Data: Some Theory for Kriging, Springer Science & Business Media, New York. (2012).

23. Chen M.H., Shao Q.M., Ibrahim J. Monte Carlo Methods in Bayesian Computation, Springer, New York. (2000).

24. Sturtz S., Ligges U., Gelman A.  R2winbugs: A Package for Running Winbugs from R. J. Stat. Softw. 12: 1-16 (2005).

25. R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Austria, Vienna. (2013).

26. Geweke J. Evaluating the Accuracy of Sampling Based Approaches to the Calculation of Posterior Moments. Federal Reserve Bank of Minneapolis, Research Department Minneapolis, MN, USA. 196: (1991).

27. Huang X., Li G., Elashoff R.M. A Joint Model of Longitudinal and Competing Risks Survival Data with Heterogeneous Random Effects and Outlying Longitudinal Measurements. Stat. Its Interface. 3: 185 (2010).