Document Type : Original Article


1 Department of Mathematics, College of Science, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq

2 Department of Mathematics, College of Basic Education, University Of Sulaimani, P.O. Box: 46, Sulaimani, Kurdistan Region, Iraq


The beta-binomial model that is generated by a simple mixture model has been commonly applied in the health, physical, and social sciences. In clinical and public health, overdispersion occurs due to biological variation between the subjects of interest. Both the binomial and beta-binomial models are applied to different problems occurring in rational test theory. In this study, we focused on modeling overdispersion for binomial distribution. The main aim was to show a complete and extensive understanding of the beta-binomial model and updated form by broaden its practical applications in the field of breast cancer with hormone medication. It is observed in different independent Bernoulli trials yes/no (xi=1, 0) experiments with success probabilities 0<p i< 1 and compare the model in a sequence of ni. The performance of the maximum likelihood estimates technique that is used in moderate and small samples ni by a Newton-Raphson iterative method using Matlab package. We have found that using hormones for other treatments have complication leading to breast cancer. We took 20 investigational testers in Hiwa Hospital for cancer treatment in Sulaymaniyah province, with proportion p i is varying from 9.7% to 50 %. In addition, we concluded that the beta-binomial theory is a good alternative of binomial model. This is due to the fact that the beta-binomial model has provided a robust estimate for events from heterogeneous binomial studies. 


  1. Anderson, D. A.: Some models for overdispersed binomial data, Aust. J. Stat., 30, 125–148, (1988). 
  2. Kleinman, J. C.: Proportions with Extraneous Variance: Single and Independent Samples, J. Am. Stat. Assoc., 68, 46–54, (1973). 
  3. Ennis, D. M. and Bi, J.: The beta-binomial model: Accounting for inter-trial variation in replicated difference and preference tests, J. Sens. Stud., 13, 389–412, (1998). 
  4. Lee, J. C. and Sabavala, D. J.: Bayesian Estimation and Prediction for the Beta-Binomial Model, J. Bus. Econ. Stat., 5, 357, (1987). 
  5. Lee, J. and Lio, Y. L.: A note on bayesian estimation and prediction for the beta-binomial model, J. Stat. Comput. Simul., 63, 73–91, (1999). 
  6. Chuang-Stein, C.: An Application of the Beta-Binomial Model to Combine and Monitor Medical Event Rates in Clinical Trials, Drug Inf. J., 27, 515–523, (1993). 
  7. Young-Xu, Y. and Chan, K. A.: Pooling overdispersed binomial data to estimate event rate, BMC Med. Res. Methodol., 8, 58, (2008). 
  8. Griffiths, D. A.: Maximum Likelihood Estimation for the Beta-Binomial Distribution and an Application to the Household Distribution of the Total Number of Cases of a Disease, Biometrics, 29, 637, (1973). 
  9. Guimarães, P.: A simple approach to fit the beta-binomial model, Stata J., 5, 385–394, (2005). 
  10. Garren, S. T., Smith, R. L., and Piegorsch, W. W.: On a Likelihood-Based Goodness-of-Fit Test of the Beta-Binomial Model, Biometrics, 56, 947–949, (2000). 
  11. Tripathi, R. C., Gupta, R. C., and Gurland, J.: Estimation of parameters in the beta binomial model, Ann. Inst. Stat. Math., 46, 317–331, (1994). 
  12. Kapourani, C. A.: Beta Binomial for overdispersion, (2008). [Online]. Available: 
  13. Bodhisuwan, W. and Saengthong, P.: The Negative Binomial – Weighted Garima Distribution: Model, Properties and Applications, Pakistan J. Stat. Oper. Res., 16, 1–10, (2020). 
  14. Azimi, S. S., Bahrami Samani, E., and Ganjali, M.: Random Effects Models for Analyzing Mixed Overdispersed Binomial and Normal Longitudinal Responses With Application to Kidney Function Data of Cancer Patients, Stat. Biopharm. Res., 0, 1–18, (2020). 
  15. Zhang, Y.-Y., Xie, Y.-H., Song, W.-H., and Zhou, M.-Q.: The Bayes rule of the parameter in (0,1) under Zhang’s loss function with an application to the beta-binomial model, Commun. Stat. - Theory Methods, 49, 1904–1920, (2020).