On Some Mixture Models for Over-dispersed Binary Data

Abstract

In this paper, we consider several binomial mixture models for fitting over-dispersed binary data. The models range from the binomial itself, to the beta-binomial (BB), the Kumaraswamy distributions I and II (KPI \& KPII) as well as the McDonald generalized beta-binomial mixed model (McGBB). The models are applied to five data sets that have received attention in various literature. Because of convergence issues, several optimization methods ranging from the Newton-Raphson to the quasi-Newton optimization algorithms were employed with SAS PROC NLMIXED using the Adaptive Gaussian Quadrature as the integral approximation method within PROC NLMIXED. Our results differ from those presented in Li, Huang and Zhao (2011) for the example data sets in that paper but agree with those presented in Manoj, Wijekoon and Yapa (2013). We also applied these models to the case where we have a $k$ vector of covariates $(x_1, x_2, \ldots, x_k)^{'}$. Our results here suggest that the McGBB performs better than the other models in the GLM framework. All computations in this paper employed PROC NLMIXED in SAS. We present in the appendix a sample of the SAS program employed for implementing the McGBB model for one of the examples.