Employing the Double, Multiplicative and The Com-Poisson Binomial Distributions for modeling Over and Under-dispersed Binary Data

In this paper, we compare the performances of several models for fitting over-dispersed binary data. The distribution models considered in this study include the binomial (BN), the beta- binomial (BB), the multiplicative binomial (MBM), the Com-Poisson binomial (CPB) and the double binomial (DBM) models. Applications of these models to several well known data sets exhibiting under-dispersion and over-dispersion were considered in this paper. We applied these models to two frequency data sets and two data sets with covariates that have been variously analysed in the literature. The first relates to the Portuguese version of Duke Religiosity Index in a sample of 273 (202 women, 71 Male) postgraduate students of the faculty of Medicine of University of Sao Paulo. The second set that employs the Generalize Linear Model (GLM) is the correlated binary data which studies the cardiotoxic effects of doxorubicin chemoteraphy on the treatment of acute lymphoblastic leukemia in childhood. In the first data set, we have a single
covariate, Sex (0,1) and two covariates in the second data set (dose and time).

Our results indicate that all the models considered here (excluding the binomial) behave reasonably well in modeling over-dispersed binary data with or without covariates, although both the multiplicative binomial and the double binomial models slightly behave better for these specific data sets. While this result may not be necessarily generalized to other variety of over and under-dispersed data, we would however, encourage the investigation of all possible models so that the right applicable model can be employed for a given data set under consideration. All analyses were carried out using PROC NLMIXED in SAS.