Testing equality of two beta binomial proportions in the presence of unequal extra-dispersion parameters

Data in the form of proportions with extra-dispersion (over/under) arise in many biomedical, epidemiological, and toxicological applications. In some situations, two samples of data in the form of proportions with extra-dispersion arise in which the problem is to test the equality of the proportions in the two groups with unspecified and possibly unequal extra-dispersion parameters. This problem is analogous to the traditional Behrens-Fisher problem in which two normal population means with possibly unequal variances are compared. To deal with this problem we develop eight tests and compare them in terms of empirical size and power, using a simulation study. Simulations show that a C() test based on extended quasi-likelihood estimates of the nuisance parameters holds nominal level most effectively (close to the nominal level) and it is at least as powerful as any other statistic that is not liberal. It has the simplest formula, is based on estimates of the nuisance parameters only under the null hypothesis, and is easiest to calculate. Also, it is robust in the sense that no distributional assumption is required to develop this statistic.