Confidence interval of the difference between two proportions with overdispersion

In confidence interval estimation of the difference between two proportions with overdispersion due to positive correlations, the usual asymptotic normality-based method generally has lower coverage rates than desired, especially when sample size is moderate. Applying the concept of effective sample size to existing methods for independent data, we propose three new asymptotic normality-based methods. It is demonstrated through an extensive Monte Carlo study that the proposed methods generally perform better than the usual method. The proposed methods are illustrated in the application to a motivating data example.