Smooth bootstrap-based confidence intervals for one binomial proportion and difference of two proportions

Constructing confidence intervals (CIs) for a binomial proportion and the difference between two binomial proportions is a fundamental and well-studied problem with respect to the analysis of binary data. In this note, we propose a new bootstrap procedure to estimate the CIs by resampling from a newly developed smooth quantile function in [11 Newcombe, R. G. 1998. Two-sided confidence intervals for the single proportion: Comparison of seven methods. Stat. Med., 17: 857–872. (doi:10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E)[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]] for discrete data. We perform a variety of simulation studies in order to illustrate the strong performance of our approach. The coverage probabilities of our CIs in the one-sample setting are superior than or comparable to other well-known approaches. The true utility of our new and novel approach is in the two-sample setting. For the difference of two proportions, our smooth bootstrap CIs provide better coverage probabilities almost uniformly over the interval (?1, 1), particularly in the tail region as compared than other published methods included in our simulation. We illustrate our methodology via an application to several different binary data sets.