Fully specified bootstrap confidence intervals for the difference of two independent binomial proportions based on the median unbiased estimator

Abstract

In studies in which a binary response for each subject is observed, the success probability and functions of this quantity are of interest. The use of confidence intervals has been increasingly encouraged as complementary to, and indeed preferable to, p-values as the primary expression of the impact of sampling uncertainty on the findings. The asymptotic confidence interval, based on a normal approximation, is often considered, but this interval can have poor statistical properties when the sample size is small and/or when the success probability is near 0 or 1. In this paper, an estimate of the risk difference based on median unbiased estimates (MUEs) of the two group probabilities is proposed. A corresponding confidence interval is derived using a fully specified bootstrap sample space. The proposed method is compared with Chen's quasi-exact method, Wald intervals and Agresti and Caffo's method with regard to mean square error and coverage probability. For a variety of settings, the MUE-based estimate of risk difference has mean square error uniformly smaller than maximum likelihood estimate within a certain range of risk difference. The fully specified bootstrap had better coverage probability in the tail area than Chen's quasi-exact method, Wald intervals and Agresti and Caffo's intervals.