Exact confidence intervals for the difference of two proportions based on partially observed binary data

In a matched pairs experiment, two binary variables are typically observed on all subjects in the experiment. However, when one of the variables is missing on some subjects, we have so called the partially observed binary data that consist of two parts: a multinomial from the subjects with a pair of observed variables and two independent binomials from the subjects with only one observed variable. The goal of this paper is to construct exact confidence intervals for the difference of two (success) proportions of the two binary variables. We first derive a new test by combining two score tests for the two parts of the data and invert it to an asymptotic confidence interval. Since asymptotic intervals do not achieve the nominal level, this interval and three other existing intervals are improved to be exact by the general $h$-function method. We compare the infimum coverage probability and average interval length of these intervals and recommend the exact intervals that are improved from the newly proposed interval. Two real data sets are used to illustrate the intervals.