Confidence interval (CI) construction with respect to proportion/rate difference for paired binary data has become a standard procedure in many clinical trials and medical studies. When the sample size is small and incomplete data are present, asymptotic CIs may be dubious and exact CIs are not yet available. In this article, we propose exact and approximate unconditional test-based methods for constructing CI for proportion/rate difference in the presence of incomplete paired binary data. Approaches based on one- and two-sided Wald's tests will be considered. Unlike asymptotic CI estimators, exact unconditional CI estimators always guarantee their coverage probabilities at or above the pre-specified confidence level. Our empirical studies further show that (i) approximate unconditional CI estimators usually yield shorter expected confidence width (ECW) with their coverage probabilities being well controlled around the pre-specified confidence level; and (ii) the ECWs of the unconditional two-sided-test-based CI estimators are generally narrower than those of the unconditional one-sided-test-based CI estimators. Moreover, ECWs of asymptotic CIs may not necessarily be narrower than those of two-sided-based exact unconditional CIs. Two real examples will be used to illustrate our methodologies.
We construct several explicit asymptotic two-sided confidence intervals (CIs) for the difference between two correlated proportions using the method of variance of estimates recovery (MOVER). The basic idea is to recover variance estimates required for the proportion difference from the confidence limits for single proportions. The CI estimators for a single proportion, which are incorporated with the MOVER, include the Agresti-Coull, the Wilson, and the Jeffreys CIs. Our simulation results show that the MOVER-type CIs based on the continuity corrected Phi coefficient and the Tango score CI perform satisfactory in small sample designs and spare data structures. We illustrate the proposed CIs with several real examples.
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