Confidence intervals for difference in proportions for matched pairs compatible with exact McNemar's or sign tests

For testing with paired data (eg, twins randomized between two treatments), a simple test is the sign test, where we test if the distribution of the sign of the differences in responses between the two treatments within pairs is more often positive (favoring one treatment) or negative (favoring the other). When the responses are binary, this reduces to a McNemar-type test, and the calculations are the same. Although it is easy to calculate an exact P-value by conditioning on the total number of discordant pairs, the accompanying confidence interval on a parameter of interest (proportion positive minus proportion negative) is not straightforward. Effect estimates and confidence intervals are important for interpretation because it is possible that the treatment helps a very small proportion of the population yet gives a highly significant effect. We construct a confidence interval that is compatible with an exact sign test, meaning the 100 $(1-\alpha )\%$ interval excludes the null hypothesis of equality of proportions if and only if the associated exact sign test rejects at level $\alpha$ . We conjecture that the proposed confidence intervals guarantee nominal coverage, and we support that conjecture with extensive numerical calculations, but we have no mathematical proof to show guaranteed coverage. We have written and made available the function mcnemarExactDP in the exact2x2 R package and the function signTest in the asht R package to perform the methods described in this article.