Dealing with discreteness: making 'exact' confidence intervals for proportions,differences of proportions,and odds ratios more exact

'Exact' methods for categorical data are exact in terms of using probability distributions that do not depend on unknown parameters. However, they are conservative inferentially. The actual error probabilities for tests and confidence intervals are bounded above by the nominal level. This article examines the conservatism for interval estimation and describes ways of reducing it. We illustrate for confidence intervals for several basic parameters, including the binomial parameter, the difference between two binomial parameters for independent samples, and the odds ratio and relative risk. Less conservative behavior results from devices such as (1) inverting tests using statistics that are 'less discrete', (2) inverting a single two-sided test rather than two separate one-sided tests each having size at least half the nominal level, (3) using unconditional rather than conditional methods (where appropriate) and (4) inverting tests using alternative p-values. The article concludes with recommendations for selecting an interval in three situations-when one needs to guarantee a lower bound on a coverage probability, when it is sufficient to have actual coverage probability near the nominal level, and when teaching in a classroom or consulting environment.