An exact confidence set for two binomial proportions and exact unconditional confidence intervals for the difference and ratio of proportions

An exact joint confidence set is proposed for two binomial parameters estimated from independent samples. Its construction relies on inverting the minimum volume test, a two-dimensional analogue of Sterne’s test for a single probability. The algorithm involves computer-intensive exact computation based on binomial probabilities. The proposed confidence set has good coverage properties and it performs much better than the likelihood-based confidence set for the same problem. Applying the principle of intersection-union tests, the method can be used to derive exact tests and confidence intervals for functions of the two binomial parameters. Based on this, new exact unconditional two-sided confidence intervals are proposed for the risk difference and risk ratio. The performance of the new intervals is comparable to that of certain well-known confidence intervals in small samples. Extension of the methods described to two hypergeometric or two Poisson variables is straightforward.