Accurate Likelihood-Based Limits for a Probability when the Risk Difference is Specified

For clinical trials that address questions of non-inferiority or superiority, the null hypothesis specifies a non-zero difference between response probabilities p₀ p 0 ( p₁ p 1 ) under control(treatment) regimens. A confidence interval for p₀ p 0 under this hypothesis provides a summary of our knowledge of the baseline probability p₀ p 0 , which can easily be converted into an interval for the treatment probability p₁ p 1 . When the hypothesised difference between p₀ p 0 and p₁ p 1 is zero, there is an exact interval for p₀ p 0 based on the total number of responses known as the Clopper-Pearson interval. Unfortunately, this construction does not extend to the hypothesis of non-zero difference. There is no quickly computed accurate method currently available. In this paper, we propose and investigate the properties of a conceptually simple and computationally feasible interval and demonstrate that it has reasonable one-sided and excellent two-sided coverage properties. This is not only useful as a summary of post-experimental knowledge, but is a key input into the testing procedure of Berger and Boos (1994).