Problems with existing procedures to calculate exact unconditional P-values for non-inferiority/superiority and confidence intervals for two binomials and how to resolve them

Recently several papers have been published that deal with the construction of exact unconditional tests for non-inferiority and confidence intervals based on the approximative unconditional restricted maximum likelihood test for two binomial random variables. Soon after the papers have been published the commercially available software for exact tests StatXact has incorporated the new methods. There are however gaps in the proofs which since have not been resolved adequately. Further it turned out that the methods for testing non-inferiority are not coherent and test for non-inferiority can easily come to different conclusions compared to the confidence interval inclusion rule. In this paper, a proposal is made how to resolve the open problems. Berger and Boos (1994) developed the confidence interval method for testing equality of two proportions. StatXact (Version 5) has extended this method for shifted hypotheses. It is shown that at least for unbalanced designs (i.e. largely different sample sizes) the Berger and Boos method can lead to controversial results.