Testing general hypotheses under binomial sampling: the two sample caseasymptotic theory and exact procedures
A unified exact approach is given for testing hypotheses of the form H0:?1?h(?2) H 0 : ? 1 ? h ( ? 2 ) versus H1:?1<h(?2) H 1 : ? 1 < h ( ? 2 ) where ?1 ? 1 and ?2 ? 2 are the failure rates of two independent groups and h  is a monotone function. This includes the classical problem of testing equality of ?1 ? 1 and ?2 ? 2 as well as the problem of showing non-inferiority and superiority in clinical trials with respect to various commonly used measures, such as the difference, the relative risk or the odds ratio. Conditions on h are given in order to guarantee uniqueness of the restricted maximum likelihood estimator. Further, it is shown that the likelihood ratio test for H0 versus H1 follows asymptotically a 1 2 + 1 2 F ? 1 2 -law, provided h  is a smooth curve. However, it is found that this asymptotics often is not sufficiently reliable for small and moderate sample sizes, n1,n2?100 n 1 , n 2 ? 100 , say, and an exact modification will be given. This test is compared to competitors from the literature and its superiority with respect to power will be shown numerically. Procedures will be illustrated by a clinical trial.
non-inferiority superiority restricted maximum likelihood shifted null hypothesis unconditional test order-restricted inference exact test ordered alternatives equivalence testing