Unconditional Non‐Asymptotic One‐Sided Tests for Independent Binomial Proportions When the Interest Lies in Showing Non‐Inferiority and/or Superiority

For two independent binomial proportions B arnard (1947) has introduced a method to construct a non‐asymptotic unconditional test by maximisation of the probabilities over the ‘classical’ null hypothesis H 0 = {(θ 1 , θ 2 ) ∈ [0, 1] 2 : θ 1 = θ 2 }. It is shown that this method is also useful when studying test problems for different null hypotheses such as, for example, shifted null hypotheses of the form H 0 = {(θ 1 , θ 2 ) ∈ [0, 1] 2 : θ 2 ≤ θ 1 ± Δ } for non‐inferiority and 1‐sided superiority problems (including the classical null hypothesis with a 1‐sided alternative hypothesis). We will derive some results for the more general ‘shifted’ null hypotheses of the form H 0 = {(θ 1 , θ 2 ) ∈ [0, 1] 2 : θ 2 ≤ g (θ 1 )} where g is a non decreasing curvilinear function of θ 1 . Two examples for such null hypotheses in the regulatory setting are given. It is shown that the usual asymptotic approximations by the normal distribution may be quite unreliable. Non‐asymptotic unconditional tests (and the corresponding p ‐values) may, therefore, be an alternative, particularly because the effort to compute non‐asymptotic unconditional p ‐values for such more complex situations does not increase as compared to the classical situation. For ‘classical’ null hypotheses it is known that the number of possible p ‐values derived by the unconditional method is very large, albeit finite, and the same is true for the null hypotheses studied in this paper. In most of the situations investigated it becomes obvious that Barnard's CSM test (1947) when adapted to the respective null space is again a very powerful test. A theorem is provided which in addition to allowing fast algorithms to compute unconditional non‐asymptotical p ‐values fills a methodological gap in the calculation of exact unconditional p ‐values as it is implemented, for example, in S tat X act 3 for Windows (1995).