Exact confidence limits compatible with the result of a sequential trial

Abstract

Sequential (or adaptive) designs are common in acceptance sampling and pharmaceutical trials. This is because they can achieve the same type 1 and type 2 error rate with fewer subjects on average than fixed sample trials. After the trial is completed and the test result decided, we need full inference on the main parameter Δ. In this paper, we are interested in exact one-sided lower and upper limits.Unlike standard trials, for sequential trials there need not be an explicit test statistic, nor even p-value. This motivates the more general approach of defining an ordering on the sample space and using the construction of Buehler (1957). This is guaranteed to produce exact limits, however, there is no guarantee that the limits will agree with the test. For instance, we might reject Δ≤Δ0 at level α but have a lower 1−α limit being less then Δ0. This paper gives a very simple condition to ensure that this unfortunate feature does not occur. When the condition fails, the ordering is easily modified to ensure compatibility.