Sequential clinical trials for normal variates using interval composite hypotheses

Sequential methods have become increasingly important for the monitoring of patient safety during clinical trials. However, the typical Wald sequential probability ratio test (SPRT), which compares two simple hypotheses, often presents anomalies which can be attributed to an inadequate representation of the parameter space. The use of composite null and alternative hypothesis in sequential clinical trials is explored and the resulting sequential rules are examined. It is shown that the SPRT and the Bayes formulations using Bayes odds ratios are equivalent in terms of the weighted likelihood ratio (WLR). The WLR is obtained for normal variates when the null hypothesis restricts the mean to (i) an interval and (ii) a point, in each case with complementary alternatives, as well as the one-sided formulation with a half-open interval. Applications to clinical trials include large-samples procedures, the comparative binomial trial and the comparison of survival distributions. Illustrative sequential boundaries are presented and the features of these different formulations are compared and discussed. Mixed sequential rules are considered within the framework for ethical stopping rules proposed by Meier (1979, Clinical Pharmacology and Therapeutics 25, 633--640).