Challenge of multiple co-primary endpoints: a new approach

There are many disorders where regulatory agencies have required a new treatment to demonstrate efficacy on multiple co-primary endpoints, all significant at the one-sided 2.5 per cent level, before accepting the treatment's effect for the disorder. This requirement, rooted in the intersection-union (IU) test, has led many researchers to increase the study sample size to make up for the reduction in the statistical power at the study level. Unfortunately, the increase in sample size could be substantial when the endpoints are minimally correlated and the treatment effects on the multiple endpoints are comparable. In this paper, we demonstrate that the frequentist concept of controlling the maximum false positive rate, even when applied to a restricted null space, has only limited success in keeping the sample size increase at a reasonable level. We therefore propose an approach that is based on the notion of controlling an average type I error rate. By employing an upper bound for the average type I error rate, the new approach provides an adjustment to the significance level that depends only on the correlation among the endpoints. For the most common case of two or three co-primary endpoints, the adjusted significance level is at most 5 per cent (one-sided) when the endpoints are moderately correlated. We show how sample size could be calculated under the proposed approach and contrast the needed sample size with that required under the IU test. We provide additional comments and discuss why the new approach is consistent with the principle requiring evidence of significance in the drug development and approval process.