Testing Hypotheses about an Identified Treatment when there are Multiple Endpoints

Abstract

The problem of comparing an identified treatment with K other treatments is considered in a multivariate setting. Many formulations of composite alternative hypotheses are possible. For example, one might wish to examine whether the identified treatment is superior to the other treatments on all components of the response vector (i.e., is uniformly best) or whether the identified treatment is better than each treatment on at least one component (i.e., is admissible). For testing whether the identified treatment is uniformly best, the known optimality of the min test in the univariate case is extended to the multivariate case. If the distribution is multivariate normal, then the min test is shown to be a likelihood ratio test. For testing whether the identified treatment is admissible, a min test based on the Bonferroni inequality is suggested. For the multivariate normal with unknown covariance matrix, the likelihood ratio test is also a min test, but it has less stable power characteristics than does the Bonferroni-based test.

Key Words:

• Combination treatment
• Identified treatment
• Min test
• Multivariate
• Optimal