Summary: This paper considers the problem of comparing a new treatment with a control based on multiple endpoints. The hypotheses are formulated with the goal of showing that the treatment is equivalent, i.e., not inferior, on all endpoints and superior on at least one endpoint compared to the control, where thresholds for equivalence and superiority are specified for each endpoint. Roy’s union-intersection and Berger’s intersection-union principles are employed to derive the basic test. It is shown that the critical constants required for the union-intersection test of superiority can be sharpened by a careful analysis of its type I error rate. The composite UI-IU test is illustrated by an example and compared in a simulation study to alternative tests proposed by D. A. Bloch et al. [Biometrics 57, 1039–1047 (2001)] and M. D. Perlman and L. Wu [Biometrics 60, No. 1, 276–280 (2004; Zbl 1109.62048)]. The Bloch et al. test does not control the type I error rate because of its nonmonotone nature, and is hence not recommended. The UI-IU and the Perlman and Wu tests both control the type I error rate, but the latter test generally has a slightly higher power.