Name: Tamhane

Firstname: Ajit

Title: Professor

Institution: Northwestern University

City: Evanston, IL

Zip-Code: 60208-3119

Country: USA

Phone: 847-491-3577

Fax: 847-491-8005

Email: ajit@iems.northwestern.edu

Authors: Ajit C. Tamhane and Brent Logan

Title: A superiority-equivalence approach to one-sided tests on multiple endpoints

Abstract: When comparing a treatment with a passive control on multiple primary endpoints, it is often of interest to test a one-sided alternative hypothesis on the difference between the mean vectors for the treatment and the control. Exact and approximate likelihood ratio (LR) tests have been proposed by various authors (Kud\^{o} 1963, Perlman 1969, Tang, Gnecco and Geller 1989). However, these tests are not unbiased and monotone when the endpoints are positively correlated (Silvapulle 1997). The cone order montone test approach of Cohen and Sackrowitz (1998) does not have these drawbacks, but still has an undesirable feature in that it rejects the null hypothesis even if there are large negative differences between the means of the treatment and the control for some endpoints as long as the differences for other endpoints are sufficiently large and positive. To avoid these difficulties, we propose an approach that requires the treatment to be superior than the control ! on a specified number of endpoints and equivalent (non-inferior) on the others. Tests are derived using a combination of the union-intersection (UI) and intersection-union (IU) principles. It is shown that critical constants required by the UI-IU approach can be sharpened in some cases. Simulation results for type I error and power are given.

References: 1. Cohen, A. and Sackrowitz, H. B. (1998). Directional tests for one-sided alternatives in multivariate models. {\em Annals of Statistics} {\bf 26}, 2321?-2338. 2. Kud\^{o}, A. (1963). A multivariate analogue of the one-sided test. {\em Biometrika} {\bf 50}, 403-418. 3. Perlman, M. D. (1969). One-sided testing problems in multivariate analysis. {\em Annals of Mathematical Statistics} {\bf 40}, 549-567. 4. Silvapulle, M. J. (1997). A curious example involving the likelihood ratio test against one-sided hypotheses. {\em The American Statistician} {\bf 51}, 178?-180. 5.Tang, D. I., Gnecco, C., and Geller, N. L. (1989). An approximate likelihood ratio test for a normal mean vector with nonnegative components with application to clinical trials. {\em Biometrika} {\bf 76}, 577-583.