On O’Brien’s OLS and GLS Tests for Multiple Endpoints Brent R. Logan and Ajit C. Tamhane Division of Biostatistics Department of Statistics Medical College of Wisconsin Northwestern University 8701 Watertown Plank Rd. 2006 Sheridan Road Milwaukee, WI 53226, USA Evanston, IL 60208, USA E-mail: blogan@mcw.edu E-mail: ajit@iems.northwestern.edu Abstract: In this article we obtain some new results and extensions of the OLS and GLS tests proposed by O’Brien (1984) for the one-sided multivariate testing problem. In particular, we empirically obtain an accurate small sample approximation to the critical point of the OLS test. Next we show that a competing test proposed by La¨ uter (1996) is less powerful in general than the OLS test. Lastly, we extend the OLS and GLS tests to the heteroscedastic setup where the control and treatment populations have different covariance matrices. Keywords and Phrases: Clinical trials, One-sided multivariate test, Homoscedasic, Het- eroscedastic 1. Introduction Most clinical trials are conducted to compare a treatment group with a control group on multiple endpoints. Often, the treatment is expected to have a positive effect on all endpoints. O’Brien (1984) proposed two global tests, known as the ordinary least squares (OLS) and generalized least squares (GLS) tests, to demonstrate such an overall treatment effect. In this article we obtain some new results and extensions of these tests.