Adjusting OBriens Test to Control Type I Error for the Generalized Nonparametric Behrens-Fisher Problem

OʹBrien (1984, Biometrics40, 1079-1087) introduced a simple nonparametric test procedure for testing whether multiple outcomes in one treatment group have consistently larger values than outcomes in the other treatment group. We first explore the theoretical properties of OʹBrienʹs test. We then extend it to the general nonparametric Behrens–Fisher hypothesis problem when no assumption is made regarding the shape of the distributions. We provide conditions when OʹBrienʹs test controls its error probability asymptotically and when it fails. We also provide adjusted tests when the conditions do not hold. Throughout this article, we do not assume that all outcomes are continuous. Simulations are performed to compare the adjusted tests to OʹBrienʹs test. The difference is also illustrated using data from a Parkinsonʹs disease clinical trial.