ON THE DISTRIBUTION OF ROY’S LARGEST ROOT TEST IN MANOVA AND IN SIGNAL DETECTION IN NOISE By

Roy’s largest root is a common test in multivariate analysis of variance (MANOVA), with applications in several other problems, such as signal detection in noise. In this paper, assuming multivariate Gaussian observations, we derive a simple yet accurate approximation for the distribution of Roy’s largest root test, in the extreme case of concentrated non-centrality, where the signal or difference between groups is concentrated in a single direction. Our main result is that in the MANOVA setting, up to centering and scaling, Roy’s largest root test approximately follows a non-central F distribution whereas in the signal detection application, it approximately follows a modified central F distribution (of the form (s + χ2a)/χ 2 b). Our results allow power calculations for Roy’s test, as well as estimates of sample size required to detect a given (rank-one) group difference by this test, both of which are important quantities in hypothesis-driven research.