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1. Summary and introduction. The largest characteristic root has been proposed in [2] as a test statistic in (i) the multivariate analysis of variance test, and (ii) testing that two sets of variates are independent. In this paper it is shown that, in each case, the power function is a monotonically increasing function of each non-centrality parameter, separately. This property was stated in [2] without proof. This provides a stronger result than would be obtained by any direct use of Anderson's Theorem [1] which implies that the power function increases when all the roots are simultaneously increased in the same ratio. The proof of the monotonocity property for the multivariate analysis of variance is given in Section 3, and in Section 4 it is shown how the proof is modified for testing independence between two sets of variates. 2. Preliminaries for the multivariate analysis of variance situation. Let u, s and n denote respectively the "effective" number of variates, the degrees of freedom of the hypothesis and the degrees of freedom for the error and let t = min (u, s). Then, with X _ [xij]:u X s and Y- [yij]:u X n, the canonical form for the d.f. in the multivariate analysis of variance model was obtained in [2, p. 86] to be

On the Monotonic Character of the Power Functions of Two Multivariate Tests