For testing independence between two sets of normally distributed variates we consider the class of test procedures which are invariant under certain groups of transformations and depend only on the sample canonical correlation coefficients . The power function of such a test depends only on the population canonical correlation coefficients as parameters, which may be regarded as measures of deviation from the hypothesis. In this paper sufficient conditions on the invariant procedure for the power function to be a monotonically increasing function of each of the parameters are obtained. The likelihood-ratio test  and Roy's maximum root test  satisfy these conditions. In  only the unbiasedness of the maximum root test was proved, although the authors claimed to prove the monotonicity property.
"Monotonicity of the Power Functions of Some Tests of Independence between Two Sets of Variates." Ann. Math. Statist. 35 (1) 206 - 208, March, 1964. https://doi.org/10.1214/aoms/1177703743