In this paper we give further reduction for a general MANOVA problem, and show that the general MANOVA problem can be treated as the MANOVA problem with the conditional setup. The last result is used to prove that the power functions of some tests in the general MANOVA models are monotone increasing functions of each of the noncentrality parameters, and, in particular, the likelihood-ratio test has the monotonicity property.
"Monotonicity of the Power Functions of Some Tests in General Manova Models." Ann. Statist. 1 (2) 388 - 391, March, 1973. https://doi.org/10.1214/aos/1176342379