Consider the commonly occurring situation where the goal is to compare two independent groups and there are two covariates. Let Mj(X) be some conditional measure of location for the jth group associated with some random variable Y given X = (X1, X2). The goal is to H0: M1(X) = M2(X) for each X Ω in a manner that controls the probability of one or more Type I errors. An extant technique (method M1 here) addresses this goal without making any parametric assumption about Mj(X). However, a practical concern is that it does not provide enough detail regarding where the regression surfaces differ, due to using a very small number of covariate points, which can result in relatively low power. Method M2 was proposed for testing the global hypothesis H0: M1(X) = M2(X) for all X Ω, which offers a distinct power advantage over method M1. It uses the deepest half of the covariate points rather than small number of points used by method M1. However, method M2 does not provide any details about which covariate points yield a significant result. A multiple comparison procedure is proposed that deals with this shortcoming of method M2, and simultaneously it can provide higher power than method M1.
Wilcox, R. (2017). Robust ANCOVA: Confidence intervals that have some specified simultaneous probability coverage when there is curvature and two covariates. Journal of Modern Applied Statistical Methods, 16(1), 3-19. doi: 10.22237/jmasm/1493596800