ANCOVA: A global test based on a robust measure of location or quantiles when there is curvature

For two independent groups, let $M_j(x)$ be some conditional measure of location for the $j$th group associated with some random variable $Y$, given that some covariate $X=x$. When $M_j(x)$ is a robust measure of location, or even some conditional quantile of $Y$, given $X$, methods have been proposed and studied that are aimed at testing $H_0$: $M_1(x)=M_2(x)$ that deal with curvature in a flexible manner. In addition, methods have been studied where the goal is to control the probability of one or more Type I errors when testing $H_0$ for each $x \in \{x_1, \ldots, x_p\}$. This paper suggests a method for testing the global hypothesis $H_0$: $M_1(x)=M_2(x)$ for $\forall x \in \{x_1, \ldots, x_p\}$ when using a robust or quantile location estimator. An obvious advantage of testing $p$ hypotheses, rather than the global hypothesis, is that it can provide information about where regression lines differ and by how much. But the paper summarizes three general reasons to suspect that testing the global hypothesis can have more power. 2 Data from the Well Elderly 2 study illustrate that testing the global hypothesis can make a practical difference.