  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.
