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For two independent groups, let $M_j(\mathbf{X})$ be some conditional measureof location for the $j$th group associated with some random variable $Y$ given$\mathbf {X}=(X_1, X_2)$. Let $\Omega=\{\mathbf{X}_1, \ldots, \mathbf{X}_K\}$be a set of $K$ points to be determined. An extant technique can be used totest $H_0$:  $M_1(\mathbf{X})=M_2(\mathbf{X})$ for each $\mathbf{X} \in \Omega$ withoutmaking any parametric assumption about $M_j(\mathbf{X})$. But there are twogeneral reasons to suspect that the method can have relatively low power.  The paper reports simulation results on an alternative approach that isdesigned to test the global hypothesis $H_0$: $M_1(\mathbf{X})=M_2(\mathbf{X})$for all $\mathbf{X} \in \Omega$. The main result is that the new method offersa distinct power advantage.  Using data from the Well Elderly 2 study, it is illustrated that thealternative method can make a practical difference in terms of detecting adifference between two groups.

ANCOVA: A heteroscedastic global test when there is curvature and two covariates