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

For two independent groups, let $M_j(\mathbf{X})$ be some conditional measure of 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 to test $H_0$: $M_1(\mathbf{X})=M_2(\mathbf{X})$ for each $\mathbf{X} \in \Omega$ without making any parametric assumption about $M_j(\mathbf{X})$. But there are two general reasons to suspect that the method can have relatively low power. The paper reports simulation results on an alternative approach that is designed 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 offers a distinct power advantage. Using data from the Well Elderly 2 study, it is illustrated that the alternative method can make a practical difference in terms of detecting a difference between two groups.