TABLE 4.

Comparison of powers of the regression-based test and the Hotelling T² test under the third setting (runs:1000).

n	Test	Variance	The number of levels of Z
			K = 2		K = 3		K = 4		K = 5
			${\sigma }_X^2={\sigma }_Y^2$	${\sigma }_X^2\ne {\sigma }_Y^2$	${\sigma }_X^2={\sigma }_Y^2$	${\sigma }_X^2\ne {\sigma }_Y^2$	${\sigma }_X^2={\sigma }_Y^2$	${\sigma }_X^2\ne {\sigma }_Y^2$	${\sigma }_X^2={\sigma }_Y^2$	${\sigma }_X^2\ne {\sigma }_Y^2$
30	Regression	Approximate	0.300	0.272	0.121	0.103	0.230	0.197	0.326	0.342
		Bootstrapping	0.186	0.188	0.059	0.054	0.140	0.126	0.205	0.223
	Hotelling	Approximate	0.266	0.257	0.234	0.248	0.497	0.484	0.534	0.539
		Bootstrapping	0.156	0.155	0.110	0.104	0.279	0.279	0.273	0.302
60	Regression	Approximate	0.457	0.460	0.126	0.113	0.305	0.303	0.516	0.510
		Bootstrapping	0.381	0.387	0.091	0.094	0.238	0.234	0.452	0.436
	Hotelling	Approximate	0.443	0.449	0.386	0.382	0.741	0.729	0.780	0.785
		Bootstrapping	0.347	0.360	0.274	0.296	0.630	0.621	0.668	0.687
100	Regression	Approximate	0.688	0.644	0.162	0.172	0.458	0.451	0.737	0.705
		Bootstrapping	0.642	0.601	0.139	0.148	0.417	0.410	0.687	0.673
	Hotelling	Approximate	0.683	0.634	0.587	0.578	0.934	0.943	0.959	0.961
		Bootstrapping	0.626	0.586	0.516	0.514	0.902	0.909	0.934	0.939
150	Regression	Approximate	0.855	0.839	0.213	0.198	0.600	0.611	0.870	0.888
		Bootstrapping	0.831	0.822	0.204	0.174	0.585	0.595	0.848	0.866
	Hotelling	Approximate	0.852	0.836	0.759	0.745	0.986	0.993	0.998	0.996
		Bootstrapping	0.824	0.816	0.715	0.706	0.986	0.990	0.998	0.992

Note: The data were generated more generally, in which ${\theta }_k$s are different but do not follow any functional trend as the covariate Z varies.