TABLE 3.

Comparison of powers of the regression-based test and the Hotelling T² test under the second 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	0.114	0.114	0.288	0.256	0.510	0.511	0.765	0.773
		Bootstrapping	0.074	0.075	0.199	0.188	0.402	0.415	0.679	0.687
	Hotelling	Approximate	0.112	0.109	0.231	0.240	0.390	0.415	0.608	0.636
		Bootstrapping	0.069	0.072	0.146	0.144	0.228	0.240	0.401	0.428
60	Regression	Approximate	0.151	0.144	0.383	0.415	0.761	0.796	0.959	0.956
		Bootstrapping	0.119	0.120	0.330	0.363	0.721	0.743	0.948	0.938
	Hotelling	Approximate	0.149	0.141	0.325	0.336	0.620	0.639	0.886	0.859
		Bootstrapping	0.120	0.117	0.257	0.274	0.538	0.559	0.821	0.784
100	Regression	Approximate	0.210	0.224	0.598	0.614	0.955	0.950	1.000	1.000
		Bootstrapping	0.183	0.191	0.565	0.569	0.935	0.940	1.000	1.000
	Hotelling	Approximate	0.208	0.221	0.534	0.518	0.850	0.880	1.000	1.000
		Bootstrapping	0.176	0.186	0.463	0.452	0.790	0.810	0.985	1.000
150	Regression	Approximate	0.225	0.235	0.770	0.782	0.980	0.982	1.000	1.000
		Bootstrapping	0.190	0.200	0.760	0.768	0.980	0.983	1.000	1.000
	Hotelling	Approximate	0.225	0.230	0.705	0.712	0.965	0.965	1.000	1.000
		Bootstrapping	0.190	0.200	0.650	0.635	0.950	0.960	1.000	1.000

Note: The data were generated under the assumptions of regression analysis with a logit link function, in which $\log \left({\theta }_k=\left(1+{\theta }_k\right)\right)$ increases by the same amount as the level of Z is increased by one unit.