Table 2.

Results of the simulation study described in Section 5. Displayed are the empirical size (c = 0) and power (c = 2,…, 8) of the proposed garrote kernel machine (GKM) test using the two-way interaction kernel (K_Int), the Gaussian kernel (K_Gauss), the PCA based test statistics $T_n^{\ast }$ and $T_n^{\ast \ast }$ (K_Gauss,PCA1 and K_Gauss,PCA2) using the Gaussian kernel as in Section 3.3, and the usual F-test for different settings with n = 100 and M = 8. Also presented, for the linear cases (h₁(·) and h₂(·)), are the results for the ideal classical score tests assuming the direction of the alternative is known. The nominal level was set be α = 0.05.

	Test	r = 0.0					r = 0.2					r = 0.5
		c = 0	c = 2	c = 4	c = 6	c = 8	c = 0	c = 2	c = 4	c = 6	c = 8	c = 0	c = 2	c = 4	c = 6	c = 8
h1(·)	F-test	0.057	0.160	0.606	0.934	0.996	0.054	0.173	0.546	0.911	0.995	0.053	0.116	0.372	0.739	0.940
	KInt	0.053	0.198	0.658	0.952	0.993	0.052	0.233	0.691	0.954	0.997	0.055	0.149	0.429	0.773	0.948
	KGauss	0.050	0.402	0.943	1.000	1.000	0.051	0.443	0.938	0.998	1.000	0.051	0.332	0.803	0.983	0.999
	KGauss,PCA1	0.051	0.396	0.912	0.998	1.000	0.058	0.385	0.895	0.994	1.000	0.051	0.315	0.796	0.954	0.998
	KGauss,PCA2	0.051	0.406	0.922	0.998	1.000	0.054	0.405	0.904	0.996	1.000	0.051	0.321	0.799	0.961	0.991
	Score test	0.050	0.441	0.947	1.000	1.000	0.054	0.454	0.943	0.996	1.000	0.048	0.335	0.862	0.988	0.998
h2(·)	F-test	0.057	0.197	0.713	0.981	0.999	0.054	0.259	0.808	0.993	1.000	0.053	0.228	0.736	0.970	1.000
	KInt	0.053	0.223	0.758	0.967	0.998	0.052	0.431	0.916	0.998	1.000	0.055	0.453	0.93	0.998	1.000
	KGauss	0.050	0.398	0.921	0.995	0.999	0.051	0.456	0.918	0.995	0.998	0.051	0.469	0.816	0.983	0.998
	KGauss,PCA1	0.051	0.389	0.910	0.992	1.000	0.058	0.420	0.901	0.992	0.999	0.051	0.426	0.857	0.991	0.999
	KGauss,PCA2	0.051	0.407	0.912	0.997	1.000	0.054	0.445	0.914	0.991	1.000	0.051	0.421	0.849	0.983	0.992
	Score test	0.050	0.556	0.961	0.998	1.000	0.054	0.805	0.995	0.999	1.000	0.048	0.991	1.000	1.000	1.000
h3(·)	F-test	0.051	0.095	0.214	0.462	0.701	0.057	0.069	0.138	0.248	0.443	0.051	0.053	0.085	0.123	0.203
	KInt	0.045	0.099	0.234	0.513	0.758	0.046	0.103	0.213	0.380	0.612	0.043	0.069	0.093	0.167	0.280
	KGauss	0.052	0.174	0.535	0.848	0.968	0.050	0.196	0.474	0.748	0.931	0.051	0.155	0.332	0.559	0.740
	KGauss,PCA1	0.053	0.173	0.548	0.820	0.966	0.051	0.197	0.443	0.706	0.914	0.052	0.135	0.295	0.567	0.720
	KGauss,PCA2	0.053	0.171	0.534	0.833	0.971	0.051	0.189	0.451	0.712	0.921	0.052	0.143	0.292	0.541	0.731
h4(·)	F-test	0.051	0.068	0.162	0.360	0.662	0.057	0.069	0.115	0.244	0.414	0.051	0.060	0.09	0.132	0.219
	KInt	0.045	0.067	0.180	0.428	0.700	0.046	0.097	0.196	0.407	0.639	0.043	0.084	0.119	0.260	0.409
	KGauss	0.052	0.143	0.454	0.798	0.963	0.050	0.162	0.389	0.654	0.870	0.051	0.134	0.259	0.464	0.638
	KGauss,PCA1	0.053	0.143	0.402	0.744	0.947	0.051	0.142	0.344	0.603	0.828	0.052	0.116	0.234	0.415	0.602
	KGauss,PCA2	0.053	0.142	0.442	0.787	0.952	0.051	0.136	0.371	0.623	0.837	0.052	0.121	0.255	0.485	0.646