Table 1.

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 = 50 and M = 5. 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	0.049	0.142	0.409	0.723	0.940	0.042	0.109	0.356	0.703	0.931	0.047	0.095	0.223	0.494	0.765
	KInt	0.049	0.141	0.435	0.757	0.933	0.042	0.164	0.446	0.771	0.955	0.049	0.127	0.294	0.572	0.819
	KGauss	0.048	0.233	0.685	0.926	0.991	0.050	0.274	0.678	0.941	0.990	0.051	0.236	0.541	0.847	0.977
	KGauss,PCA1	0.049	0.228	0.665	0.915	0.993	0.055	0.267	0.661	0.927	0.990	0.050	0.197	0.502	0.794	0.948
	KGauss,PCA2	0.049	0.238	0.675	0.917	0.992	0.058	0.277	0.671	0.924	0.991	0.055	0.207	0.552	0.804	0.958
	Score test	0.054	0.279	0.726	0.940	0.993	0.054	0.255	0.685	0.925	0.994	0.051	0.221	0.594	0.864	0.970
h2(·)	F-test	0.049	0.143	0.417	0.793	0.945	0.042	0.134	0.446	0.794	0.963	0.047	0.107	0.332	0.661	0.881
	KInt	0.049	0.154	0.474	0.807	0.921	0.042	0.208	0.582	0.857	0.967	0.049	0.189	0.509	0.777	0.925
	KGauss	0.048	0.248	0.665	0.933	0.982	0.050	0.263	0.682	0.917	0.987	0.051	0.230	0.548	0.808	0.938
	KGauss,PCA1	0.049	0.242	0.658	0.920	0.988	0.055	0.278	0.658	0.909	0.982	0.050	0.227	0.580	0.828	0.945
	KGauss,PCA2	0.049	0.248	0.665	0.927	0.982	0.058	0.267	0.681	0.926	0.981	0.055	0.237	0.582	0.824	0.932
	Score test	0.054	0.354	0.881	0.972	0.991	0.054	0.453	0.889	0.983	0.997	0.051	0.637	0.954	0.996	1.000
h3(·)	F-test	0.058	0.083	0.178	0.356	0.544	0.049	0.081	0.137	0.249	0.404	0.056	0.059	0.077	0.149	0.217
	KInt	0.044	0.081	0.213	0.377	0.562	0.046	0.077	0.187	0.330	0.487	0.046	0.067	0.113	0.192	0.290
	KGauss	0.047	0.112	0.351	0.615	0.821	0.051	0.131	0.319	0.581	0.793	0.050	0.117	0.258	0.423	0.599
	KGauss,PCA1	0.052	0.128	0.341	0.583	0.801	0.052	0.132	0.325	0.545	0.749	0.050	0.107	0.232	0.380	0.539
	KGauss,PCA2	0.052	0.130	0.344	0.593	0.820	0.052	0.130	0.328	0.555	0.759	0.050	0.116	0.242	0.401	0.559
h4(·)	F-test	0.058	0.076	0.134	0.293	0.483	0.049	0.071	0.116	0.225	0.389	0.056	0.067	0.083	0.107	0.194
	KInt	0.044	0.066	0.145	0.314	0.527	0.046	0.082	0.152	0.301	0.493	0.046	0.074	0.100	0.155	0.279
	KGauss	0.047	0.100	0.268	0.571	0.804	0.051	0.122	0.274	0.540	0.734	0.050	0.108	0.225	0.357	0.510
	KGauss,PCA1	0.052	0.112	0.254	0.541	0.795	0.052	0.122	0.273	0.498	0.702	0.050	0.101	0.205	0.322	0.498
	KGauss,PCA2	0.052	0.111	0.264	0.551	0.799	0.052	0.121	0.264	0.501	0.713	0.050	0.102	0.209	0.337	0.508