Table 5. Empirical type I error rates of F-distributed statistics and LRT statistics at different α-levels based on ${10}^6$ simulated data sets, when the causal variants are only rare.

			F-distributed statistics				LRT statistics
			Basis of both GVF and ${\beta }_{\ell }\left(t\right)$		Basis of β-smooth only		Basis of both GVF and ${\beta }_{\ell }\left(t\right)$		Basis of β-smooth only
Type of tests	Scenario	Level α	B-spline	Fourier	B-spline	Fourier	B-spline	Fourier	B-spline	Fourier
Het-F and Het-LRT	1	0.05	0.049876	0.049922	0.050093	0.049924	0.050611	0.050895	0.050819	0.050916
		0.01	0.009932	0.010006	0.009987	0.010029	0.010173	0.010407	0.010225	0.010422
		0.001	0.000991	0.000974	0.001000	0.000971	0.001055	0.001056	0.001065	0.001056
		0.0001	0.000089	0.000097	0.000091	0.000095	0.000094	0.000107	0.000097	0.000105
	2	0.05	0.049838	0.050189	0.050077	0.050194	0.050546	0.051163	0.050789	0.051164
		0.01	0.009944	0.009848	0.009998	0.009851	0.010239	0.010251	0.010305	0.010253
		0.001	0.001024	0.001021	0.001036	0.001025	0.001079	0.001082	0.001090	0.001088
		0.0001	0.000094	0.000101	0.000094	0.000102	0.000103	0.000118	0.000105	0.000118
	3	0.05	0.049886	0.050002	0.050081	0.049940	0.050593	0.050934	0.050789	0.050906
		0.01	0.009948	0.010084	0.009989	0.010090	0.010255	0.010454	0.010294	0.010446
		0.001	0.000981	0.001044	0.000985	0.001035	0.001029	0.001104	0.001033	0.001098
		0.0001	0.000106	0.000093	0.000108	0.000097	0.000116	0.000105	0.000118	0.000108
Hom-F and Hom-LRT	1	0.05	0.049834	0.049795	0.049948	0.049906	0.050131	0.050221	0.050240	0.050337
		0.01	0.009932	0.009901	0.009896	0.010018	0.010050	0.010062	0.010012	0.010216
		0.001	0.000987	0.001039	0.001030	0.000996	0.001000	0.001070	0.001054	0.001022
		0.0001	0.000091	0.000102	0.000077	0.000108	0.000098	0.000104	0.000078	0.000112
	2	0.05	0.050140	0.050340	0.050057	0.050050	0.050459	0.050784	0.050349	0.050475
		0.01	0.009995	0.010131	0.010001	0.009911	0.010103	0.010308	0.010141	0.010078
		0.001	0.000965	0.001029	0.000977	0.000998	0.000984	0.001061	0.001001	0.001031
		0.0001	0.000095	0.000106	0.000085	0.000092	0.000099	0.000111	0.000088	0.000097
	3	0.05	0.049900	0.049757	0.050173	0.049742	0.050201	0.050213	0.050453	0.050180
		0.01	0.010043	0.010068	0.010047	0.009950	0.010157	0.010260	0.010161	0.010138
		0.001	0.001025	0.001002	0.001010	0.001017	0.001045	0.001023	0.001035	0.001060
		0.0001	0.000090	0.000121	0.000098	0.000118	0.000092	0.000128	0.000100	0.000125

The results of “Basis of both GVF and ${\beta }_{\ell }\left(t\right)$” were based on smoothing both GVF and genetic effect functions ${\beta }_{\ell }\left(t\right)$ of model (3), and the results of “Basis of β-smooth only” were based on the smoothing ${\beta }_{\ell }\left(t\right)$ only approach of model (5). GVF, genetic variant function.