Table 2.

Empirical Type I Error Rates of the Approximate F-distribution Tests based on Pillai-Bartlett Trace of Six Traits and Moderate Correlation, When the Variants Are Only Rare. 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 (5), the results of “Basis of beta-Smooth Only” were based on the smoothing ${\beta }_{\ell }\left(t\right)$ only approach of model (7). The order of B-spline basis was 4, and the number of basis functions of B-spline was K = K_β = 15; the number of Fourier basis functions was K = K_β = 21.

Region Size	Sample Size	Nominal	Basis of both GVF and ${\beta }_{\ell }\left(t\right)$		Basis of beta-Smooth Only		MANOVA Model (1)
		Level α	B-sp Basis	Fourier Basis	B-sp Basis	Fourier Basis
6 kb	500	0.001	0.000906	0.000919	0.000906	0.000916	0.000921
		0.0001	0.000091	0.000088	0.000093	0.000090	0.000085
	1000	0.001	0.000996	0.000930	0.000998	0.000930	0.000918
		0.0001	0.000096	0.000091	0.000096	0.000091	0.000089
	1500	0.001	0.000985	0.000991	0.000985	0.000991	0.000984
		0.0001	0.000094	0.000099	0.000094	0.000099	0.000095
9 kb	500	0.001	0.000940	0.000925	0.000940	0.000923	0.000912
		0.0001	0.000090	0.000095	0.000090	0.000095	0.000100
	1000	0.001	0.000906	0.000969	0.000906	0.000969	0.000900
		0.0001	0.000092	0.000086	0.000092	0.000086	0.000092
	1500	0.001	0.000981	0.000980	0.000981	0.000980	0.000952
		0.0001	0.000111	0.000091	0.000111	0.000091	0.000076
12 kb	500	0.001	0.000930	0.000901	0.000930	0.000901	0.000909
		0.0001	0.000086	0.000089	0.000086	0.000089	0.000078
	1000	0.001	0.000905	0.000930	0.000905	0.000930	0.000946
		0.0001	0.000094	0.000085	0.000094	0.000085	0.000094
	1500	0.001	0.000965	0.000983	0.000965	0.000983	0.000984
		0.0001	0.000099	0.000099	0.000099	0.000099	0.000097
15 kb	500	0.001	0.000950	0.000947	0.000950	0.000947	0.000940
		0.0001	0.000093	0.000099	0.000093	0.000099	0.000093
	1000	0.001	0.000951	0.000946	0.000951	0.000946	0.000965
		0.0001	0.000103	0.000094	0.000103	0.000094	0.000098
	1500	0.001	0.000925	0.000966	0.000925	0.000966	0.000987
		0.0001	0.000098	0.000089	0.000098	0.000089	0.000104
18 kb	500	0.001	0.000896	0.000957	0.000896	0.000957	0.000913
		0.0001	0.000077	0.000088	0.000077	0.000088	0.000105
	1000	0.001	0.000979	0.000955	0.000979	0.000955	0.000946
		0.0001	0.000093	0.000078	0.000093	0.000078	0.000105
	1500	0.001	0.000969	0.000985	0.000969	0.000985	0.000962
		0.0001	0.000083	0.000114	0.000083	0.000114	0.000105
21 kb	500	0.001	0.000888	0.000929	0.000888	0.000929	0.000936
		0.0001	0.000086	0.000085	0.000086	0.000085	0.000077
	1000	0.001	0.000879	0.000940	0.000879	0.000940	0.001018
		0.0001	0.000092	0.000095	0.000092	0.000095	0.000093
	1500	0.001	0.000919	0.000932	0.000919	0.000932	0.000989
		0.0001	0.000086	0.000079	0.000086	0.000079	0.000086
24 kb	500	0.001	0.000943	0.000846	0.000943	0.000846	0.000931
		0.0001	0.000087	0.000091	0.000087	0.000091	0.000076
	1000	0.001	0.000968	0.000986	0.000968	0.000986	0.000975
		0.0001	0.000085	0.000084	0.000085	0.000084	0.000085
	1500	0.001	0.000989	0.000990	0.000989	0.000990	0.001014
		0.0001	0.000110	0.000096	0.000110	0.000096	0.000090
27 kb	500	0.001	0.000935	0.000960	0.000935	0.000960	0.000946
		0.0001	0.000105	0.000107	0.000105	0.000107	0.000092
	1000	0.001	0.000988	0.000974	0.000988	0.000974	0.000984
		0.0001	0.000105	0.000106	0.000105	0.000106	0.000098
	1500	0.001	0.000999	0.000993	0.000999	0.000993	0.000966
		0.0001	0.000097	0.000113	0.000097	0.000113	0.000097
30 kb	500	0.001	0.000900	0.000916	0.000900	0.000916	0.000942
		0.0001	0.000069	0.000082	0.000069	0.000082	0.000083
	1000	0.001	0.000953	0.000940	0.000953	0.000940	0.000938
		0.0001	0.000109	0.000083	0.000109	0.000083	0.000104
	1500	0.001	0.000997	0.000940	0.000997	0.000940	0.000980
		0.0001	0.000095	0.000098	0.000095	0.000098	0.000097