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. Author manuscript; available in PMC: 2017 Aug 23.
Published in final edited form as: Genet Epidemiol. 2016 Jul 4;40(8):702–721. doi: 10.1002/gepi.21984

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

Region
size		 Sample
size		 Nominal
Level α 		Basis of both GVF and βℓ(t) Basis of beta-smooth only MANOVA
Model (1)
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
1,000 0.001 0.000996 0.000930 0.000998 0.000930 0.000918
0.0001 0.000096 0.000091 0.000096 0.000091 0.000089
1,500 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
1,000 0.001 0.000906 0.000969 0.000906 0.000969 0.000900
0.0001 0.000092 0.000086 0.000092 0.000086 0.000092
1,500 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
1,000 0.001 0.000905 0.000930 0.000905 0.000930 0.000946
0.0001 0.000094 0.000085 0.000094 0.000085 0.000094
1,500 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
1,000 0.001 0.000951 0.000946 0.000951 0.000946 0.000965
0.0001 0.000103 0.000094 0.000103 0.000094 0.000098
1,500 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
1,000 0.001 0.000979 0.000955 0.000979 0.000955 0.000946
0.0001 0.000093 0.000078 0.000093 0.000078 0.000105
1,500 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
1,000 0.001 0.000879 0.000940 0.000879 0.000940 0.001018
0.0001 0.000092 0.000095 0.000092 0.000095 0.000093
1,500 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
1,000 0.001 0.000968 0.000986 0.000968 0.000986 0.000975
0.0001 0.000085 0.000084 0.000085 0.000084 0.000085
1,500 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
1,000 0.001 0.000988 0.000974 0.000988 0.000974 0.000984
0.0001 0.000105 0.000106 0.000105 0.000106 0.000098
1,500 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
1,000 0.001 0.000953 0.000940 0.000953 0.000940 0.000938
0.0001 0.000109 0.000083 0.000109 0.000083 0.000104
1,500 0.001 0.000997 0.000940 0.000997 0.000940 0.000980
0.0001 0.000095 0.000098 0.000095 0.000098 0.000097

The results of ‘Basis of both GVF and βℓ(t)” were based on smoothing both GVF and genetic effect functions βℓ(t) of model (5), the results of ‘Basis of beta-smooth only” were based on the smoothing βℓ(t) only approach of model (7). The order of B-spline basis was 4, and the number of basis functions of B-spline was K = Kβ = 15; the number of Fourier basis functions was K = Kβ = 21.