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. Author manuscript; available in PMC: 2015 May 26.
Published in final edited form as: Genet Epidemiol. 2015 Mar 23;39(4):259–275. doi: 10.1002/gepi.21895

Table 4.

Empirical type I error rates of the approximate F-distribution tests based on Pillai–Bartlett trace, when the causal variants are both rare and common

Sample
size		 Nominal
level α		 Basis of both GVF and βℓ(t) Basis of beta-smooth only
Traits B-sp basis Fourier basis FPCA B-sp basis Fourier basis
(y1, y2, y3) 500 0.05 0.049276 0.049309 0.049354 0.049172 0.049258
0.01 0.009762 0.009775 0.009700 0.009746 0.009790
0.001 0.000932 0.000948 0.000916 0.000958 0.000954
0.0001 0.000084 0.000077 0.000075 0.000091 0.000076
1,000 0.05 0.049608 0.049845 0.049651 0.049669 0.049811
0.01 0.009775 0.009781 0.009818 0.009771 0.009812
0.001 0.000947 0.000963 0.001013 0.000943 0.000971
0.0001 0.000089 0.000102 0.000081 0.000090 0.000104
1,500 0.05 0.049501 0.050344 0.049806 0.049521 0.050273
0.01 0.009954 0.009984 0.009865 0.009942 0.009987
0.001 0.000988 0.000977 0.000962 0.000990 0.000993
0.0001 0.000107 0.000101 0.000093 0.000106 0.000095
2,000 0.05 0.049660 0.049636 0.049661 0.049672 0.049679
0.01 0.009869 0.010023 0.009904 0.009872 0.010014
0.001 0.000957 0.001042 0.000968 0.000956 0.001045
0.0001 0.000097 0.000107 0.000103 0.000097 0.000109
(y1, y2) 500 0.05 0.049599 0.049487 0.049746 0.049512 0.049432
0.01 0.009807 0.009784 0.009733 0.009821 0.009825
0.001 0.000982 0.000956 0.000945 0.000977 0.000978
0.0001 0.000109 0.000090 0.000080 0.000109 0.000091
1,000 0.05 0.049727 0.049620 0.050035 0.049759 0.049642
0.01 0.009847 0.009777 0.009820 0.009851 0.009731
0.001 0.001011 0.000976 0.000938 0.001010 0.000972
0.0001 0.000098 0.000126 0.000093 0.000099 0.000126
1,500 0.05 0.049868 0.049992 0.049918 0.049875 0.049984
0.01 0.010013 0.009943 0.009946 0.010005 0.009948
0.001 0.001009 0.000997 0.000966 0.001002 0.001002
0.0001 0.000089 0.000100 0.000096 0.000090 0.000099
2,000 0.05 0.049785 0.050148 0.050102 0.049811 0.050090
0.01 0.010006 0.009923 0.009908 0.009999 0.009932
0.001 0.001037 0.001016 0.000979 0.001036 0.001015
0.0001 0.000106 0.000094 0.000097 0.000106 0.000095
(y1, y3) 500 0.05 0.049691 0.049598 0.049754 0.049757 0.049519
0.01 0.009734 0.009874 0.009773 0.009755 0.009925
0.001 0.000914 0.000923 0.000976 0.000925 0.000920
0.0001 0.000095 0.000083 0.000089 0.000097 0.000089
1,000 0.05 0.049754 0.050020 0.049954 0.049743 0.050016
0.01 0.010023 0.010007 0.009983 0.010012 0.009965
0.001 0.000972 0.001001 0.001023 0.000976 0.001014
0.0001 0.000085 0.000096 0.000088 0.000083 0.000108
1,500 0.05 0.049688 0.050607 0.049659 0.049719 0.050506
0.01 0.009953 0.010013 0.009880 0.009956 0.010013
0.001 0.000966 0.001021 0.000992 0.000966 0.001013
0.0001 0.000100 0.000107 0.000103 0.000099 0.000105
2000 0.05 0.049685 0.049387 0.049816 0.049686 0.049378
0.01 0.009975 0.009861 0.009697 0.009965 0.009863
0.001 0.000972 0.000994 0.000991 0.000974 0.000987
0.0001 0.000082 0.000108 0.000101 0.000083 0.000106

Notes: The results of “Basis of both GVF and βℓ(t)” were based on smoothing both GVF and genetic effect functions βℓ(t) of model (7), the results of “FPCA approach” were based on the FPCA approach of model (8), and the results of “Basis of beta-smooth only” were based on the smoothing βℓ(t) only approach of model (3).