Skip to main content

Table 2 Simulation results when the multivariate phenotypes come from a mixture of two multivariate normal distributions

From: An efficient genome-wide association test for multivariate phenotypes based on the Fisher combination function

ϱ MANOVA PCA GEE TATES FC-\(\chi _{2m}^{2}\) FC-Permutation FC-Pearson FC-Kendall
  β=(0,0,0,0,0)
0 0.0535 0.0543 0.0135 0.0481 0.0487 0.0482 0.0461 0.0477
  (0.0023) (0.0023) (0.0012) (0.0021) (0.0022) (0.0021) (0.0021) (0.0021)
0.25 0.0553 0.0514 0.0771 0.0496 0.0627 0.0465 0.0458 0.0469
  (0.0023) (0.0022) (0.0027) (0.0022) (0.0024) (0.0021) (0.0021) (0.0021)
0.5 0.0537 0.0501 0.1505 0.0522 0.0895 0.0480 0.0491 0.0501
  (0.0023) (0.0022) (0.0036) (0.0022) (0.0029) (0.0021) (0.0022) (0.0022)
0.75 0.0525 0.0538 0.2206 0.0481 0.1296 0.0493 0.0526 0.0513
  (0.0022) (0.0023) (0.0041) (0.0021) (0.0034) (0.0022) (0.0022) (0.0022)
  β=(0.3,0.3,0.3,0.3,0.3)
0 0.5943 0.3299 0.8172 0.5683 0.7677 0.7633 0.7595 0.7619
  (0.0049) (0.0047) (0.0039) (0.0050) (0.0042) (0.0043) (0.0043) (0.0043)
0.25 0.3038 0.5414 0.7487 0.5003 0.6779 0.6330 0.6333 0.6332
  (0.0046) (0.0050) (0.0043) (0.0050) (0.0047) (0.0048) (0.0048) (0.0048)
0.5 0.2073 0.3981 0.7135 0.4402 0.6168 0.4989 0.5083 0.5082
  (0.0041) (0.0049) (0.0045) (0.0050) (0.0049) (0.0050) (0.0050) (0.0050)
0.75 0.1601 0.3135 0.6847 0.3870 0.5779 0.4038 0.4111 0.4116
  (0.0037) (0.0046) (0.0046) (0.0049) (0.0049) (0.0049) (0.0049) (0.0049)
  β=(0.1,0.2,0.3,0.4,0.5)
0 0.6972 0.4002 0.8087 0.7328 0.8451 0.8425 0.8379 0.8408
  (0.0046) (0.0049) (0.0039) (0.0044) (0.0036) (0.0036) (0.0037) (0.0037)
0.25 0.4766 0.5579 0.7427 0.6698 0.7656 0.7269 0.7236 0.7259
  (0.0050) (0.0050) (0.0044) (0.0047) (0.0042) (0.0045) (0.0045) (0.0045)
0.5 0.4728 0.4083 0.7073 0.6237 0.7036 0.5766 0.5855 0.5862
  (0.0050) (0.0049) (0.0046) (0.0048) (0.0046) (0.0049) (0.0049) (0.0049)
0.75 0.6576 0.3172 0.6799 0.5976 0.6394 0.4532 0.4624 0.4617
  (0.0047) (0.0047) (0.0047) (0.0049) (0.0048) (0.0050) (0.0050) (0.0050)
  1. The three different effect sizes are: no effect β=(0,0,0,0,0); moderate effects β=(0.3,0.3,0.3,0.3,0.3); and varied effects β=(0.1,0.2,0.3,0.4,0.5). The correlation between genes is ϱ ranging from 0 to 0.75. The competing methods are MANOVA (Multivariate analysis of variance), PCA (Principal component analysis), GEE (Generalized estimating equations), TATES (Trait-based association test involving the extended Simes procedure), FC-\(\chi _{2m}^{2}\) (the chi-squared distribution with 2m degrees of freedom under the independence assumption), FC-Permutation (the permutation method based on 1,000 permutes), FC-Pearson (the proposed method with the correlation \(\hat {\rho }_{\text {\textit {j,k}}}\) being estimated by the Pearson’s sample correlation coefficient), and FC-Kendall (the proposed method with \(\hat {\rho }_{\text {\textit {j,k}}}\) being estimated by the Kendall’s τ). The numbers in each cell are the mean (standard deviation) of the indicator variable for p-value <0.05 among the 10,000 replications