ϱ

MANOVA

PCA

GEE

TATES

FC\(\chi _{2m}^{2}\)

FCPermutation

FCPearson

FCKendall



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)

 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 (Traitbased association test involving the extended Simes procedure), FC\(\chi _{2m}^{2}\) (the chisquared distribution with 2m degrees of freedom under the independence assumption), FCPermutation (the permutation method based on 1,000 permutes), FCPearson (the proposed method with the correlation \(\hat {\rho }_{\text {\textit {j,k}}}\) being estimated by the Pearson’s sample correlation coefficient), and FCKendall (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 pvalue <0.05 among the 10,000 replications