compare these two methods with the other methods in terms of the statistical power. Table 1Simulation results when the multivariate phenotypes come from a multivariate normal distribution ϱ MANOVAPCAGEETATESFC-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\chi _{2m}^{2}$\end{document}χ2m2 FC-PermutationFC-PearsonFC-Kendall β=(0,0,0,0,0)′ 00.04770.05140.01090.04870.04680.04550.04550.0451(0.0021)(0.0022)(0.0010)(0.0022)(0.0021)(0.0021)(0.0021)(0.0021)0.250.04770.04990.07630.04980.06310.04880.04820.0477(0.0021)(0.0022)(0.0027)(0.0022)(0.0024)(0.0022)(0.0021)(0.0021)0.50.04770.04960.15180.05060.09420.04730.04820.0484(0.0021)(0.0022)(0.0036)(0.0022)(0.0029)(0.0021)(0.0021)(0.0021)0.750.04770.04960.22020.04940.12630.04670.04890.0485(0.0021)(0.0022)(0.0041)(0.0022)(0.0033)(0.0021)(0.0022)(0.0021) β=(0.3,0.3,0.3,0.3,0.3)′ 00.75950.56790.93330.73590.90670.90580.90470.9040(0.0043)(0.0050)(0.0025)(0.0044)(0.0029)(0.0029)(0.0029)(0.0029)0.250.40860.70750.85700.64060.80760.77480.77490.7749(0.0049)(0.0045)(0.0035)(0.0048)(0.0039)(0.0042)(0.0042)(0.0042)0.50.26550.52950.81130.56680.74110.63140.64200.6421(0.0044)(0.0050)(0.0039)(0.0050)(0.0044)(0.0048)(0.0048)(0.0048)0.750.20110.41440.78270.49490.69270.51690.52720.5278(0.0040)(0.0049)(0.0041)(0.0050)(0.0046)(0.0050)(0.0050)(0.0050) β=(0.1,0.2,0.3,0.4,0.5)′ 00.85500.66460.92720.87310.94570.94540.94480.9445(0.0035)(0.0047)(0.0026)(0.0033)(0.0023)(0.0023)(0.0023)(0.0023)0.250.63340.72430.85000.82370.88640.86040.86310.8621(0.0048)(0.0045)(0.0036)(0.0038)(0.0032)(0.0035)(0.0034)(0.0034)0.50.62030.54370.80430.77580.82830.72520.73340.7333(0.0049)(0.0050)(0.0040)(0.0042)(0.0038)(0.0045)(0.0044)(0.0044)0.750.81770.42270.77560.75120.77210.58210.59420.5941(0.0039)(0.0049)(0.0042)(0.0043)(0.0042)(0.0049)(0.0049)(0.0049)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-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\chi _{2m}^{2}$\end{document}χ2m2 (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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat {\rho }_{\text {\textit {j,k}}}$\end{document}ρ^j,k being estimated by the Pearson’s sample correlation coefficient), and FC-Kendall (the proposed method with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat {\rho }_{\text {\textit {j,k}}}$\end{document}ρ^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
