Table 2 shows the simulation results when the multivariate phenotypes come from a mixture of two multivariate normal distributions. In comparison to the corresponding settings under the multivariate normal distributions in Table 1, all the competing methods tended to have lower power under these long tailed distributions. Yet, Table 2 demonstrates similar patterns to the ones observed in Table 1 in general. Table 2Simulation results when the multivariate phenotypes come from a mixture of two multivariate normal distributions 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)′ 00.05350.05430.01350.04810.04870.04820.04610.0477(0.0023)(0.0023)(0.0012)(0.0021)(0.0022)(0.0021)(0.0021)(0.0021)0.250.05530.05140.07710.04960.06270.04650.04580.0469(0.0023)(0.0022)(0.0027)(0.0022)(0.0024)(0.0021)(0.0021)(0.0021)0.50.05370.05010.15050.05220.08950.04800.04910.0501(0.0023)(0.0022)(0.0036)(0.0022)(0.0029)(0.0021)(0.0022)(0.0022)0.750.05250.05380.22060.04810.12960.04930.05260.0513(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)′ 00.59430.32990.81720.56830.76770.76330.75950.7619(0.0049)(0.0047)(0.0039)(0.0050)(0.0042)(0.0043)(0.0043)(0.0043)0.250.30380.54140.74870.50030.67790.63300.63330.6332(0.0046)(0.0050)(0.0043)(0.0050)(0.0047)(0.0048)(0.0048)(0.0048)0.50.20730.39810.71350.44020.61680.49890.50830.5082(0.0041)(0.0049)(0.0045)(0.0050)(0.0049)(0.0050)(0.0050)(0.0050)0.750.16010.31350.68470.38700.57790.40380.41110.4116(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)′ 00.69720.40020.80870.73280.84510.84250.83790.8408(0.0046)(0.0049)(0.0039)(0.0044)(0.0036)(0.0036)(0.0037)(0.0037)0.250.47660.55790.74270.66980.76560.72690.72360.7259(0.0050)(0.0050)(0.0044)(0.0047)(0.0042)(0.0045)(0.0045)(0.0045)0.50.47280.40830.70730.62370.70360.57660.58550.5862(0.0050)(0.0049)(0.0046)(0.0048)(0.0046)(0.0049)(0.0049)(0.0049)0.750.65760.31720.67990.59760.63940.45320.46240.4617(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 (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),
