Table 1.
Exchangeable, low correlation |
Exchangeable, high correlation |
AR(1) correlation |
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PC | N | M | Q | MMR | MCM | KMR | SiMMR-D | SiMMR-PC(3) | MMR | MCM | KMR | SiMMR-D | SiMMR-PC(3) | MMR | MCM | KMR | SiMMR-D | SiMMR-PC(3) |
5 | 0.054 | 0.045 | 0.043 | 0.047 | 0.049 | 0.054 | 0.032 | 0.051 | 0.048 | 0.049 | 0.054 | 0.042 | 0.048 | 0.048 | 0.051 | |||
2 | 100 | 0.017 | 0.039 | 0.032 | 0.034 | 0.017 | 0.032 | 0.032 | 0.035 | 0.036 | 0.037 | 0.037 | 0.042 | |||||
5 | 0.034 | 0.062 | 0.053 | 0.042 | 0.014 | 0.058 | 0.051 | 0.05 | 0.032 | 0.05 | 0.048 | 0.033 | ||||||
25 | 10 | 100 | 0.007 | 0.051 | 0.044 | 0.052 | 0.005 | 0.047 | 0.041 | 0.054 | 0.05 | 0.042 | 0.041 | 0.04 | ||||
5 | 0.051 | 0.054 | 0.053 | 0.055 | 0.048 | 0.051 | 0.038 | 0.057 | 0.058 | 0.046 | 0.051 | 0.052 | 0.057 | 0.055 | 0.05 | |||
2 | 100 | 0.033 | 0.052 | 0.043 | 0.054 | 0.028 | 0.048 | 0.045 | 0.052 | 0.044 | 0.047 | 0.046 | 0.038 | |||||
5 | 0.04 | 0.039 | 0.05 | 0.045 | 0.052 | 0.04 | 0.013 | 0.053 | 0.052 | 0.046 | 0.04 | 0.029 | 0.05 | 0.05 | 0.06 | |||
None | 100 | 10 | 100 | 0.011 | 0.05 | 0.044 | 0.046 | 0.006 | 0.042 | 0.045 | 0.047 | 0.051 | 0.055 | 0.055 | 0.045 | |||
5 | 0.54 | 0.948 | 0.968 | 0.964 | 0.89 | 0.215 | 0.664 | 0.732 | 0.732 | 0.512 | 0.233 | 0.671 | 0.714 | 0.714 | 0.521 | |||
2 | 100 | 0.104 | 0.151 | 0.138 | 0.097 | 0.041 | 0.064 | 0.065 | 0.053 | 0.157 | 0.182 | 0.182 | 0.18 | |||||
5 | 0.328 | 0.498 | 0.475 | 0.32 | 0.086 | 0.195 | 0.194 | 0.131 | 0.21 | 0.347 | 0.349 | 0.237 | ||||||
25 | 10 | 100 | 0.019 | 0.074 | 0.061 | 0.06 | 0.005 | 0.052 | 0.053 | 0.051 | 0.084 | 0.083 | 0.084 | 0.092 | ||||
5 | 1 | 1 | 1 | 1 | 1 | 0.953 | 1 | 1 | 1 | 0.997 | 0.976 | 1 | 1 | 1 | 0.999 | |||
2 | 100 | 0.459 | 0.554 | 0.524 | 0.326 | 0.171 | 0.23 | 0.223 | 0.154 | 0.689 | 0.764 | 0.755 | 0.743 | |||||
5 | 0.253 | 0.946 | 0.982 | 0.98 | 0.928 | 0.092 | 0.448 | 0.652 | 0.646 | 0.476 | 0.185 | 0.81 | 0.929 | 0.928 | 0.871 | |||
1 | 100 | 10 | 100 | 0.052 | 0.161 | 0.145 | 0.095 | 0.018 | 0.084 | 0.084 | 0.057 | 0.248 | 0.418 | 0.411 | 0.311 | |||
5 | 0.103 | 0.125 | 0.127 | 0.132 | 0.138 | 0.196 | 0.066 | 0.083 | 0.086 | 0.25 | 0.078 | 0.081 | 0.065 | 0.068 | 0.125 | |||
2 | 100 | 0.083 | 0.129 | 0.113 | 0.889 | 0.025 | 0.048 | 0.044 | 1 | 0.151 | 0.23 | 0.23 | 0.154 | |||||
5 | 0.169 | 0.168 | 0.155 | 0.363 | 0.048 | 0.078 | 0.074 | 0.951 | 0.09 | 0.114 | 0.115 | 0.106 | ||||||
25 | 10 | 100 | 0.048 | 0.183 | 0.145 | 1 | 0.009 | 0.065 | 0.065 | 1 | 0.227 | 0.787 | 0.796 | 0.517 | ||||
5 | 0.488 | 0.407 | 0.465 | 0.458 | 0.416 | 0.931 | 0.177 | 0.2 | 0.201 | 0.809 | 0.241 | 0.291 | 0.231 | 0.231 | 0.418 | |||
2 | 100 | 0.987 | 1 | 0.997 | 1 | 0.127 | 0.196 | 0.188 | 1 | 0.951 | 1 | 1 | 0.304 | |||||
5 | 0.984 | 0.892 | 0.916 | 0.898 | 0.939 | 1 | 0.43 | 0.181 | 0.183 | 1 | 0.9 | 0.409 | 0.576 | 0.568 | 0.222 | |||
25% | 100 | 10 | 100 | 0.795 | 1 | 0.999 | 1 | 0.044 | 0.188 | 0.191 | 1 | 1 | 1 | 1 | 0.92 | |||
5 | 0.272 | 0.303 | 0.264 | 0.279 | 0.339 | 0.749 | 0.124 | 0.118 | 0.123 | 0.838 | 0.616 | 0.125 | 0.117 | 0.117 | 0.574 | |||
2 | 100 | 0.318 | 0.503 | 0.436 | 0.999 | 0.042 | 0.071 | 0.069 | 1 | 0.393 | 0.635 | 0.632 | 0.317 | |||||
5 | 0.37 | 0.44 | 0.414 | 0.818 | 0.084 | 0.116 | 0.11 | 1 | 0.136 | 0.238 | 0.233 | 0.187 | ||||||
25 | 10 | 100 | 0.194 | 0.577 | 0.492 | 1 | 0.013 | 0.091 | 0.084 | 1 | 0.554 | 1 | 1 | 0.954 | ||||
5 | 0.989 | 0.947 | 0.957 | 0.957 | 0.917 | 1 | 0.655 | 0.652 | 0.64 | 1 | 1 | 0.583 | 0.651 | 0.651 | 0.987 | |||
2 | 100 | 1 | 1 | 1 | 1 | 0.475 | 0.863 | 0.836 | 1 | 1 | 1 | 1 | 0.542 | |||||
5 | 1 | 1 | 1 | 1 | 1 | 1 | 0.788 | 0.81 | 0.772 | 1 | 1 | 0.959 | 1 | 1 | 0.322 | |||
50% | 100 | 10 | 100 | 1 | 1 | 1 | 1 | 0.174 | 0.882 | 0.857 | 1 | 1 | 1 | 1 | 1 |
Rejection rates are shown across varying number of subjects (N), number of modalities (M), number of features per modality (Q), and number of principal components included in the binary covariate effect (PC). The highest power among tests within each simulation setting is bolded. MMR, multivariate multiple regression using Pillai’s trace; MCM, multiple MDMR statistics after Bonferroni correction; KMR, kernel machine regression.