TABLE 9.
Actual Statistical Power of Interaction Effect (Statistical Power = 0.7)
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Sample Size | Dist. | CPI | GAPI | UPI | LMS | |||||||||||
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ML | SB | ML | SB | ML | SB | |||||||||||
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Wald | LRT | Wald | LRT | Wald | LRT | Wald | LRT | Wald | LRT | Wald | LRT | Wald | LRT | |||
100 | Normal | 0.202 | 0.254 + | 0.285 + | 0.248 + | 0.131 # | 0.237 + | 0.187 | 0.211 + | 0.118 # | 0.242 + | 0.166 | 0.213 + | 0.202 # | 0.283 + | |
100 | Uniform | 0.124 | 0.164 + | 0.199 + | 0.180 + | 0.052 # | 0.147 + | 0.081 # | 0.155 + | 0.040 # | 0.155 + | 0.075 # | 0.163 + | 0.140 | 0.200 + | |
100 | K1 | 0.492 | 0.534 + | 0.538 + | 0.517 + | 0.411 # | 0.464 | 0.471 + | 0.332 + | 0.415 | 0.476 + | 0.484 + | 0.336 + | 0.458 # | 0.553 + | |
100 | K2 | 0.572 | 0.625 + | 0.609 + | 0.602 + | 0.468 # | 0.519 | 0.530 + | 0.400 | 0.458 | 0.504 + | 0.534 + | 0.401 + | 0.496 | 0.615 + | |
100 |
|
0.642 + | 0.682 + | 0.694 + | 0.691 + | 0.344 | 0.356 + | 0.385 | 0.419 + | 0.343 | 0.356 + | 0.407 + | 0.420 + | 0.564 + | 0.710 + | |
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200 | Normal | 0.205 | 0.233 | 0.248 + | 0.230 + | 0.175 | 0.229 | 0.196 | 0.187 | 0.159 # | 0.235 | 0.186 | 0.183 + | 0.256 | 0.308 | |
200 | Uniform | 0.146 | 0.174 | 0.205 + | 0.174 + | 0.105 # | 0.169 + | 0.123 | 0.162 + | 0.089 # | 0.178 + | 0.112 | 0.166 + | 0.165 | 0.186 + | |
200 | K1 | 0.582 + | 0.600 + | 0.536 + | 0.559 + | 0.552 | 0.564 | 0.576 + | 0.360 | 0.540 | 0.570 | 0.552 + | 0.371 | 0.571 | 0.620 + | |
200 | K2 | 0.655 + | 0.675 + | 0.594 + | 0.647 + | 0.605 | 0.634 | 0.617 + | 0.430 | 0.602 | 0.633 | 0.619 + | 0.438 + | 0.635 + | 0.683 + | |
200 |
|
0.798 + | 0.815 + | 0.795 + | 0.802 + | 0.403 | 0.405 | 0.428 | 0.460 | 0.420 | 0.429 | 0.447 + | 0.453 + | 0.819 + | 0.874 + | |
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500 | Normal | 0.245 # | 0.249 # | 0.255 # | 0.239 | 0.225 # | 0.242 | 0.226 # | 0.219 | 0.221 # | 0.248 | 0.215 # | 0.227 | 0.288 # | 0.301 # | |
500 | Uniform | 0.154 | 0.166 | 0.221 + | 0.177 | 0.153 | 0.173 | 0.153 | 0.166 | 0.142 | 0.173 | 0.148 | 0.163 | 0.192 | 0.202 | |
500 | K1 | 0.707 + | 0.709 + | 0.544 | 0.643 + | 0.693 | 0.698 | 0.693 + | 0.429 | 0.696 | 0.706 | 0.690 + | 0.424 | 0.740 + | 0.747 + | |
500 | K2 | 0.803 + | 0.807 + | 0.697 + | 0.766 + | 0.805 | 0.810 | 0.808 + | 0.505 | 0.813 | 0.819 | 0.805 + | 0.492 | 0.832 + | 0.846 + | |
500 |
|
0.963 + | 0.964 + | 0.924 + | 0.950 + | 0.532 + | 0.529 + | 0.523 + | 0.539 + | 0.546 | 0.538 + | 0.537 + | 0.497 | 0.987 + | 0.992 + | |
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1000 | Normal | 0.209 # | 0.213 # | 0.220 | 0.211 # | 0.218 | 0.221 | 0.206 | 0.197 # | 0.209 | 0.218 | 0.202 | 0.191 # | 0.284 | 0.295 | |
1000 | Uniform | 0.167 | 0.174 | 0.209 + | 0.178 | 0.167 | 0.173 | 0.158 | 0.163 | 0.164 | 0.177 | 0.155 | 0.166 | 0.203 | 0.207 | |
1000 | K1 | 0.778 + | 0.779 + | 0.570 + | 0.695 + | 0.786 | 0.787 | 0.769 + | 0.486 | 0.783 | 0.784 | 0.769 + | 0.480 | 0.790 + | 0.797 + | |
1000 | K2 | 0.871 + | 0.870 + | 0.706 + | 0.830 + | 0.878 | 0.878 | 0.873 + | 0.565 # | 0.881 | 0.881 | 0.881 + | 0.565 | 0.876 + | 0.879 + | |
1000 |
|
0.998 + | 0.998 + | 0.983 + | 0.993 + | 0.601 | 0.594 | 0.560 | 0.580 | 0.615 | 0.610 | 0.588 | 0.547 | 1.000 + | 1.000 + | |
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5000 | Normal | 0.263 | 0.265 | 0.267 | 0.263 | 0.263 | 0.265 | 0.250 | 0.245 | 0.265 | 0.265 | 0.249 | 0.245 | 0.325 | 0.326 | |
5000 | Uniform | 0.160 | 0.160 | 0.192 + | 0.164 | 0.157 | 0.161 | 0.151 | 0.152 | 0.158 | 0.162 | 0.146 | 0.148 | 0.194 | 0.195 | |
5000 | K1 | 0.878 + | 0.877 + | 0.574 | 0.810 + | 0.917 | 0.918 | 0.894 + | 0.694 # | 0.916 | 0.916 | 0.896 + | 0.679 | 0.890 + | 0.882 + | |
5000 | K2 | 0.926 + | 0.922 + | 0.662 + | 0.875 + | 0.969 + | 0.969 + | 0.959 + | 0.734 | 0.971 + | 0.971 + | 0.958 + | 0.719 | 0.937 + | 0.918 + | |
5000 |
|
1.000 + | 1.000 + | 1.000 + | 1.000 + | 0.643 + | 0.641 + | 0.580 + | 0.606 | 0.652 + | 0.651 + | 0.611 + | 0.601 | 1.000 + | 1.000 + |
Actual Statistical Power of Interaction Effect (Statistical Power = 0.9)
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Sample Size | Dist. | CPI | GAPI | UPI | LMS | |||||||||||
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ML | SB | ML | SB | ML | SB | |||||||||||
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Wald | LRT | Wald | LRT | Wald | LRT | Wald | LRT | Wald | LRT | Wald | LRT | Wald | LRT | |||
100 | Normal | 0.295 | 0.352 + | 0.374 + | 0.323 + | 0.206 # | 0.314 + | 0.248 | 0.262 + | 0.184 # | 0.319 + | 0.227 | 0.262 + | 0.346 # | 0.437 + | |
100 | Uniform | 0.185 | 0.241 + | 0.292 + | 0.251 + | 0.104 # | 0.228 + | 0.133 # | 0.228 + | 0.068 # | 0.224 + | 0.118 # | 0.231 + | 0.217 | 0.310 + | |
100 | K1 | 0.607 | 0.651 + | 0.634 + | 0.612 + | 0.525 # | 0.583 | 0.566 + | 0.447 + | 0.514 | 0.576 + | 0.549 + | 0.447 + | 0.576 # | 0.686 + | |
100 | K2 | 0.680 | 0.714 + | 0.688 + | 0.674 + | 0.564 # | 0.615 | 0.621 + | 0.477 | 0.567 | 0.613 + | 0.620 + | 0.466 + | 0.629 | 0.720 + | |
100 |
|
0.750 + | 0.780 + | 0.777 + | 0.762 + | 0.461 | 0.465 + | 0.485 | 0.518 + | 0.454 | 0.458 + | 0.504 + | 0.493 + | 0.700 + | 0.822 + | |
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200 | Normal | 0.331 | 0.356 | 0.389 + | 0.337 + | 0.286 | 0.356 | 0.319 | 0.313 | 0.265 # | 0.347 | 0.286 | 0.300 + | 0.387 | 0.440 | |
200 | Uniform | 0.227 | 0.251 | 0.309 + | 0.250 + | 0.163 # | 0.236 + | 0.179 | 0.227 + | 0.131 # | 0.247 + | 0.161 | 0.227 + | 0.259 | 0.297 + | |
200 | K1 | 0.734 + | 0.756 + | 0.679 + | 0.676 + | 0.702 | 0.733 | 0.708 + | 0.507 | 0.687 | 0.733 | 0.695 + | 0.513 | 0.752 | 0.787 + | |
200 | K2 | 0.787 + | 0.809 + | 0.720 + | 0.754 + | 0.727 | 0.753 | 0.736 + | 0.517 | 0.728 | 0.746 | 0.741 + | 0.545 + | 0.793 + | 0.826 + | |
200 |
|
0.890 + | 0.902 + | 0.866 + | 0.860 + | 0.537 | 0.529 | 0.525 | 0.569 | 0.547 | 0.556 | 0.547 + | 0.572 + | 0.907 + | 0.941 + | |
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500 | Normal | 0.376 # | 0.386 # | 0.383 # | 0.358 | 0.356 # | 0.373 | 0.349 # | 0.333 | 0.343 # | 0.376 | 0.342 # | 0.338 | 0.453 # | 0.476 # | |
500 | Uniform | 0.234 | 0.246 | 0.304 + | 0.253 | 0.232 | 0.257 | 0.227 | 0.235 | 0.221 | 0.259 | 0.217 | 0.233 | 0.288 | 0.308 | |
500 | K1 | 0.840 + | 0.848 + | 0.740 | 0.789 + | 0.827 | 0.834 | 0.824 + | 0.582 | 0.833 | 0.840 | 0.817 + | 0.575 | 0.873 + | 0.875 + | |
500 | K2 | 0.890 + | 0.893 + | 0.785 + | 0.858 + | 0.877 | 0.879 | 0.869 + | 0.640 | 0.883 | 0.891 | 0.874 + | 0.635 | 0.907 + | 0.912 + | |
500 |
|
0.986 + | 0.989 + | 0.963 + | 0.978 + | 0.693 + | 0.679 + | 0.656 + | 0.671 + | 0.703 | 0.696 + | 0.687 + | 0.652 | 0.994 + | 0.997 + | |
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1000 | Normal | 0.366 # | 0.369 # | 0.371 | 0.359 # | 0.350 | 0.362 | 0.339 | 0.328 # | 0.347 | 0.364 | 0.337 | 0.326 # | 0.454 | 0.470 | |
1000 | Uniform | 0.261 | 0.262 | 0.312 + | 0.273 | 0.253 | 0.260 | 0.239 | 0.250 | 0.247 | 0.262 | 0.237 | 0.251 | 0.314 | 0.320 | |
1000 | K1 | 0.908 + | 0.908 + | 0.797 + | 0.874 + | 0.909 | 0.909 | 0.888 + | 0.651 | 0.909 | 0.909 | 0.891 + | 0.636 | 0.924 + | 0.928 + | |
1000 | K2 | 0.956 + | 0.955 + | 0.866 + | 0.932 + | 0.950 | 0.951 | 0.942 + | 0.696 # | 0.958 | 0.958 | 0.945 + | 0.687 | 0.957 + | 0.958 + | |
1000 |
|
1.000 + | 1.000 + | 0.996 + | 0.997 + | 0.717 | 0.708 | 0.683 | 0.688 | 0.727 | 0.725 | 0.702 | 0.672 | 1.000 + | 1.000 + | |
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5000 | Normal | 0.381 | 0.382 | 0.379 | 0.376 | 0.369 | 0.371 | 0.357 | 0.354 | 0.368 | 0.370 | 0.356 | 0.350 | 0.495 | 0.494 | |
5000 | Uniform | 0.248 | 0.250 | 0.311 + | 0.263 | 0.257 | 0.261 | 0.244 | 0.244 | 0.257 | 0.262 | 0.245 | 0.245 | 0.317 | 0.316 | |
5000 | K1 | 0.954 + | 0.954 + | 0.788 | 0.927 + | 0.969 | 0.969 | 0.960 + | 0.786 # | 0.968 | 0.968 | 0.961 + | 0.784 | 0.963 + | 0.960 + | |
5000 | K2 | 0.989 + | 0.985 + | 0.888 + | 0.974 + | 0.995 + | 0.995 + | 0.992 + | 0.816 | 0.996 + | 0.996 + | 0.993 + | 0.803 | 0.988 + | 0.978 + | |
5000 |
|
1.000 + | 1.000 + | 1.000 + | 1.000 + | 0.813 + | 0.810 + | 0.765 + | 0.783 | 0.822 + | 0.819 + | 0.792 + | 0.779 | 1.000 + | 1.000 + |
Note. Two-tailed Wald test with nominal Type-I error rate = 0.05 (zcritical = ±1.96) was used to calculate the actual Type-I error rate. One-tailed likelihood ratio test with nominal Type-I error rate = 0.05 (χ2critical = 3.84) was used to calculate the actual Type-I error rate. Negative likelihood ratio test statistic by SB correction was treated as missing values.
means the actual Type-I error rate (Table 8) is smaller than the lower limit criterion 0.036.
means the actual Type-I error rate (Table 8) is larger than the upper limit criterion 0.064.
Note. Two-tailed Wald test with nominal Type-I error rate = 0.05 (zcritical = ±1.96) was used to calculate the actual Type-I error rate. One-tailed likelihood ratio test with nominal Type-I error rate = 0.05 (χ2critical = 3.84) was used to calculate the actual Type-I error rate. Negative likelihood ratio test statistic by SB correction was treated as missing values.
means the actual Type-I error rate (Table 8) is smaller than the lower limit criterion 0.036.
means the actual Type-I error rate (Table 8) is larger than the upper limit criterion 0.064.