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. Author manuscript; available in PMC: 2014 Jan 17.
Published in final edited form as: Multivariate Behav Res. 2013 Jan 17;47(6):840–876. doi: 10.1080/00273171.2012.732901

TABLE 9.

Actual Statistical Power of Interaction Effect (Statistical Power = 0.7)
Sample Size Dist. CPI GAPI UPI LMS
ML SB ML SB ML SB
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
χ12
 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 +
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
χ12
 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 +
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
χ12
 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 +
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
χ12
 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 +
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
χ12
 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)
Sample Size Dist. CPI GAPI UPI LMS
ML SB ML SB ML SB
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
χ12
 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 +
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
χ12
 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 +
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
χ12
 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 +
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
χ12
 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 +
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
χ12
 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.