Background
Methods
Multivariate permutation tests for survival matched data
Results
Simulations
Scenario* | Strata effect (w) | N. of strata (k) | TSi
1
| TLSi
1
| ||||||
---|---|---|---|---|---|---|---|---|---|---|
4 fixed | 8 fixed | perc | all | 4 fixed | 8 fixed | perc | all | |||
A) Fisher test
| ||||||||||
PH
| 0 | 30 | 0.047 | 0.046 | 0.053 | 0.046 | 0.046 | 0.052 | 0.055 | 0.043 |
0 | 50 | 0.048 | 0.041 | 0.041 | 0.049 | 0.052 | 0.047 | 0.051 | 0.045 | |
0 | 100 | 0.044 | 0.045 | 0.043 | 0.044 | 0.047 | 0.038 | 0.045 | 0.045 | |
0.22 | 30 | 0.048 | 0.047 | 0.055 | 0.050 | 0.047 | 0.052 | 0.058 | 0.053 | |
0.22 | 50 | 0.050 | 0.047 | 0.048 | 0.050 | 0.051 | 0.051 | 0.053 | 0.048 | |
0.22 | 100 | 0.056 | 0.051 | 0.052 | 0.049 | 0.054 | 0.056 | 0.051 | 0.053 | |
ED
| 0 | 30 | 0.057 | 0.056 | 0.062 | 0.062 | 0.065 | 0.061 | 0.055 | 0.054 |
0 | 50 | 0.064 | 0.063 | 0.061 | 0.061 | 0.066 | 0.062 | 0.057 | 0.049 | |
0 | 100 | 0.052 | 0.050 | 0.053 | 0.056 | 0.050 | 0.049 | 0.049 | 0.053 | |
0.22 | 30 | 0.049 | 0.056 | 0.057 | 0.058 | 0.049 | 0.052 | 0.054 | 0.050 | |
0.22 | 50 | 0.038 | 0.040 | 0.043 | 0.042 | 0.037 | 0.039 | 0.044 | 0.048 | |
0.22 | 100 | 0.048 | 0.049 | 0.045 | 0.041 | 0.049 | 0.052 | 0.041 | 0.042 | |
LD
| 0 | 30 | 0.047 | 0.047 | 0.051 | 0.055 | 0.046 | 0.050 | 0.056 | 0.053 |
0 | 50 | 0.053 | 0.054 | 0.055 | 0.056 | 0.060 | 0.059 | 0.052 | 0.054 | |
0 | 100 | 0.046 | 0.048 | 0.053 | 0.050 | 0.049 | 0.050 | 0.050 | 0.049 | |
0.22 | 30 | 0.062 | 0.063 | 0.066 | 0.063 | 0.063 | 0.065 | 0.054 | 0.062 | |
0.22 | 50 | 0.059 | 0.057 | 0.056 | 0.057 | 0.054 | 0.054 | 0.056 | 0.054 | |
0.22 | 100 | 0.054 | 0.054 | 0.050 | 0.050 | 0.055 | 0.055 | 0.048 | 0.050 | |
CH
| 0 | 30 | 0.052 | 0.049 | 0.050 | 0.052 | 0.059 | 0.051 | 0.052 | 0.057 |
0 | 50 | 0.051 | 0.050 | 0.052 | 0.048 | 0.051 | 0.047 | 0.053 | 0.055 | |
0 | 100 | 0.049 | 0.046 | 0.038 | 0.037 | 0.050 | 0.051 | 0.043 | 0.047 | |
0.22 | 30 | 0.050 | 0.044 | 0.053 | 0.054 | 0.053 | 0.049 | 0.055 | 0.053 | |
0.22 | 50 | 0.043 | 0.040 | 0.040 | 0.038 | 0.044 | 0.041 | 0.049 | 0.048 | |
0.22 | 100 | 0.054 | 0.060 | 0.063 | 0.057 | 0.061 | 0.059 | 0.052 | 0.055 | |
B) Tippett test
| ||||||||||
PH
| 0 | 30 | 0.042 | 0.054 | 0.053 | 0.061 | 0.043 | 0.067 | 0.056 | 0.059 |
0 | 50 | 0.041 | 0.048 | 0.043 | 0.047 | 0.050 | 0.056 | 0.053 | 0.049 | |
0 | 100 | 0.049 | 0.051 | 0.058 | 0.051 | 0.047 | 0.050 | 0.051 | 0.053 | |
0.22 | 30 | 0.057 | 0.059 | 0.058 | 0.065 | 0.060 | 0.069 | 0.053 | 0.071 | |
0.22 | 50 | 0.047 | 0.051 | 0.058 | 0.064 | 0.047 | 0.052 | 0.055 | 0.057 | |
0.22 | 100 | 0.051 | 0.052 | 0.056 | 0.059 | 0.055 | 0.058 | 0.050 | 0.054 | |
ED
| 0 | 30 | 0.056 | 0.055 | 0.059 | 0.065 | 0.056 | 0.060 | 0.048 | 0.063 |
0 | 50 | 0.064 | 0.069 | 0.066 | 0.070 | 0.063 | 0.065 | 0.055 | 0.054 | |
0 | 100 | 0.045 | 0.051 | 0.060 | 0.056 | 0.044 | 0.053 | 0.053 | 0.046 | |
0.22 | 30 | 0.053 | 0.062 | 0.059 | 0.066 | 0.056 | 0.054 | 0.062 | 0.064 | |
0.22 | 50 | 0.038 | 0.043 | 0.043 | 0.049 | 0.046 | 0.041 | 0.053 | 0.053 | |
0.22 | 100 | 0.042 | 0.049 | 0.038 | 0.047 | 0.045 | 0.053 | 0.047 | 0.052 | |
LD
| 0 | 30 | 0.048 | 0.046 | 0.056 | 0.069 | 0.044 | 0.063 | 0.058 | 0.065 |
0 | 50 | 0.044 | 0.051 | 0.055 | 0.049 | 0.050 | 0.061 | 0.052 | 0.059 | |
0 | 100 | 0.047 | 0.053 | 0.055 | 0.060 | 0.049 | 0.049 | 0.054 | 0.052 | |
0.22 | 30 | 0.061 | 0.058 | 0.069 | 0.066 | 0.062 | 0.065 | 0.056 | 0.056 | |
0.22 | 50 | 0.054 | 0.055 | 0.060 | 0.063 | 0.052 | 0.056 | 0.056 | 0.061 | |
0.22 | 100 | 0.057 | 0.055 | 0.057 | 0.050 | 0.062 | 0.064 | 0.061 | 0.057 | |
CH
| 0 | 30 | 0.053 | 0.047 | 0.056 | 0.060 | 0.053 | 0.047 | 0.053 | 0.055 |
0 | 50 | 0.054 | 0.049 | 0.045 | 0.054 | 0.051 | 0.046 | 0.047 | 0.063 | |
0 | 100 | 0.041 | 0.048 | 0.041 | 0.047 | 0.042 | 0.048 | 0.051 | 0.059 | |
0.22 | 30 | 0.059 | 0.047 | 0.054 | 0.066 | 0.054 | 0.055 | 0.067 | 0.062 | |
0.22 | 50 | 0.046 | 0.042 | 0.042 | 0.048 | 0.048 | 0.046 | 0.061 | 0.060 | |
0.22 | 100 | 0.061 | 0.062 | 0.063 | 0.066 | 0.061 | 0.065 | 0.055 | 0.054 |
Scenario* | Strata effect (w) | N. of Strata (k) | TSi
1
| TLSi
1
| ||||||
---|---|---|---|---|---|---|---|---|---|---|
4 fixed | 8 fixed | perc | all | 4 fixed | 8 fixed | perc | all | |||
A) Fisher test
| ||||||||||
PH
| 0 | 30 |
0.
734
|
0.
723
| 0.627 | 0.592 |
0.
744
|
0.
737
| 0.673 | 0.651 |
0 | 50 |
0.
911
|
0.
906
| 0.854 | 0.822 |
0.
909
|
0.
906
| 0.871 | 0.847 | |
0 | 100 |
0.
998
|
0.
996
| 0.993 | 0.968 |
0.
997
|
0.
996
| 0.993 | 0.978 | |
0.22 | 30 |
0.
692
|
0.
691
| 0.606 | 0.578 |
0.
706
|
0.
720
| 0.658 | 0.638 | |
0.22 | 50 |
0.
882
|
0.
873
| 0.816 | 0.795 |
0.
888
|
0.
887
| 0.839 | 0.831 | |
0.22 | 100 |
0.
995
|
0.
995
| 0.983 | 0.978 |
0.
997
|
0.
995
| 0.985 | 0.981 | |
ED
| 0 | 30 | 0.274 | 0.274 | 0.414 | 0.432 | 0.302 | 0.297 |
0.
514
|
0.534
|
0 | 50 | 0.406 | 0.408 | 0.622 | 0.641 | 0.443 | 0.452 |
0.
682
|
0.
699
| |
0 | 100 | 0.730 | 0.746 | 0.916 | 0.927 | 0.760 | 0.770 |
0.
939
|
0.
937
| |
0.22 | 30 | 0.240 | 0.252 | 0.376 | 0.409 | 0.267 | 0.285 |
0.
470
|
0.
488
| |
0.22 | 50 | 0.387 | 0.252 | 0.616 | 0.632 | 0.423 | 0.285 |
0.
671
|
0.682
| |
0.22 | 100 | 0.710 | 0.727 | 0.894 | 0.902 | 0.739 | 0.753 |
0.
924
|
0.
929
| |
LD
| 0 | 30 |
0.
425
|
0.
408
| 0.278 | 0.247 |
0.408
|
0.
400
| 0.298 | 0.266 |
0 | 50 |
0.
674
|
0.
642
| 0.464 | 0.392 |
0.
648
|
0.
627
| 0.470 | 0.406 | |
0 | 100 |
0.
949
|
0.
937
| 0.802 | 0.749 |
0.
940
|
0.
928
| 0.796 | 0.691 | |
0.22 | 30 |
0.
399
|
0.369
| 0.243 | 0.209 |
0.
387
|
0.
373
| 0.273 | 0.245 | |
0.22 | 50 |
0.
626
|
0.
590
| 0.414 | 0.359 |
0.
620
|
0.
587
| 0.440 | 0.384 | |
0.22 | 100 |
0.
910
|
0.
890
| 0.740 | 0.712 |
0.
908
|
0.
888
| 0.751 | 0.725 | |
CH
| 0 | 30 | 0.144 | 0.165 | 0.285 | 0.272 | 0.079 | 0.078 | 0.115 | 0.099 |
0 | 50 | 0.291 | 0.321 | 0.488 | 0.461 | 0.162 | 0.148 | 0.229 | 0.207 | |
0 | 100 | 0.747 | 0.799 | 0.897 | 0.881 | 0.560 | 0.596 | 0.746 | 0.729 | |
0.22 | 30 | 0.149 | 0.181 | 0.273 | 0.248 | 0.087 | 0.089 | 0.111 | 0.105 | |
0.22 | 50 | 0.292 | 0.319 | 0.477 | 0.419 | 0.178 | 0.161 | 0.237 | 0.207 | |
0.22 | 100 | 0.736 | 0.794 | 0.878 | 0.844 | 0.575 | 0.589 | 0.744 | 0.668 | |
B) Tippett test
| ||||||||||
PH
| 0 | 30 | 0.685 | 0.666 | 0.586 | 0.605 | 0.676 | 0.654 | 0.674 | 0.677 |
0 | 50 | 0.874 | 0.866 | 0.828 | 0.807 | 0.878 | 0.869 | 0.856 | 0.875 | |
0 | 100 | 0.996 | 0.995 | 0.991 | 0.979 | 0.996 | 0.994 | 0.992 | 0.996 | |
0.22 | 30 | 0.646 | 0.632 | 0.567 | 0.572 | 0.657 | 0.663 | 0.659 | 0.684 | |
0.22 | 50 | 0.834 | 0.826 | 0.768 | 0.766 | 0.850 | 0.831 | 0.816 | 0.814 | |
0.22 | 100 | 0.992 | 0.990 | 0.976 | 0.975 | 0.993 | 0.990 | 0.982 | 0.979 | |
ED
| 0 | 30 | 0.393 | 0.369 | 0.351 | 0.384 | 0.458 | 0.458 |
0.
531
|
0.
596
|
0 | 50 | 0.615 | 0.571 | 0.579 | 0.597 | 0.661 | 0.646 |
0.
679
|
0.
708
| |
0 | 100 | 0.906 | 0.910 | 0.904 | 0.914 | 0.920 | 0.924 |
0.
937
|
0.
942
| |
0.22 | 30 | 0.369 | 0.343 | 0.347 | 0.365 | 0.429 | 0.422 |
0.
501
|
0.
544
| |
0.22 | 50 | 0.597 | 0.343 | 0.560 | 0.589 | 0.652 | 0.422 |
0.676
|
0.
699
| |
0.22 | 100 | 0.887 | 0.877 | 0.896 | 0.897 | 0.910 | 0.901 |
0.923
|
0.
925
| |
LD
| 0 | 30 |
0.
472
|
0.
458
| 0.392 | 0.383 |
0.
438
|
0.
442
| 0.398 | 0.400 |
0 | 50 |
0.730
|
0.
718
| 0.644 | 0.632 |
0.
688
|
0.
683
| 0.634 | 0.615 | |
0 | 100 |
0.963
|
0.957
| 0.926 | 0.914 |
0.
956
|
0.
950
| 0.920 | 0.918 | |
0.22 | 30 |
0.
425
|
0.397
| 0.342 | 0.318 |
0.
411
|
0.
393
| 0.363 | 0.361 | |
0.22 | 50 |
0.
660
|
0.
650
| 0.572 | 0.567 |
0.
658
|
0.
634
| 0.584 | 0.566 | |
0.22 | 100 |
0.
930
|
0.
924
| 0.871 | 0.864 |
0.
927
|
0.
920
| 0.874 | 0.858 | |
CH
| 0 | 30 | 0.356 | 0.418 |
0.
475
|
0.521
| 0.183 | 0.155 | 0.191 | 0.163 |
0 | 50 | 0.584 | 0.636 |
0.
700
|
0.
729
| 0.380 | 0.335 | 0.374 | 0.347 | |
0 | 100 | 0.920 | 0.931 |
0.
951
|
0.
971
| 0.854 | 0.842 | 0.851 | 0.848 | |
0.22 | 30 | 0.335 | 0.399 |
0.
433
|
0.
484
| 0.176 | 0.165 | 0.185 | 0.163 | |
0.22 | 50 | 0.560 | 0.626 |
0.
663
|
0.
722
| 0.408 | 0.358 | 0.385 | 0.348 | |
0.22 | 100 | 0.887 | 0.921 |
0.
953
|
0.
963
| 0.830 | 0.806 | 0.850 | 0.836 |
Scenario* | Strata effect (w) | N. of strata (k) | Under H0
| Under H1
| ||||
---|---|---|---|---|---|---|---|---|
(1) | (2) | (3) | (1) | (2) | (3) | |||
PH
| 0 | 30 | 0.042 | 0.043 | 0.049 | 0.660 |
0.
734
|
0.
770
|
0 | 50 | 0.042 | 0.043 | 0.051 | 0.867 |
0.
923
|
0.
938
| |
0 | 100 | 0.054 | 0.050 | 0.060 | 0.995 |
0.
999
|
1.
000
| |
0.22 | 30 | 0.044 | 0.052 | 0.053 | 0.635 |
0.
731
|
0.
726
| |
0.22 | 50 | 0.057 | 0.051 | 0.058 | 0.827 |
0.
898
|
0.
903
| |
0.22 | 100 | 0.056 | 0.054 | 0.053 | 0.986 |
0.
997
|
0.
996
| |
ED
| 0 | 30 | 0.053 | 0.061 | 0.064 | 0.207 | 0.189 | 0.216 |
0 | 50 | 0.061 | 0.067 | 0.071 | 0.317 | 0.262 | 0.310 | |
0 | 100 | 0.049 | 0.052 | 0.061 | 0.620 | 0.531 | 0.571 | |
0.22 | 30 | 0.046 | 0.051 | 0.056 | 0.204 | 0.169 | 0.166 | |
0.22 | 50 | 0.037 | 0.044 | 0.045 | 0.292 | 0.237 | 0.253 | |
0.22 | 100 | 0.051 | 0.048 | 0.053 | 0.596 | 0.501 | 0.510 | |
LD
| 0 | 30 | 0.042 | 0.046 | 0.054 | 0.384 | 0.473 | 0.520 |
0 | 50 | 0.053 | 0.059 | 0.060 | 0.600 | 0.731 | 0.775 | |
0 | 100 | 0.052 | 0.052 | 0.118 | 0.863 | 0.965 | 0.971 | |
0.22 | 30 | 0.049 | 0.065 | 0.070 | 0.338 | 0.454 | 0.470 | |
0.22 | 50 | 0.046 | 0.051 | 0.051 | 0.553 | 0.707 | 0.705 | |
0.22 | 100 | 0.051 | 0.056 | 0.051 | 0.816 | 0.949 | 0.947 | |
CH
| 0 | 30 | 0.062 | 0.070 | 0.076 | 0.076 | 0.078 | 0.082 |
0 | 50 | 0.045 | 0.045 | 0.046 | 0.087 | 0.119 | 0.119 | |
0 | 100 | 0.041 | 0.045 | 0.048 | 0.095 | 0.174 | 0.662 | |
0.22 | 30 | 0.046 | 0.043 | 0.049 | 0.079 | 0.091 | 0.083 | |
0.22 | 50 | 0.040 | 0.042 | 0.039 | 0.101 | 0.142 | 0.138 | |
0.22 | 100 | 0.063 | 0.056 | 0.064 | 0.116 | 0.233 | 0.233 |
Application
Time points strategy | N. of time points (years) | Global p-value | |||
---|---|---|---|---|---|
TSF
2
| TLSF
2
| TST
2
| TLST
2
| ||
fixed
| 4 (0.5,1,2,3) | 0.387 | 0.375 | 0.263 | 0.245 |
fixed
| 4 (1,1.5,2,3) | 0.348 | 0.338 | 0.225 | 0.212 |
fixed
| 4 (1,2,3,4) | 0.232 | 0.221 | 0.213 | 0.199 |
perc
| 9 (0.16-1.74)* | 0.665 | 0.675 | 0.515 | 0.486 |
all
| 52 (0.19-1.71)* | 0.799 | 0.829 | 0.630 | 0.579 |