Table 2.
Power in settings with three endpoints. Power to reject all elementary hypotheses ( ), at least one union of two hypotheses (any H i∪H j), at least one elementary hypothesis (any H i), at least one pairwise intersection hypothesis (any H i∩H j), or H 1 under alternatives with standardized effects δ = (δ 1,δ 2,δ 3), assuming trivariate normal test statistics with variances equal to 1 and equal correlations ρ and global one‐sided level α = 0.025. The power is given in percent (100,000 simulation runs per scenario).
| δ | ρ | Test |
|
any H i∪H j | any H i | any H i∩H j | H 1 | |
|---|---|---|---|---|---|---|---|---|
| (3,0,0) | 0 | 2 out of 3 Fallback/Hommel | 0.1/0.1 | 1.9/1.9 | 4.0/73.4 | 4.2/73.4 | 3.8/73.0 | |
| Bonferroni–Holm/maxT | 0.0/0.0 | 1.9/1.9 | 73.3/72.0 | 73.3/72.0 | 72.9/71.5 | |||
| 0.5 | 2 out of 3 Fallback/Hommel | 0.5/0.5 | 2.4/2.4 | 4.5/72.8 | 4.6/72.8 | 4.5/72.8 | ||
| Bonferroni–Holm/maxT | 0.4/0.4 | 2.3/2.4 | 72.8/72.8 | 72.8/72.8 | 72.8/72.8 | |||
| 0.75 | 2 out of 3 Fallback/Hommel | 0.9/0.9 | 2.2/2.2 | 4.1/72.7 | 4.1/72.7 | 4.1/72.7 | ||
| Bonferroni–Holm/maxT | 0.8/0.9 | 2.1/2.5 | 72.7/75.0 | 72.7/75.0 | 72.7/75.0 | |||
| 0.85 | 2 out of 3 Fallback/Hommel | 1.3/1.3 | 2.1/2.1 | 3.8/72.6 | 3.8/72.6 | 3.8/72.6 | ||
| Bonferroni–Holm/maxT | 1.1/1.2 | 1.9/2.5 | 72.6/76.5 | 72.6/76.5 | 72.6/76.5 | |||
| 0.9 | 2 out of 3 Fallback/Hommel | 1.5/1.5 | 2.1/2.1 | 3.5/72.8 | 3.5/72.8 | 3.5/72.8 | ||
| Bonferroni–Holm/maxT | 1.2/1.3 | 1.8/2.5 | 72.8/77.3 | 72.8/77.3 | 72.8/77.3 | |||
| (3,3,0) | 0 | 2 out of 3 Fallback/Hommel | 1.9/1.9 | 60.8/60.9 | 72.4/93.2 | 73.1/93.2 | 66.5/77.0 | |
| Bonferroni–Holm/maxT | 1.7/1.6 | 60.4/58.3 | 92.6/91.8 | 92.6/91.8 | 76.5/74.9 | |||
| 0.5 | 2 out of 3 Fallback/Hommel | 2.5/2.5 | 65.5/65.5 | 75.0/87.3 | 75.9/87.3 | 70.2/76.4 | ||
| Bonferroni–Holm/maxT | 2.5/2.4 | 65.2/64.5 | 86.6/86.5 | 86.6/86.5 | 75.9/75.5 | |||
| 0.75 | 2 out of 3 Fallback/Hommel | 2.5/2.5 | 69.1/69.1 | 77.4/83.3 | 78.6/83.3 | 73.3/76.1 | ||
| Bonferroni–Holm/maxT | 2.5/2.5 | 68.7/69.7 | 82.3/83.9 | 82.3/83.9 | 75.4/76.8 | |||
| 0.85 | 2 out of 3 Fallback/Hommel | 2.4/2.4 | 70.9/70.9 | 78.4/81.2 | 79.8/81.2 | 74.6/76.1 | ||
| Bonferroni–Holm/maxT | 2.4/2.5 | 70.3/72.8 | 79.9/83.3 | 79.9/83.3 | 75.2/78.1 | |||
| 0.9 | 2 out of 3 Fallback/Hommel | 2.5/2.5 | 72.3/72.3 | 79.0/80.2 | 80.9/80.2 | 75.8/76.4 | ||
| Bonferroni–Holm/maxT | 2.5/2.5 | 71.6/74.5 | 78.7/82.9 | 78.7/82.9 | 75.3/78.7 | |||
| (2,3,3) | 0 | 2 out of 3 Fallback/Hommel | 37.3/37.3 | 75.9/75.9 | 85.0/95.8 | 85.5/95.8 | 47.6/48.3 | |
| Bonferroni–Holm/maxT | 35.7/34.2 | 74.1/72.4 | 95.1/94.5 | 95.1/94.5 | 46.7/45.5 | |||
| 0.5 | 2 out of 3 Fallback/Hommel | 45.9/45.9 | 72.1/72.1 | 79.9/88.2 | 80.9/88.2 | 49.0/49.1 | ||
| Bonferroni–Holm/maxT | 44.4/43.4 | 70.3/69.9 | 87.2/87.3 | 87.2/87.3 | 47.7/47.0 | |||
| 0.75 | 2 out of 3 Fallback/Hommel | 50.1/50.1 | 71.9/71.9 | 78.9/83.9 | 80.1/83.9 | 50.7/50.7 | ||
| Bonferroni–Holm/maxT | 48.9/48.4 | 70.4/72.1 | 82.7/84.7 | 82.7/84.7 | 49.5/49.3 | |||
| 0.85 | 2 out of 3 Fallback/Hommel | 51.3/51.3 | 72.4/72.4 | 79.0/81.5 | 80.4/81.5 | 51.5/51.4 | ||
| Bonferroni–Holm/maxT | 50.5/49.8 | 71.0/73.6 | 80.1/83.5 | 80.1/83.5 | 50.6/50.0 | |||
| 0.9 | 2 out of 3 Fallback/Hommel | 51.7/51.7 | 72.8/72.8 | 79.1/80.2 | 80.9/80.2 | 51.7/51.7 | ||
| Bonferroni–Holm/maxT | 51.2/50.5 | 71.7/75.4 | 78.6/83.5 | 78.6/83.5 | 51.2/50.6 | |||
| (3,3,3) | 0 | 2 out of 3 Fallback/Hommel | 61.4/61.4 | 88.5/88.5 | 93.7/98.4 | 94.0/98.4 | 81.3/82.8 | |
| Bonferroni–Holm/maxT | 60.0/58.1 | 86.9/85.6 | 97.9/97.7 | 97.9/97.7 | 81.7/80.6 | |||
| 0.5 | 2 out of 3 Fallback/Hommel | 69.5/69.5 | 83.0/83.0 | 88.3/92.5 | 89.0/92.5 | 80.2/81.6 | ||
| Bonferroni–Holm/maxT | 67.8/66.4 | 81.1/80.8 | 91.6/91.6 | 91.6/91.6 | 80.1/79.6 | |||
| 0.75 | 2 out of 3 Fallback/Hommel | 74.5/74.5 | 81.2/81.2 | 85.7/87.8 | 86.9/87.8 | 80.5/81.2 | ||
| Bonferroni–Holm/maxT | 72.2/71.8 | 78.7/80.2 | 86.2/88.1 | 86.2/88.1 | 79.1/80.1 | |||
| 0.85 | 2 out of 3 Fallback/Hommel | 77.2/77.2 | 81.2/81.2 | 84.8/85.7 | 86.2/85.7 | 81.0/81.4 | ||
| Bonferroni–Holm/maxT | 74.4/74.6 | 78.1/80.3 | 83.6/86.5 | 83.6/86.5 | 78.7/80.3 | |||
| 0.9 | 2 out of 3 Fallback/Hommel | 78.5/78.5 | 80.9/80.9 | 84.0/84.0 | 85.7/84.0 | 81.2/81.2 | ||
| Bonferroni–Holm/maxT | 75.2/76.6 | 77.4/80.9 | 81.3/85.9 | 81.3/85.9 | 78.0/81.2 |