Table 5.
Treatment balancing properties of 1:2 sequence balance minimisation with 0 (treatment totals only) to 10 prognostic factors with 2 levels, treatment totals weighted as total number of other minimisation factors, sample size 30, 60 and 120, p = 0.95 to 0.5
| Random element | Number of factors | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||
| Sample size (n = 30), the expected number of participants allocated to the treatment with smallest-allocation ratio (n = 10) | |||||||||||
| p = 0.5 | Mean (SE) | 10.8 (0.06) | 10.5 (0.03) | 10.2 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) |
| Median (p1–p99) | 11 (7–15) | 10 (8–13) | 10 (8–13) | 10 (8–13) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | |
| p = 0.6 | Mean (SE) | 10.7 (0.06) | 10.5 (0.03) | 10.2 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.2 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) |
| Median (p1–p99) | 11 (7–15) | 10 (8–13) | 10 (8–13) | 10 (8–13) | 10 (8–12) | 10 (8–12) | 10 (8–12.5) | 10 (8–13) | 10 (8–13) | 10 (8–12) | |
| p = 0.7 | Mean (SE) | 10.5 (0.05) | 10.3 (0.03) | 10.2 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.0 (0.03) |
| Median (p1–p99) | 10 (7–14) | 10 (8–13) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12.5) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | |
| p = 0.8 | Mean (se) | 10.4 (0.03) | 10.2 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.0 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) |
| Median (p1–p99) | 10 (8–13) | 10 (8–12) | 10 (8–13) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | |
| p = 0.85 | Mean (SE) | 10.3 (0.03) | 10.2 (0.03) | 10.1 (0.03) | 10.0 (0.03) | 10.0 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) |
| Median (p1–p99) | 10 (8–13) | 10 (8–13) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | |
| p = 0.9 | Mean (SE) | 10.2 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.0 (0.03) | 10.1 (0.03) | 10.0 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) |
| Median (p1–p99) | 10 (8–13) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | |
| p = 0.95 | Mean (SE) | 10.1 (0.02) | 10.1 (0.03) | 10.1 (0.03) | 10.0 (0.03) | 10.0 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) |
| Median (p1–p99) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | |
| Sample size (n = 60), the expected number of participants allocated to the treatment with smallest-allocation ratio (n = 20) | |||||||||||
| p = 0.5 | Mean (SE) | 21.4 (0.07) | 21.0 (0.06) | 20.7 (0.05) | 20.4 (0.04) | 20.2 (0.04) | 20.1 (0.04) | 20.0 (0.04) | 19.9 (0.04) | 20.0 (0.04) | 20.0 (0.04) |
| Median (p1–p99) | 21 (16–27) | 21 (17–25.5) | 21 (17–25) | 20 (18–23) | 20 (17.5–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | |
| p = 0.6 | Mean (SE) | 21.1 (0.07) | 20.8 (0.05) | 20.5 (0.05) | 20.3 (0.04) | 20.2 (0.04) | 20.1 (0.04) | 20.1 (0.04) | 20.1 (0.04) | 20.1 (0.04) | 20.1 (0.04) |
| Median (p1–p99) | 21 (16–27) | 21 (17–25) | 20 (17–24) | 20 (17.5–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (18–23) | |
| p = 0.7 | Mean (SE) | 21.0 (0.06) | 20.6 (0.05) | 20.4 (0.05) | 20.3 (0.04) | 20.1 (0.04) | 20.1 (0.04) | 20.0 (0.04) | 20.1 (0.04) | 20.1 (0.04) | 20.0 (0.04) |
| Median (p1–p99) | 21 (17–26) | 21 (17–24) | 20 (17–24) | 20 (17–23.5) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | |
| p = 0.8 | Mean (SE) | 20.6 (0.05) | 20.4 (0.04) | 20.2 (0.04) | 20.2 (0.04) | 20.1 (0.04) | 20.0 (0.04) | 20.1 (0.04) | 20.1 (0.04) | 20.0 (0.04) | 20.1 (0.04) |
| Median (p1–p99) | 21 (17–24) | 20 (17.5–24) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | |
| p = 0.85 | Mean (SE) | 20.5 (0.04) | 20.4 (0.04) | 20.2 (0.04) | 20.2 (0.03) | 20.1 (0.04) | 20.0 (0.04) | 20.0 (0.04) | 20.1 (0.04) | 20.1 (0.04) | 20.1 (0.04) |
| Median (p1–p99) | 20 (18–24) | 20 (17.5–23) | 20 (17–23) | 20 (18–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17.5–23) | 20 (17–23) | 20 (17–23) | |
| p = 0.9 | Mean (SE) | 20.3 (0.04) | 20.3 (0.04) | 20.2 (0.04) | 20.1 (0.04) | 20.0 (0.04) | 20.0 (0.04) | 20.0 (0.04) | 20.1 (0.04) | 20.1 (0.04) | 20.1 (0.04) |
| Median (p1–p99) | 20 (17–23.5) | 20 (18–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17.5–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | |
| p = 0.95 | Mean (SE) | 20.2 (0.03) | 20.1 (0.03) | 20.2 (0.04) | 20.1 (0.04) | 20.0 (0.04) | 20.0 (0.04) | 20.0 (0.04) | 20.0 (0.04) | 20.1 (0.04) | 20.0 (0.04) |
| Median (p1–p99) | 20 (18–23) | 20 (18–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | |
| Sample size (n = 120), the expected number of participants allocated to the treatment with smallest-allocation ratio (n = 40) | |||||||||||
| p = 0.5 | Mean (SE) | 42.7 (0.09) | 41.5 (0.07) | 40.8 (0.06) | 40.4 (0.06) | 40.2 (0.06) | 40.1 (0.06) | 40.2 (0.06) | 40.1 (0.06) | 40.2 (0.06) | 40.2 (0.06) |
| Median (p1–p99) | 43 (36–49.5) | 41 (36–47) | 41 (36.5–45) | 40 (36–45) | 40 (36–45) | 40 (36–44) | 40 (36–45) | 40 (36–44) | 40 (36–45) | 40 (36–44) | |
| p = 0.6 | Mean (SE) | 42.1 (0.09) | 41.1 (0.07) | 40.6 (0.06) | 40.3 (0.06) | 40.0 (0.06) | 40.1 (0.06) | 40.1 (0.06) | 40.1 (0.06) | 40.2 (0.06) | 40.0 (0.06) |
| Median (p1–p99) | 42 (36–49) | 41 (36–46) | 41 (37–45) | 40 (36–45) | 40 (36–44) | 40 (36–44) | 40 (36–44) | 40 (36–44) | 40 (36–45) | 40 (36–45) | |
| p = 0.7 | Mean (SE) | 41.6 (0.08) | 40.8 (0.06) | 40.4 (0.06) | 40.2 (0.06) | 40.1 (0.06) | 40.1 (0.06) | 40.1 (0.06) | 40.0 (0.06) | 40.1 (0.06) | 40.1 (0.06) |
| Median (p1–p99) | 42 (36–47.5) | 41 (36–45.5) | 40 (36–45) | 40 (36–44) | 40 (36–44) | 40 (36–44) | 40 (36–45) | 40 (36–45) | 40 (36–44) | 40 (36–45) | |
| p = 0.8 | Mean (SE) | 41.1 (0.07) | 40.5 (0.06) | 40.3 (0.06) | 40.2 (0.06) | 40.1 (0.06) | 40.2 (0.06) | 40.1 (0.06) | 40.0 (0.06) | 40.2 (0.06) | 40.1 (0.06) |
| Median (p1–p99) | 41 (36–46) | 40 (36–45) | 40 (36–44) | 40 (36–45) | 40 (35.5–44) | 40 (36–45) | 40 (36–44) | 40 (36–45) | 40 (36–44) | 40 (36–44) | |
| p = 0.85 | Mean (SE) | 40.8 (0.06) | 40.4 (0.06) | 40.2 (0.05) | 40.3 (0.06) | 40.1 (0.06) | 40.1 (0.06) | 40.1 (0.06) | 40.2 (0.06) | 40.2 (0.06) | 40.2 (0.06) |
| Median (p1–p99) | 41 (37–45) | 40 (36–45) | 40 (36–44) | 40 (36–44) | 40 (36–44) | 40 (36–44) | 40 (36–44) | 40 (36–45) | 40 (36–44) | 40 (36–44.5) | |
| p = 0.9 | Mean (SE) | 40.5 (0.06) | 40.3 (0.06) | 40.1 (0.05) | 40.2 (0.06) | 40.1 (0.06) | 40.1 (0.06) | 40.2 (0.06) | 40.0 (0.06) | 40.2 (0.06) | 40.1 (0.06) |
| Median (p1–p99) | 40 (36–45) | 40 (36–44.5) | 40 (36–44) | 40 (36–44.5) | 40 (36–44) | 40 (36–44) | 40 (36–45) | 40 (36–45) | 40 (36–44) | 40 (36–45) | |
| p = 0.95 | Mean (SE) | 40.3 (0.05) | 40.1 (0.05) | 40.0 (0.05) | 40.1 (0.06) | 40.1 (0.05) | 40.2 (0.05) | 40.1 (0.06) | 40.1 (0.05) | 40.2 (0.06) | 40.1 (0.06) |
| Median (p1–p99) | 40 (37–44) | 40 (36–44) | 40 (36–44) | 40 (36–45) | 40 (36–44) | 40 (36–44) | 40 (36–44.5) | 40 (36–44) | 40 (36–44) | 40 (36–45) | |
Summary statistics from 1000 simulations. Treatment balance under best case scenario: mean (se): 10 (0), 20 (0), 40 (0) and median (p1 – p99): 10 (10 - 10), 20 (20 – 20), 40 (40 – 40) for sample size 30, 60 and 120 respectively