Table 4 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 one, sample size 30, 60 and 120, p = 0.95 to 0.5
From: Sequence balance minimisation: minimising with unequal treatment allocations
Random element | Â | Treatment totals only | 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) | 10.9 (0.07) | 10.8 (0.06) | 10.5 (0.03) | 10.2 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.0 (0.04) | 10.1 (0.03) | 10.0 (0.03) | 10.1 (0.03) |
|  | Median (p1–p99) | 11 (6–16) | 11 (7–15) | 10 (8–13) | 10 (8–13) | 10 (8–13) | 10 (8–12) | 10 (8–13) | 10 (7–13) | 10 (8–13) | 10 (8–13) | 10 (8–13) |
p = 0.6 | Mean (SE) | 10.7 (0.06) | 10.7 (0.05) | 10.4 (0.03) | 10.2 (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) |
|  | Median (p1–p99) | 11 (7–15) | 11 (7–15) | 10 (8–13) | 10 (8–13) | 10 (8–13) | 10 (8–13) | 10 (8–13) | 10 (7–12) | 10 (8–12.5) | 10 (8–13) | 10 (8–13) |
p = 0.7 | Mean (SE) | 10.5 (0.05) | 10.5 (0.04) | 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) | 10.1 (0.03) |
|  | Median (p1–p99) | 10 (6–14) | 10 (7–14) | 10 (8–13) | 10 (8–12.5) | 10 (8–12.5) | 10 (8–12) | 10 (8–12) | 10 (8–13) | 10 (8–13) | 10 (8–13) | 10 (8–13) |
p = 0.8 | Mean (SE) | 10.3 (0.04) | 10.4 (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) | 10.2 (0.03) | 10.1 (0.04) |
|  | Median (p1–p99) | 10 (7–14) | 10 (8–13) | 10 (8–13) | 10 (8–12) | 10 (8–13) | 10 (8–12) | 10 (8–12.5) | 10 (8–12) | 10 (8–13) | 10 (8–13) | 10 (7–13) |
p = 0.85 | Mean (SE) | 10.3 (0.03) | 10.3 (0.03) | 10.2 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.0 (0.03) | 10.0 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.0 (0.03) |
|  | Median (p1–p99) | 10 (8–13) | 10 (8–13) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12.5) | 10 (8–12) | 10 (8–13) | 10 (7.5–12.5) | 10 (7–13) |
p = 0.9 | Mean (SE) | 10.2 (0.03) | 10.2 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.1 (0.03) | 10.0 (0.03) | 10.0 (0.03) | 10.0 (0.03) | 10.1 (0.03) | 10.0 (0.04) | 10.1 (0.03) |
|  | Median (p1–p99) | 10 (8–12) | 10 (8–13) | 10 (8–12) | 10 (8–12.5) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (7.5–13) | 10 (7–13) | 10 (8–12) |
p = 0.95 | Mean (SE) | 10.1 (0.02) | 10.1 (0.02) | 10.0 (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 (9–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–12) | 10 (8–13) | 10 (8–13) | 10 (8–13) |
Sample size (n = 60), the expected number of participants allocated to the treatment with smallest-allocation ratio (n = 20) | ||||||||||||
p = 0.5 | Mean (SE) | 21.7 (0.09) | 21.4 (0.07) | 20.9 (0.06) | 20.6 (0.06) | 20.4 (0.05) | 20.3 (0.05) | 20 (0.05) | 20.1 (0.05) | 20.1 (0.05) | 20.1 (0.05) | 20.1 (0.05) |
|  | Median (p1–p99) | 22 (15–29) | 21 (16–27) | 21 (17–25) | 21 (17–26) | 20 (17–24) | 20 (17–24) | 20 (17–23) | 20 (17–23.5) | 20 (17–23.5) | 20 (17–24) | 20 (16.5–24) |
p = 0.6 | Mean (SE) | 21.3 (0.09) | 21.1 (0.07) | 20.9 (0.06) | 20.5 (0.05) | 20.4 (0.04) | 20.3 (0.05) | 20.1 (0.04) | 20.1 (0.05) | 20.1 (0.05) | 20.2 (0.05) | 20.1 (0.05) |
|  | Median (p1–p99) | 21 (16–28) | 21 (16–27) | 21 (17–26) | 20 (17–24) | 20 (17–24) | 20 (17–23.5) | 20 (17–24) | 20 (17–24) | 20 (17–23) | 20 (17–24) | 20 (17–24) |
p = 0.7 | Mean (SE) | 20.9 (0.07) | 21.0 (0.06) | 20.7 (0.05) | 20.4 (0.05) | 20.4 (0.04) | 20.2 (0.04) | 20.1 (0.04) | 20.1 (0.05) | 20.0 (0.05) | 20.1 (0.05) | 20.2 (0.05) |
|  | Median (p1–p99) | 21 (16–26) | 21 (17–26) | 21 (17–25) | 20 (17–24) | 20 (17–24) | 20 (17–23) | 20 (17–24) | 20 (17–23) | 20 (17–23) | 20 (17–24) | 20 (17–24) |
p = 0.8 | Mean (SE) | 20.8 (0.06) | 20.6 (0.05) | 20.4 (0.04) | 20.3 (0.05) | 20.2 (0.04) | 20.2 (0.04) | 20.1 (0.04) | 20.0 (0.04) | 20.0 (0.05) | 20.1 (0.05) | 20.1 (0.05) |
|  | Median (p1–p99) | 21 (17–25) | 21 (17–24) | 20 (17–24) | 20 (17–24) | 20 (17–23) | 20 (17–24) | 20 (17–23) | 20 (17–23) | 20 (17–24) | 20 (17–24) | 20 (17–24) |
p = 0.85 | Mean (SE) | 20.5 (0.05) | 20.5 (0.04) | 20.3 (0.04) | 20.2 (0.04) | 20.2 (0.04) | 20.1 (0.04) | 20.1 (0.05) | 20.1 (0.04) | 20.1 (0.05) | 20.1 (0.05) | 20.2 (0.05) |
|  | Median (p1–p99) | 21 (17–24) | 20 (18–24) | 20 (17–23) | 20 (17–24) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–24) | 20 (17–24) | 20 (17–24) |
p = 0.9 | Mean (SE) | 20.4 (0.04) | 20.3 (0.04) | 20.1 (0.04) | 20.2 (0.04) | 20.2 (0.04) | 20.1 (0.04) | 20.1 (0.04) | 20.1 (0.05) | 20.2 (0.05) | 20.1 (0.05) | 20.1 (0.05) |
|  | Median (p1–p99) | 20 (17–24) | 20 (17–23.5) | 20 (17–23) | 20 (17–23.5) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–24) | 20 (17–24) | 20 (17–24) | 20 (16–24) |
p = 0.95 | Mean (SE) | 20.2 (0.03) | 20.2 (0.03) | 20.1 (0.04) | 20.1 (0.04) | 20.1 (0.04) | 20.2 (0.04) | 20.1 (0.04) | 20.1 (0.04) | 20.2 (0.05) | 20.1 (0.05) | 20.1 (0.05) |
|  | Median (p1–p99) | 20 (18–23) | 20 (18–23) | 20 (17–23) | 20 (17–23) | 20 (17–23) | 20 (17–23.5) | 20 (16.5–23) | 20 (17–23) | 20 (17–24) | 20 (16.5–24) | 20 (17–24) |
Sample size (n = 120), the expected number of participants allocated to the treatment with smallest-allocation ratio (n = 40) | ||||||||||||
p = 0.5 | Mean (SE) | 43.4 (0.14) | 42.7 (0.09) | 41.2 (0.08) | 40.8 (0.07) | 40.5 (0.07) | 40.2 (0.07) | 40.3 (0.07) | 40.1 (0.07) | 40.3 (0.07) | 40.2 (0.07) | 40.3 (0.07) |
|  | Median (p1–p99) | 43 (33–54) | 43 (36–49.5) | 41 (36–47) | 41 (36–46) | 40 (36–45) | 40 (35–46) | 40 (35–45) | 40 (35–45.5) | 40 (35–45) | 40 (35–46) | 40 (36–45) |
p = 0.6 | Mean (SE) | 42.5 (0.12) | 42.1 (0.09) | 41.1 (0.07) | 40.7 (0.07) | 40.3 (0.07) | 40.2 (0.07) | 40.3 (0.07) | 40.2 (0.07) | 40.2 (0.07) | 40.3 (0.07) | 40.4 (0.07) |
|  | Median (p1–p99) | 42 (34–51.5) | 42 (36–49) | 41 (36–46) | 41 (36–46) | 40 (36–45) | 40 (35–45) | 40 (35–45) | 40 (35–45) | 40 (35–45) | 40 (35–46) | 40 (36–45) |
p = 0.7 | Mean (SE) | 42.0 (0.10) | 41.6 (0.08) | 40.8 (0.07) | 40.6 (0.07) | 40.2 (0.07) | 40.3 (0.07) | 40.2 (0.07) | 40.3 (0.07) | 40.2 (0.07) | 40.2 (0.07) | 40.2 (0.07) |
|  | Median (p1–p99) | 42 (35–49) | 42 (36–47.5) | 41 (36–46) | 41 (36–45) | 40 (35–45) | 40 (36–45) | 40 (35–45) | 40 (35.5–45) | 40 (35–45) | 40 (35–45) | 40 (36–45) |
p = 0.8 | Mean (SE) | 41.3 (0.08) | 41.1 (0.07) | 40.6 (0.06) | 40.4 (0.06) | 40.3 (0.06) | 40.1 (0.07) | 40.2 (0.06) | 40.2 (0.07) | 40.2 (0.07) | 40.2 (0.07) | 40.3 (0.07) |
|  | Median (p1–p99) | 41 (36–47) | 41 (36–46) | 41 (36–45) | 40 (36–45) | 40 (35.5–45) | 40 (35–45) | 40 (36–45) | 40 (35–45) | 40 (35–45) | 40 (35.5–45) | 40 (36–45) |
p = 0.85 | Mean (SE) | 41.0 (0.07) | 40.8 (0.06) | 40.5 (0.06) | 40.4 (0.06) | 40.3 (0.07) | 40.3 (0.07) | 40.2 (0.07) | 40.4 (0.07) | 40.2 (0.07) | 40.3 (0.07) | 40.2 (0.07) |
|  | Median (p1–p99) | 41 (36–46) | 41 (37–45) | 40 (36–45) | 40 (36–45) | 40 (36–45) | 40 (35–45) | 40 (35–45) | 40 (35–46) | 40 (35–45) | 40 (35–45) | 40 (35–45) |
p = 0.9 | Mean (SE) | 40.7 (0.06) | 40.5 (0.06) | 40.2 (0.06) | 40.2 (0.06) | 40.2 (0.06) | 40.1 (0.06) | 40.2 (0.07) | 40.2 (0.07) | 40.2 (0.07) | 40.2 (0.07) | 40.2 (0.07) |
|  | Median (p1–p99) | 41 (36–45) | 40 (36–45) | 40 (35–45) | 40 (36–44) | 40 (35–45) | 40 (36–45) | 40 (36–45.5) | 40 (36–45) | 40 (35–46) | 40 (35–45.5) | 40 (35–45.5) |
p = 0.95 | Mean (SE) | 40.4 (0.04) | 40.3 (0.05) | 40.1 (0.06) | 40.2 (0.06) | 40.1 (0.06) | 40.2 (0.07) | 40.3 (0.06) | 40.1 (0.07) | 40.3 (0.07) | 40.2 (0.07) | 40.3 (0.07) |
|  | Median (p1–p99) | 40 (37.5–43.5) | 40 (37–44) | 40 (36–44) | 40 (36–44) | 40 (36–45) | 40 (35–46) | 40 (35–45) | 40 (35–45) | 40 (35–45) | 40 (35–45) | 40 (36–45) |