Table 7.
Factor balancing properties of 1:2 sequence balance minimisation with 0 (treatment totals only) to 10prognostic factors with 2 levels, treatment totals weighted as one, sample size 30, 60 and 120, p = 0.95 to 0.5. Summary statistics from 1000 simulations
| Random element | Number of factors | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Treatment totals only (worst-case scenario) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||
| Sample size (n = 30), the expected number of participants allocated with the particular factor level n = 5 | ||||||||||||
| p = 0.5 | Mean (SE) | 5.5 (0.05) | 5.3 (0.04) | 5.4 (0.03) | 5.0 (0.03) | 5.2 (0.03) | 5.1 (0.03) | 5.1 (0.03) | 5.0 (0.03) | 5.1 (0.04) | 5.0 (0.04) | 5.0 (0.04) |
| Median (p1–p99) | 5 (2–10) | 5 (2–8) | 5 (3–8) | 5 (2–7) | 5 (3–7.5) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (2–8) | 5 (2.5–8) | 5 (2–8) | |
| p = 0.6 | Mean (SE) | 5.3 (0.05) | 5.3 (0.04) | 5.3 (0.03) | 5.0 (0.03) | 5.3 (0.03) | 5.2 (0.03) | 5.2 (0.04) | 5.0 (0.03) | 5.1 (0.04) | 5.0 (0.04) | 5.0 (0.04) |
| Median (p1–p99) | 5 (2–9) | 5 (3–9) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (2.5–8) | |
| p = 0.7 | Mean (SE) | 5.3 (0.05) | 5.2 (0.04) | 5.3 (0.03) | 5.1 (0.03) | 5.2 (0.04) | 5.2 (0.03) | 5.1 (0.03) | 5.1 (0.03) | 5.1 (0.04) | 5.0 (0.04) | 5.0 (0.03) |
| Median (p1–p99) | 5 (2–9) | 5 (2.5–8) | 5 (3–8) | 5 (3–8) | 5 (2–8) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (2.5–8) | 5 (2.5–8) | |
| p = 0.8 | Mean (SE) | 5.2 (0.04) | 5.2 (0.03) | 5.3 (0.03) | 5.0 (0.03) | 5.2 (0.03) | 5.1 (0.03) | 5.1 (0.03) | 5.1 (0.03) | 5.0 (0.03) | 5.0 (0.04) | 4.9 (0.04) |
| Median (p1–p99) | 5 (2–8) | 5 (3–7) | 5 (3–7) | 5 (3–7) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (2–8) | 5 (2.5–8) | 5 (2.5–8) | 5 (2–8) | |
| p = 0.85 | Mean (SE) | 5.1 (0.04) | 5.1 (0.03) | 5.2 (0.03) | 5.1 (0.03) | 5.2 (0.03) | 5.1 (0.03) | 5.0 (0.04) | 4.9 (0.04) | 5.1 (0.04) | 5.1 (0.03) | 5.0 (0.04) |
| Median (p1–p99) | 5 (2–8) | 5 (3–7) | 5 (3–7.5) | 5 (3–7) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (2–8) | 5 (2.5–8) | 5 (2–8) | 5 (2.5–8) | |
| p = 0.9 | Mean (SE) | 5.1 (0.04) | 5.1 (0.02) | 5.2 (0.03) | 5.1 (0.03) | 5.2 (0.03) | 5.2 (0.03) | 5.1 (0.03) | 5.0 (0.03) | 5.0 (0.04) | 5.0 (0.04) | 5.1 (0.04) |
| Median (p1–p99) | 5 (2–8) | 5 (3–7) | 5 (3–7) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (3–7) | 5 (3–7.5) | 5 (2.5–8) | 5 (2–8) | 5 (3–8) | |
| p = 0.95 | Mean (SE) | 5.0 (0.04) | 5.0 (0.02) | 5.1 (0.03) | 5.0 (0.03) | 5.2 (0.03) | 5.2 (0.03) | 5.1 (0.03) | 5.0 (0.03) | 5.1 (0.04) | 5.0 (0.04) | 5.0 (0.04) |
| Median (p1–p99) | 5 (2–8) | 5 (3–6.5) | 5 (3–7) | 5 (3–7) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (3–8) | 5 (2–7.5) | 5 (2–8) | |
| Sample size (n = 60), the expected number of participants allocated with the particular factor level n = 10 | ||||||||||||
| p = 0.5 | Mean (SE) | 10.9 (0.07) | 10.6 (0.05) | 10.4 (0.05) | 10.3 (0.05) | 10.1 (0.05) | 10.1 (0.05) | 10.0 (0.05) | 9.9 (0.05) | 10.0 (0.05) | 10.0 (0.05) | 10.1 (0.05) |
| Median (p1–p99) | 11 (6–17) | 11 (7–14.5) | 10 (7–14) | 10 (7–14) | 10 (7–14) | 10 (7–13.5) | 10 (6–13.5) | 10 (7–14) | 10 (6–14) | 10 (6–14) | 10 (7–14) | |
| p = 0.6 | Mean (SE) | 10.6 (0.07) | 10.5 (0.05) | 10.4 (0.04) | 10.2 (0.05) | 10.1 (0.05) | 10.0 (0.05) | 10.0 (0.05) | 9.9 (0.05) | 10.0 (0.05) | 10.1 (0.05) | 10.2 (0.05) |
| Median (p1–p99) | 11 (6–16) | 10 (7–14) | 10 (7.5–14) | 10 (7–14) | 10 (7–13) | 10 (7–14) | 10 (6–14) | 10 (6–14) | 10 (6–14) | 10 (6–14) | 10 (6.5–14) | |
| p = 0.7 | Mean (SE) | 10.4 (0.06) | 10.4 (0.04) | 10.4 (0.04) | 10.2 (0.05) | 10.1 (0.05) | 10.0 (0.05) | 10.0 (0.05) | 10.0 (0.05) | 10.0 (0.05) | 10.1 (0.05) | 10.2 (0.05) |
| Median (p1–p99) | 10 (6–15) | 10 (7–14) | 10 (8–14) | 10 (7–14) | 10 (7–13.5) | 10 (7–13) | 10 (6.5–13) | 10 (7–14) | 10 (6–14) | 10 (6–14) | 10 (6.5–14.5) | |
| p = 0.8 | Mean (SE) | 10.4 (0.06) | 10.2 (0.04) | 10.3 (0.04) | 10.1 (0.04) | 10.0 (0.04) | 10.0 (0.05) | 10.1 (0.05) | 9.9 (0.05) | 10.0 (0.05) | 10.1 (0.05) | 10.1 (0.05) |
| Median (p1–p99) | 10 (6–14) | 10 (8–13) | 10 (8–13) | 10 (7–13) | 10 (7–13) | 10 (7–14) | 10 (7–14) | 10 (6–14) | 10 (7–14) | 10 (6–14) | 10 (7–14) | |
| p = 0.85 | Mean (SE) | 10.2 (0.05) | 10.2 (0.03) | 10.2 (0.04) | 10.1 (0.04) | 10.0 (0.04) | 10.0 (0.05) | 10.0 (0.05) | 10.0 (0.05) | 10.1 (0.05) | 10.1 (0.05) | 10.2 (0.05) |
| Median (p1–p99) | 10 (6–14) | 10 (8–13) | 10 (7–13) | 10 (7–13) | 10 (7–14) | 10 (7–14) | 10 (6.5–14) | 10 (7–14) | 10 (6–14) | 10 (6.5–14) | 10 (7–14) | |
| p = 0.9 | Mean (SE) | 10.1 (0.05) | 10.1 (0.03) | 10.2 (0.04) | 10.0 (0.04) | 9.9 (0.04) | 10.0 (0.05) | 10.0 (0.05) | 10.0 (0.05) | 10.1 (0.05) | 10.1 (0.05) | 10.2 (0.05) |
| Median (p1–p99) | 10 (6–14) | 10 (8–12) | 10 (7–13) | 10 (7–13) | 10 (7–13) | 10 (6–14) | 10 (7–14) | 10 (6–14) | 10 (6–14) | 10 (6–14) | 10 (6–14) | |
| p = 0.95 | Mean (SE) | 10.1 (0.05) | 10.0 (0.03) | 10.0 (0.03) | 10.0 (0.04) | 9.9 (0.05) | 10.0 (0.05) | 10.0 (0.05) | 9.9 (0.05) | 10.1 (0.05) | 10.1 (0.05) | 10.2 (0.05) |
| Median (p1–p99) | 10 (6–13.5) | 10 (8–12) | 10 (7–12) | 10 (7–13) | 10 (7–13) | 10 (7–13) | 10 (7–13) | 10 (6–14) | 10 (6–14) | 10 (6.5–14) | 10 (6–14) | |
| Sample size (n = 120), the expected number of participants allocated with the particular factor level n = 20 | ||||||||||||
| p = 0.5 | Mean (SE) | 22.0 (0.11) | 21.7 (0.06) | 20.6 (0.07) | 20.7 (0.07) | 20.3 (0.07) | 20.1 (0.08) | 20.1 (0.08) | 20.1 (0.08) | 20.4 (0.08) | 20.5 (0.08) | 20.5 (0.08) |
| Median (p1–p99) | 22 (14–30) | 22 (17–26) | 20 (16–26) | 21 (15–26) | 20 (15–26) | 20 (14–26) | 20 (15–26) | 20 (14–26) | 20 (14–26) | 21 (15–27) | 20 (14–26) | |
| p = 0.6 | Mean (SE) | 21.6 (0.10) | 21.3 (0.07) | 20.5 (0.07) | 20.6 (0.07) | 20.1 (0.07) | 20.0 (0.08) | 20.1 (0.08) | 20.1 (0.08) | 20.3 (0.08) | 20.4 (0.08) | 20.5 (0.08) |
| Median (p1–p99) | 22 (14–30) | 21 (17–26) | 20 (16–26) | 21 (16–26) | 20 (15–25) | 20 (14–25.5) | 20 (14–26) | 20 (14–26) | 20 (13.5–26) | 20 (15–26) | 21 (14–26.5) | |
| p = 0.7 | Mean (SE) | 21.1 (0.10) | 21.0 (0.06) | 20.4 (0.06) | 20.6 (0.07) | 20.1 (0.07) | 20.1 (0.08) | 20.1 (0.08) | 20.1 (0.08) | 20.4 (0.08) | 20.4 (0.09) | 20.5 (0.08) |
| Median (p1–p99) | 21 (14–29) | 21 (17–26) | 20 (16–25) | 21 (16–25) | 20 (15–25) | 20 (15–26) | 20 (14–26) | 20 (14–26) | 20 (15–26) | 20 (14–27) | 20 (14.5–27) | |
| p = 0.8 | Mean (SE) | 20.8 (0.09) | 20.7 (0.05) | 20.3 (0.06) | 20.6 (0.06) | 20.2 (0.07) | 20.0 (0.08) | 20.0 (0.08) | 20.1 (0.08) | 20.3 (0.08) | 20.4 (0.08) | 20.5 (0.09) |
| Median (p1–p99) | 21 (14–28) | 21 (17–24) | 20 (16–25) | 21 (16–26) | 20 (15–25) | 20 (15–25.5) | 20 (14–26) | 20 (15–26) | 20 (15–26) | 20 (14–26) | 20 (14–27) | |
| p = 0.85 | Mean (SE) | 20.4 (0.09) | 20.5 (0.05) | 20.2 (0.06) | 20.5 (0.07) | 20.2 (0.07) | 20.2 (0.08) | 20.1 (0.08) | 20.2 (0.08) | 20.4 (0.08) | 20.4 (0.08) | 20.5 (0.08) |
| Median (p1–p99) | 20 (14–27) | 20 (17–24) | 20 (16–25) | 20 (16–26) | 20 (15–25) | 20 (14.5–26) | 20 (15–26) | 20 (14–26) | 20 (15–27) | 20 (14–26) | 21 (15–27) | |
| p = 0.9 | Mean (SE) | 20.4 (0.09) | 20.2 (0.04) | 20.0 (0.06) | 20.4 (0.07) | 20.2 (0.07) | 20.1 (0.08) | 20.0 (0.08) | 20.1 (0.08) | 20.3 (0.08) | 20.5 (0.08) | 20.4 (0.08) |
| Median (p1–p99) | 20 (14–27) | 20 (17–24) | 20 (16–24) | 20 (16–25) | 20 (15–25) | 20 (15–26) | 20 (14–26) | 20 (14–26) | 20 (14.5–26) | 21 (14–26) | 20 (14–26) | |
| p = 0.95 | Mean (SE) | 20.2 (0.09) | 20.2 (0.04) | 20.0 (0.06) | 20.4 (0.06) | 20.2 (0.07) | 20.1 (0.08) | 20.1 (0.08) | 20.2 (0.08) | 20.4 (0.08) | 20.4 (0.08) | 20.5 (0.08) |
| Median (p1–p99) | 20 (14–27) | 20 (17–23) | 20 (16–24) | 20 (16–25) | 20 (15–25) | 20 (14–26) | 20 (15–26) | 20 (14–26) | 20 (15–26.5) | 20.5 (15–26) | 20 (14–26) | |