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Table 8 Factor 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, p = 0.95 to 0.5. Summary statistics from 1000 simulations

From: Sequence balance minimisation: minimising with unequal treatment allocations

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)

5.2 (0.03)

5.1 (0.03)

5.0 (0.04)

5.0 (0.04)

5.0 (0.04)

5.0 (0.04)

 

Median (p1–p99)

5 (2–10)

5 (2–8)

5 (3–8)

5 (2–8)

5 (3–8)

5 (3–8)

5 (3–8)

5 (2–8)

5 (3–8)

5 (3–8)

5 (3–8)

p = 0.6

Mean (SE)

5.3 (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)

5.0 (0.04)

5.1 (0.04)

5.1 (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 (3–8)

p = 0.7

Mean (SE)

5.3 (0.05)

5.2 (0.04)

5.2 (0.03)

5.1 (0.03)

5.2 (0.03)

5.2 (0.04)

5.1 (0.04)

5.0 (0.04)

5.0 (0.04)

5.1 (0.04)

5.0 (0.04)

 

Median (p1–p99)

5 (2–9)

5 (2.5–8)

5 (3–8)

5 (2.5–7)

5 (3–8)

5 (3–8)

5 (3–8)

5 (2–8)

5 (2–8)

5 (2.5–8)

5 (3–8)

p = 0.8

Mean (SE)

5.2 (0.04)

5.2 (0.03)

5.3 (0.03)

5.1 (0.03)

5.1 (0.03)

5.0 (0.03)

5.2 (0.03)

5.1 (0.04)

5.1 (0.04)

5.0 (0.03)

5.1 (0.04)

 

Median (p1–p99)

5 (2–8)

5 (3–7)

5 (3–8)

5 (3–8)

5 (3–8)

5 (3–8)

5 (3–8)

5 (2–8)

5 (3–8)

5 (2–7)

5 (2–8)

p = 0.85

Mean (SE)

5.1 (0.04)

5.1 (0.03)

5.3 (0.03)

5.0 (0.03)

5.2 (0.03)

5.1 (0.03)

5.1 (0.03)

5.0 (0.03)

5.1 (0.03)

5.0 (0.04)

5.0 (0.04)

 

Median (p1–p99)

5 (2–8)

5 (3–7)

5 (3–8)

5 (3–8)

5 (3–8)

5 (3–8)

5 (2–8)

5 (3–8)

5 (3–8)

5 (3–8)

5 (3–8)

p = 0.9

Mean (SE)

5.1 (0.04)

5.1 (0.02)

5.2 (0.03)

5.1 (0.03)

5.2 (0.04)

5.2 (0.03)

5.1 (0.03)

5.1 (0.04)

5.0 (0.04)

5.1 (0.04)

5.1 (0.04)

 

Median (p1–p99)

5 (2–8)

5 (3–7)

5 (3–7)

5 (3–8)

5 (2–8)

5 (3–8)

5 (3–8)

5 (3–8)

5 (2–8)

5 (3–8)

5 (2–8)

p = 0.95

Mean (SE)

5.0 (0.04)

5.0 (0.02)

5.2 (0.03)

5.1 (0.03)

5.2 (0.03)

5.1 (0.03)

5.0 (0.03)

5.0 (0.04)

5.1 (0.03)

5.1 (0.04)

5.0 (0.04)

 

Median (p1–p99)

5 (2–8)

5 (3–6.5)

5 (3–7.5)

5 (2.5–8)

5 (3–8)

5 (3–8)

5 (3–8)

5 (2–8)

5 (3–8)

5 (2.5–8)

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)

10.1 (0.05)

9.9 (0.05)

9.8 (0.05)

9.9 (0.05)

9.7 (0.05)

9.8 (0.05)

9.9 (0.05)

9.9 (0.05)

 

Median (p1–p99)

11 (6–17)

11 (7–14.5)

10 (7–14)

10 (7–14)

10 (7–13.5)

10 (7–13)

10 (7–13)

10 (7–13)

10 (6–13)

10 (6–13)

10 (6–13)

p = 0.6

Mean (SE)

10.6 (0.07)

10.5 (0.05)

10.3 (0.04)

10.0 (0.05)

9.8 (0.05)

9.9 (0.05)

9.8 (0.05)

9.8 (0.05)

9.9 (0.05)

9.9 (0.05)

9.9 (0.05)

 

Median (p1–p99)

11 (6–16)

10 (7–14)

10 (7–14)

10 (7–13)

10 (6–13)

10 (6–14)

10 (6–13)

10 (7–14)

10 (7–13.5)

10 (6–14)

10 (7–13)

p = 0.7

Mean (SE)

10.4 (0.06)

10.4 (0.04)

10.2 (0.04)

10.1 (0.05)

9.9 (0.05)

9.8 (0.05)

9.8 (0.05)

9.7 (0.05)

9.9 (0.05)

9.8 (0.05)

9.9 (0.05)

 

Median (p1–p99)

10 (6–15)

10 (7–14)

10 (7–13)

10 (7–14)

10 (7–14)

10 (6–13)

10 (6–13)

10 (6–13)

10 (6–13)

10 (6–14)

10 (6–14)

p = 0.8

Mean (SE)

10.4 (0.06)

10.2 (0.04)

10.2 (0.04)

9.9 (0.05)

9.9 (0.04)

9.8 (0.05)

9.8 (0.05)

9.8 (0.05)

9.8 (0.05)

9.9 (0.05)

9.9 (0.05)

 

Median (p1–p99)

10 (6–14)

10 (8–13)

10 (7–13)

10 (6–13)

10 (7–13)

10 (6.5–13)

10 (6–13.5)

10 (6–14)

10 (6.5–13)

10 (7–13)

10 (6.5–14)

p = 0.85

Mean (SE)

10.2 (0.05)

10.2 (0.03)

10.2 (0.04)

9.9 (0.04)

9.8 (0.04)

9.8 (0.05)

9.8 (0.05)

9.8 (0.05)

9.9 (0.05)

9.9 (0.05)

10.0 (0.05)

 

Median (p1–p99)

10 (6–14)

10 (8–13)

10 (7–13)

10 (7–13)

10 (7–13)

10 (7–13)

10 (6–13)

10 (6–13)

10 (7–14)

10 (6–14)

10 (6–14)

p = 0.9

Mean (SE)

10.1 (0.05)

10.1 (0.03)

10.1 (0.04)

9.9 (0.04)

9.7 (0.04)

9.8 (0.05)

9.8 (0.05)

9.7 (0.05)

9.8 (0.05)

9.8 (0.05)

10.0 (0.05)

 

Median (p1–p99)

10 (6–14)

10 (8–12)

10 (8–13)

10 (7–13)

10 (7–13)

10 (7–13)

10 (7–13)

10 (6–13.5)

10 (6–13)

10 (6–14)

10 (6–14)

p = 0.95

Mean (SE)

10.1 (0.05)

10.0 (0.03)

10.1 (0.03)

9.9 (0.04)

9.8 (0.04)

9.7 (0.05)

9.8 (0.05)

9.8 (0.05)

9.8 (0.05)

9.9 (0.05)

9.9 (0.05)

 

Median (p1–p99)

10 (6–13.5)

10 (8–12)

10 (8–13)

10 (7–13)

10 (7–13)

10 (6–14)

10 (6–13)

10 (7–13)

10 (6–13)

10 (6–14)

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)

20.8 (0.07)

20.7 (0.07)

20.3 (0.08)

19.9 (0.08)

20.2 (0.08)

20.2 (0.08)

20.3 (0.08)

20.5 (0.08)

20.4 (0.08)

 

Median (p1–p99)

22 (14–30)

22 (17–26)

21 (16–26)

21 (16–26)

20 (15–26)

20 (14–25.5)

20 (15–26)

20 (14–27)

20 (15–26)

20 (15–26.5)

20 (15–27)

p = 0.6

Mean (SE)

21.6 (0.10)

21.3 (0.07)

20.6 (0.07)

20.6 (0.07)

20.3 (0.07)

20.1 (0.08)

20.2 (0.08)

20.2 (0.08)

20.4 (0.08)

20.4 (0.08)

20.5 (0.09)

 

Median (p1–p99)

22 (14–30)

21 (17–26)

21 (16–25.5)

20 (16–26)

20 (15–25)

20 (14–26)

20 (15–26)

20 (14–26)

20.5 (14–26)

20 (14–26)

21 (14.5–27)

p = 0.7

Mean (SE)

21.1 (0.10)

21.0 (0.06)

20.5 (0.07)

20.5 (0.07)

20.3 (0.08)

19.9 (0.07)

20.2 (0.08)

20.1 (0.08)

20.5 (0.08)

20.4 (0.08)

20.5 (0.09)

 

Median (p1–p99)

21 (14–29)

21 (17–26)

20 (16–26)

20 (16–26)

20 (15–26)

20 (14.5–25)

20 (14.5–26)

20 (14–26)

20 (15–26.5)

20 (14.5–26)

21 (14.5–27)

p = 0.8

Mean (SE)

20.8 (0.09)

20.7 (0.05)

20.3 (0.06)

20.4 (0.07)

20.1 (0.08)

20.1 (0.08)

20.2 (0.08)

20.2 (0.08)

20.2 (0.08)

20.5 (0.08)

20.5 (0.09)

 

Median (p1–p99)

21 (14–28)

21 (17–24)

20 (16–25)

20 (15–25)

20 (15–25)

20 (14.5–26)

20 (14–26)

20 (14–27)

20 (15–27)

20 (14.5–27)

20 (14.5–27)

p = 0.85

Mean (SE)

20.4 (0.09)

20.5 (0.05)

20.3 (0.06)

20.4 (0.07)

20.3 (0.07)

20.2 (0.08)

20.0 (0.08)

20.2 (0.08)

20.3 (0.08)

20.5 (0.08)

20.4 (0.08)

 

Median (p1–p99)

20 (14–27)

20 (17–24)

20 (16–25)

20 (16–26)

20 (15–26)

20 (15–26)

20 (15–26)

20 (14–26)

20 (14–26)

21 (14.5–26.5)

21 (14–26)

p = 0.9

Mean (SE)

20.4 (0.09)

20.2 (0.04)

20.2 (0.06)

20.2 (0.07)

20.3 (0.07)

20.0 (0.08)

20.2 (0.08)

20.2 (0.08)

20.2 (0.08)

20.5 (0.08)

20.5 (0.09)

 

Median (p1–p99)

20 (14–27)

20 (17–24)

20 (16–25)

20 (15–25)

20 (15–26)

20 (14–26.5)

20 (15–26)

20 (14.5–26)

20 (15–27)

20 (14.5–27)

21 (14.5–27)

p = 0.95

Mean (SE)

20.2 (0.09)

20.2 (0.04)

20.1 (0.06)

20.3 (0.07)

20.2 (0.08)

20.0 (0.08)

20.1 (0.08)

20.1 (0.08)

20.6 (0.08)

20.5 (0.08)

20.5 (0.09)

 

Median (p1–p99)

20 (14–27)

20 (17–23)

20 (16–24)

20 (15.5–25)

20 (14.5–26)

20 (14–27)

20 (15–26)

20 (14–26)

21 (15–27)

20 (14.5–27)

21 (14.5–27)