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Table 1 Optimal, minimax and admissible design for Simon’s two-stage design

From: Optimal, minimax and admissible two-stage design for phase II oncology clinical trials

p0

p1

α

β

αT

1-βT

n1

r1

PETS1

n

r

ENS

Type

q

0.05

0.25

0.05

0.20

0.047

0.812

9

0

0.630

17

2

12.0

Optimal

[0.000, 0.643]

    

0.043

0.801

12

0

0.540

16

2

13.8

Minimax

[0.643, 1.000]

0.10

0.30

0.05

0.20

0.047

0.805

10

1

0.736

29

5

15.0

Optimal

[0.000, 0.286]

    

0.040

0.806

11

1

0.697

27

5

15.8

Admissible

[0.286, 0.500]

    

0.036

0.805

12

1

0.659

26

5

16.8

Admissible

[0.500, 0.730]

    

0.033

0.802

15

1

0.549

25

5

19.5

Minimax

[0.730, 1.000]

0.30

0.50

0.05

0.20

0.050

0.803

15

5

0.722

46

18

23.6

Optimal

[0.000, 0.216]

    

0.044

0.801

18

6

0.722

42

17

24.7

Admissible

[0.216, 0.250]

    

0.046

0.804

19

6

0.666

39

16

25.7

Minimax

[0.250, 1.000]

0.05

0.25

0.10

0.10

0.093

0.903

9

0

0.630

24

2

14.6

Optimal

[0.000, 0.091]

    

0.083

0.905

10

0

0.599

22

2

14.8

Admissible

[0.091, 0.333]

    

0.078

0.905

11

0

0.569

21

2

15.3

Admissible

[0.333, 0.524]

    

0.074

0.903

13

0

0.513

20

2

16.4

Minimax

[0.524, 1.000]

0.10

0.30

0.10

0.10

0.098

0.901

12

1

0.659

35

5

19.8

Optimal

[0.000, 0.032]

    

0.10

0.904

18

2

0.734

26

4

20.1

Admissible

[0.032, 0.231]

    

0.095

0.903

16

1

0.515

25

4

20.4

Minimax

[0.231, 1.000]

0.30

0.50

0.10

0.10

0.097

0.905

22

7

0.671

46

17

29.9

Optimal

[0.000, 0.111]

    

0.090

0.901

21

6

0.551

42

16

30.4

Admissible

[0.111, 0.605]

    

0.094

0.900

28

7

0.365

39

15

35.0

Minimax

[0.605, 1.000]

0.05

0.25

0.05

0.10

0.049

0.902

9

0

0.630

30

3

16.8

Optimal

[0.000, 0.167]

    

0.047

0.913

10

0

0.599

29

3

17.6

Inadmissible

    

0.043

0.906

10

0

0.599

28

3

17.2

Admissible

[0.167, 0.375]

    

0.040

0.908

11

0

0.569

27

3

18.0

Inadmissible

    

0.037

0.905

12

0

0.540

26

3

18.4

Admissible

[0.375, 0.667]

    

0.034

0.901

15

0

0.463

25

3

20.4

Minimax

[0.667, 1.000]

0.10

0.30

0.05

0.10

0.047

0.902

18

2

0.734

35

6

22.5

Optimal

[0.000, 0.474]

    

0.044

0.901

19

2

0.705

34

6

23.4

Admissible

[0.474, 0.737]

    

0.041

0.902

22

2

0.620

33

6

26.2

Minimax

[0.737, 1.000]

0.30

0.50

0.05

0.10

0.050

0.903

24

8

0.725

63

24

34.7

Optimal

[0.000, 0.114]

    

0.044

0.902

22

7

0.671

62

24

35.2

Inadmissible

    

0.045

0.902

20

6

0.608

59

23

35.3

Inadmissible

    

0.046

0.903

23

7

0.618

56

22

35.6

Admissible

[0.114, 0.250]

    

0.047

0.902

24

7

0.565

53

21

36.6

Minimax

[0.250, 1.000]

  1. (p1, p0) = (0.05, 0.25), (0.10, 0.30), (0.30, 0.50) are considered in Simon’s two-stage design. Each (p1, p0) gives three type I & II error constraints, (0.05, 0.20), (0.10, 0.10) and (0.05, 0.10), respectively. (αT, 1-βT) denotes the true type I error and power. n1 and n is the sample size required in the first stage and in the whole trial, respectively. (r1, r) are the critical values. PET1 denotes the probability of early termination at first stage. EN represents expected sample size