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Table 2 Optimal, minimax and admissible design for Fleming’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

a1

r1

PETF1

n

r

ENF

Type

q

0.05

0.25

0.05

0.20

0.047

0.812

9

0

3

0.639

17

3

11.9

Optimal

[0.000, 0.655]

    

0.043

0.801

12

0

3

0.560

16

3

13.8

Minimax

[0.655, 1.000]

0.10

0.30

0.05

0.20

0.047

0.805

10

1

5

0.738

29

6

15.0

Optimal

[0.000, 0.200]

    

0.048

0.812

11

1

4

0.716

27

6

15.5

Admissible

[0.200, 0.474]

    

0.049

0.817

12

1

4

0.685

26

6

16.4

Admissible

[0.474, 0.661]

    

0.043

0.802

19

2

5

0.741

24

6

20.3

Minimax

[0.661, 1.000]

0.30

0.50

0.05

0.20

0.046

0.803

18

6

11

0.728

42

18

24.5

Optimal

[0.000, 0.250]

    

0.049

0.807

19

6

11

0.676

39

17

25.5

Admissible

[0.250, 0.634]

    

0.049

0.800

27

8

14

0.592

36

16

30.7

Minimax

[0.634, 1.000]

    

0.049

0.800

21

0

12

0.009

36

16

35.9

Inadmissible

0.05

0.25

0.10

0.10

0.093

0.903

9

0

3

0.639

24

3

14.4

Optimal

[0.000, 0.130]

    

0.083

0.905

10

0

3

0.610

22

3

14.7

Admissible

[0.130, 0.333]

    

0.078

0.905

11

0

3

0.584

21

3

15.2

Admissible

[0.333, 0.500]

    

0.074

0.903

13

0

3

0.538

20

3

16.2

Minimax

[0.500, 1.000]

0.10

0.30

0.10

0.10

0.085

0.900

13

1

4

0.656

31

6

19.2

Optimal

[0.000, 0.123]

    

0.099

0.904

18

2

5

0.762

26

5

19.9

Admissible

[0.123, 0.231]

    

0.095

0.903

16

1

5

0.532

25

5

20.2

Minimax

[0.231, 1.000]

0.30

0.50

0.10

0.10

0.097

0.901

20

6

10

0.656

47

19

29.3

Optimal

[0.000, 0.048]

    

0.091

0.900

20

6

11

0.625

45

18

29.4

Admissible

[0.048, 0.117]

    

0.093

0.900

23

7

12

0.640

42

17

29.8

Admissible

[0.117, 0.492]

    

0.097

0.901

26

7

13

0.486

39

16

32.7

Minimax

[0.492, 1.000]

0.05

0.25

0.05

0.10

0.049

0.902

9

0

4

0.631

30

4

16.8

Optimal

[0.000, 0.091]

    

0.047

0.913

10

0

4

0.600

29

4

17.6

Inadmissible

    

0.047

0.907

10

0

3

0.610

28

4

17.0

Inadmissible

    

0.046

0.911

11

0

3

0.584

27

4

17.7

Inadmissible

    

0.042

0.901

11

0

3

0.584

26

4

17.2

Admissible

[0.091, 0.565]

    

0.045

0.903

13

0

3

0.538

25

4

18.5

Minimax

[0.565, 1.000]

0.10

0.30

0.05

0.10

0.048

0.901

17

2

5

0.784

41

8

22.2

Optimal

[0.000, 0.032]

    

0.048

0.902

18

2

6

0.740

35

7

22.4

Admissible

[0.032, 0.444]

    

0.047

0.900

16

1

5

0.532

33

7

24.0

Minimax

[0.444, 1.000]

0.30

0.50

0.05

0.10

0.047

0.900

25

8

14

0.683

56

23

34.8

Optimal

[0.000, 0.250]

    

0.049

0.901

26

8

14

0.637

53

22

35.8

Admissible

[0.250, 0.670]

    

0.049

0.900

37

11

18

0.579

50

21

42.5

Minimax

[0.670, 1.000]

    

0.05

0.900

28

0

15

0.008

50

21

49.8

Inadmissible

  1. (p1, p0) = (0.05, 0.25), (0.10, 0.30), (0.30, 0.50) are considered in Fleming’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. (a1, r1, r) are the critical values. PET1 denotes the probability of early termination at first stage. EN represents expected sample size