General discussions on Phase II clinical trials are presented in PhaseII.php and not repeated here
This page provides explanations, calculations, and sample size tables for Phase II trials, where the sample size needed is determined by Simon's procedure.
Simon's is a two stage procedure.
The first stage requires a small sample size (n1), and sets a bench mark number of successes (r1) above which the
trial enters the second stage. If that bench mark (r1) is not surpassed at the end of stage 1 (n1), then the trial ends
with the treatment considered inadequate and abandoned (rejection).
In the second stage, the total sample size, including those already
collected in stage 1, is defined (nTot), and a second and final bench mark for the total number of
successes, including those already collected in stage 1, is defined (rTot). Once
the number of successes surpassed rTot, the trial can terminate and the treatment
considered worthy of further evaluation at the Phase III or control trial level (acceptance).
If rTot is not surpassed after nTot cases, then the trial terminates, and the
treatment considered inadequate and abandoned (rejection).
Please note: In Simon's procedure, acceptance and rejection refers to whether the treatment being tested is successful enough for further evaluation, but not the acceptance or rejection of the null hypothesis
Simon's procedure therefore has advantages over Fleming's procedure and Gehan's Procedure, in that the sample size is not fixed, and the
trial can terminate early if the results are obvious. It is particularly effective in rejecting new treatments with
below expectation proportion of successes.
Parameters : The following parameters are required
The success rate below which the treatment is considered inadequate and rejected (π0)
The success rate above which the treatment is accepted as worthy of further evaluatuion (π1)
Probability of Type I Error (α), the probability of error for rejecting the null hypothesis, that of wrongly accepting the treatment as worthy of further evaluation. In most cases α of 0.1 or 0.05 is used
Power (1-β) where β is the Probability of Type II Error. Power is therefore the probability of accepting the treatment as worthy of further evaluation, and in most cases power of 0.8 or 0.9 is used
Results : The programs produces the following results
The maximum sample size for stage 1 (n1)
The number of success required in stage 1, above which stage 2 is entered, and at or below which after n1 cases results in
terminating the trial with rejection of the treatment
The total maximum sample size (nTot), inclusive of both stages
The total number of success required (rTot), inclusive of both stages, above which resulting in terminating the trial with
acceptance of the treatment for further evaluation, and at or below which after nTot cases results in rejection of the treatment
The average expected number of cases (EN) for a decision
The probability of early termination of the study (PET) if the true success rate is below requirements
Models : Simon's Procedure produces two alternative results, based on different statistical assumptions.
The Optimal model has a smaller sample size for stage 1 and is more likely to terminate early (PET) if the true success rate
is below requirement (<π0), so is preferred for early screening of new treatments to exclude those without potentials from expensive further study.
The minimax model requires a smaller overall sample size (EN), so is preferred if the researcher
is optimistic about the treatment being tested, hoping to require a smaller overall sample size to validate the acceptance of the treatment for further trials
References
Simon R (1989) Optimal two-stage designs for phase II clinical trials.
Control Clin Trials 10:1-10
Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for
Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 256-257
Technical Considerations
Algorithm
I created the Javascript program on this page to calculate the sample sizes for Simon's procedure. I followed the formulae as described in Machin's text book (see references). In checking my results against the table in the text book (Table 10.5 page 285), I found that I was able to replicate all the contents of the table in the Optimal model, but in nearly a third of the calculations for the Minimax model, the results of my calculations differed from that in the text book. After examining the formulae in detail, and some trial and errors I think I have found out the reason and the solution.
The formula for calculating the number to use when starting interation (the starting point), produced by formula 10.2 (page 257) produces the correct result in the majority of situations, but is too large when the precision required is too high (higher power or lower α) when the rejection rate (π0) is close to 0.5, or when the gap between acceptance and rejection rates (π1-π0) is narrow. In these situation the starting point number need to be smaller.
After some trial and error, I reduce the starting number by 10, then iterate the whole algorithm, increasing the starting number with each iteration, until the starting point calculated by formula 10.2 is reached, choosing the results where the total sample size (nTot) for the Minimax model is the smallest. By doing this, I was able to replicate exactly all the numbers in table 10.5 of the text book.
My conclusions are that the algorithm in StatsToDo produces the correct answer within the tested ranges. These are
π0 in a range between 0.05 (5%) and 0.8 (80%)
(π1 - π0)>=0.1 (10%)
α >= 0.01
power <= 0.9
Users should however note that, although the reference values can be replicated, the original formulation has been altered to achieve this. Inexperienced user should therefore seek advice before using the results.
The algorithm I used, translated into R code, is presented in the R code section on this page. The coding is deliberately long handed to make it simple for others to follow. This algorithm is able to reproduce all the numbers in table 10.5 of the text book by Machins. For those who are interested, feedback, criticism, correction, and advice are very much welcomed.
Limitations
The algorithm in this web page is written in Javascript and has the following limitations
To calculate the best Optimal results, the program is iterated forewards by 30 (finish = start + 30). If this sample size value exceeds 120, the program will be aborted as it would take too long to run.
To calculate the Minimax model with the least total sample size, the starting value is reduce by 10, then iterate forwards until it reaches the original starting number. The iteration with the minuimum total sample size for Minimax model is then selected as the result. This is sufficient for most cases, but may still produce suboptimal values (Total sample size larger than necessary) if α is low (α<0.01), power is large (pw>=0.9), π0 is close to 0.5 and the difference (π1-π0) less than 0.1
For special cases where the range of search need to be wider than that set in the web page program, the user will need to download the R orogram, adjust the range of search (start and finish) manually, and run the program in R.
A particular cancer, with the current available treatment, has a five year survival rate of 10%.
A new drug is developed which looks promising, and we wish to conduct a Phase II trial. We decided that
if the new drug can improve survival to 40% in a phase II trial, then it is worth the expense
of develop this drug and test it in a large phase III trial (acceptance of the treatment). However, if the
survival rate is no better than 40%, then the new drug should be rejected from
further development (rejection of the treatment). We decided to use α of 0.05, and power of 0.8 in such a study.
Optimal
Minimax
r1
0
1
n1
4
8
rTot
3
3
nTot
15
13
EN
8
9
PET
0.66
0.81
The parameters are therefore α=0.05, (1-β)=0.8, π0 = 0.1, and π1=0.4. The results
are as in the table to the right.
The following are the possible scenarios after, the minimax model is used to demonstrate
Stage 1. n1 = 8, r1 = 1
If no more than 1 case survived (success<=1) in the first 8 cases,
the treatment is abandoned (treatment rejection) at the end of stage 1. End of study
With the second survival (success>1) anytime within the first 8 cases, the trial enters the second stage
Stage 2. nTot = 13, rTot = 3
Including the data from stage 1, if there are 3 or less survivals (successes<=3) at the end of 13 cases,
the treatment is abandoned (treatment rejection) as not worthy of further development. End of study.
On the 4th survival (>3 successes, 2 in stage 2 plus the 2 from stage 1) any time before 13 cases are studied,
the treatment is declared worthy of further development, go onto Phase III trial, etc, (treatment acceptance). End of study.
We expect that, if the true success rate is 40% or more, we need an average of 9 (EN=9) cases to decide accepting the treatment.
If the true success rate is less than 40%, we have 81% chance (PET=0.81) of rejecting the treatment
Sample Size Tables
α=0.1 pw=0.8
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.05
0.15
1
20
4
56
29.5
0.74
1
29
4
44
35.4
0.57
0.1
0.8
0.05
0.2
0
9
2
24
14.5
0.63
0
12
2
21
16.1
0.54
0.1
0.8
0.05
0.25
0
6
2
23
10.5
0.74
0
12
2
16
13.8
0.54
0.1
0.8
0.05
0.3
0
5
1
12
6.6
0.77
0
8
1
9
8.3
0.66
0.1
0.8
0.05
0.35
0
4
1
11
5.3
0.81
0
6
1
8
6.5
0.74
0.1
0.8
0.05
0.4
0
4
1
8
4.7
0.81
0
5
1
7
5.5
0.77
0.1
0.8
0.05
0.45
0
3
1
8
3.7
0.86
0
4
1
6
4.4
0.81
0.1
0.8
0.05
0.5
0
3
1
6
3.4
0.86
0
3
1
6
3.4
0.86
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.1
0.2
2
24
9
65
41.9
0.56
2
29
8
56
44.3
0.43
0.1
0.8
0.1
0.25
1
13
5
34
21
0.62
1
16
5
31
23.3
0.51
0.1
0.8
0.1
0.3
0
7
3
18
12.7
0.48
0
7
3
18
12.7
0.48
0.1
0.8
0.1
0.35
1
8
2
13
8.9
0.81
1
9
2
12
9.7
0.77
0.1
0.8
0.1
0.4
0
4
2
11
6.4
0.66
0
5
2
10
7
0.59
0.1
0.8
0.1
0.45
0
3
2
11
5.2
0.73
0
4
2
9
5.7
0.66
0.1
0.8
0.1
0.5
0
3
1
6
3.8
0.73
0
4
1
5
4.3
0.66
0.1
0.8
0.1
0.55
0
3
1
5
3.5
0.73
0
3
1
5
3.5
0.73
0.1
0.8
0.1
0.6
0
2
1
5
2.6
0.81
0
3
1
4
3.3
0.73
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.15
0.25
4
29
16
84
53.4
0.56
6
46
15
75
61.8
0.45
0.1
0.8
0.15
0.3
3
19
8
39
25.3
0.68
2
18
8
37
27.9
0.48
0.1
0.8
0.15
0.35
1
9
5
23
14.6
0.6
1
10
5
22
15.5
0.54
0.1
0.8
0.15
0.4
1
7
4
18
10.1
0.72
1
9
4
16
11.8
0.6
0.1
0.8
0.15
0.45
1
6
3
13
7.6
0.78
0
4
3
12
7.8
0.52
0.1
0.8
0.15
0.5
0
4
2
8
5.9
0.52
0
4
2
8
5.9
0.52
0.1
0.8
0.15
0.55
0
3
2
7
4.5
0.61
0
3
2
7
4.5
0.61
0.1
0.8
0.15
0.6
0
2
2
8
3.7
0.72
0
3
2
6
4.2
0.61
0.1
0.8
0.15
0.65
0
2
1
4
2.6
0.72
0
2
1
4
2.6
0.72
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.2
0.3
8
39
24
99
61.5
0.62
8
45
21
84
66.8
0.44
0.1
0.8
0.2
0.35
2
13
12
46
29.4
0.5
2
13
12
46
29.4
0.5
0.1
0.8
0.2
0.4
2
12
7
25
17.7
0.56
2
14
7
24
19.5
0.45
0.1
0.8
0.2
0.45
1
7
5
17
11.2
0.58
1
10
5
16
13.7
0.38
0.1
0.8
0.2
0.5
1
6
4
13
8.4
0.66
1
8
4
12
10
0.5
0.1
0.8
0.2
0.55
1
5
3
10
6.3
0.74
0
4
3
9
7
0.41
0.1
0.8
0.2
0.6
0
3
2
6
4.5
0.51
0
3
2
6
4.5
0.51
0.1
0.8
0.2
0.65
0
2
2
6
3.4
0.64
0
2
2
6
3.4
0.64
0.1
0.8
0.2
0.7
0
2
2
5
3.1
0.64
0
2
2
5
3.1
0.64
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.25
0.35
11
43
33
112
69.6
0.61
12
52
29
96
76.3
0.45
0.1
0.8
0.25
0.4
7
25
16
52
32.4
0.73
5
24
15
46
36.7
0.42
0.1
0.8
0.25
0.45
4
15
9
27
18.8
0.69
3
15
9
26
20.9
0.46
0.1
0.8
0.25
0.5
2
8
7
21
12.2
0.68
2
9
6
17
12.2
0.6
0.1
0.8
0.25
0.55
1
5
5
14
8.3
0.63
1
6
5
13
9.3
0.53
0.1
0.8
0.25
0.6
1
5
4
10
6.8
0.63
1
5
4
10
6.8
0.63
0.1
0.8
0.25
0.65
0
2
3
8
4.6
0.56
0
2
3
8
4.6
0.56
0.1
0.8
0.25
0.7
0
2
2
5
3.3
0.56
0
2
2
5
3.3
0.56
0.1
0.8
0.25
0.75
0
2
2
5
3.3
0.56
0
2
2
5
3.3
0.56
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.3
0.4
12
40
42
122
74.7
0.58
20
67
36
102
82.8
0.55
0.1
0.8
0.3
0.45
6
20
20
55
33.7
0.61
6
23
18
48
37
0.44
0.1
0.8
0.3
0.5
5
15
12
32
19.7
0.72
3
12
11
28
20.1
0.49
0.1
0.8
0.3
0.55
2
8
8
20
13.4
0.55
2
8
8
20
13.4
0.55
0.1
0.8
0.3
0.6
1
5
6
14
9.2
0.53
1
5
6
14
9.2
0.53
0.1
0.8
0.3
0.65
2
6
4
9
6.8
0.74
2
6
4
9
6.8
0.74
0.1
0.8
0.3
0.7
2
5
3
7
5.3
0.84
2
5
3
7
5.3
0.84
0.1
0.8
0.3
0.75
1
3
3
7
3.9
0.78
0
2
3
6
4
0.49
0.1
0.8
0.3
0.8
0
2
2
4
3
0.49
0
2
2
4
3
0.49
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.35
0.45
19
52
52
132
79.9
0.65
18
55
44
109
86.2
0.42
0.1
0.8
0.35
0.5
7
20
24
58
35.2
0.6
10
31
21
49
40.8
0.46
0.1
0.8
0.35
0.55
4
12
14
32
20.3
0.58
4
12
14
32
20.3
0.58
0.1
0.8
0.35
0.6
4
10
10
23
13.2
0.75
4
13
8
17
15
0.5
0.1
0.8
0.35
0.65
1
4
7
15
8.8
0.56
2
7
7
14
10.3
0.53
0.1
0.8
0.35
0.7
0
2
5
10
6.6
0.42
0
2
5
10
6.6
0.42
0.1
0.8
0.35
0.75
1
3
5
10
5
0.72
0
2
4
8
5.5
0.42
0.1
0.8
0.35
0.8
1
3
3
6
3.8
0.72
1
3
3
6
3.8
0.72
0.1
0.8
0.35
0.85
0
1
3
6
2.8
0.65
0
1
3
6
2.8
0.65
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.4
0.5
23
55
60
135
82
0.66
27
71
51
112
94.9
0.42
0.1
0.8
0.4
0.55
9
22
28
60
36.3
0.62
11
30
24
50
41.4
0.43
0.1
0.8
0.4
0.6
5
12
18
38
20.7
0.67
6
16
14
28
21.7
0.53
0.1
0.8
0.4
0.65
5
11
10
20
13.2
0.75
3
9
10
19
14.2
0.48
0.1
0.8
0.4
0.7
2
5
8
16
8.5
0.68
2
6
7
13
9.2
0.54
0.1
0.8
0.4
0.75
2
5
5
9
6.3
0.68
2
5
5
9
6.3
0.68
0.1
0.8
0.4
0.8
1
3
4
7
4.4
0.65
1
3
4
7
4.4
0.65
0.1
0.8
0.4
0.85
0
1
4
7
3.4
0.6
1
3
3
5
3.7
0.65
0.1
0.8
0.4
0.9
0
1
3
5
2.6
0.6
0
1
3
5
2.6
0.6
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.45
0.55
22
48
66
133
81.6
0.6
27
61
58
115
87.6
0.51
0.1
0.8
0.45
0.6
9
20
31
60
36.3
0.59
11
26
27
51
39.2
0.47
0.1
0.8
0.45
0.65
8
16
18
34
20.6
0.74
5
12
17
31
21
0.53
0.1
0.8
0.45
0.7
5
10
11
20
12.6
0.74
3
8
11
19
13.8
0.48
0.1
0.8
0.45
0.75
2
5
7
12
7.8
0.59
2
5
7
12
7.8
0.59
0.1
0.8
0.45
0.8
1
3
6
10
6
0.57
1
3
6
10
6
0.57
0.1
0.8
0.45
0.85
2
4
5
8
5
0.76
2
4
5
8
5
0.76
0.1
0.8
0.45
0.9
1
2
4
7
3
0.8
0
1
4
6
3.3
0.55
0.1
0.8
0.45
0.95
0
1
2
3
1.9
0.55
0
1
2
3
1.9
0.55
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.5
0.6
32
61
71
130
82
0.7
52
96
63
114
99.2
0.82
0.1
0.8
0.5
0.65
12
23
34
60
35.5
0.66
10
22
29
50
38.4
0.42
0.1
0.8
0.5
0.7
6
12
19
32
19.7
0.61
7
15
17
28
21.5
0.5
0.1
0.8
0.5
0.75
5
9
13
22
12.3
0.75
3
7
12
19
13
0.5
0.1
0.8
0.5
0.8
4
7
8
13
8.4
0.77
4
7
8
13
8.4
0.77
0.1
0.8
0.5
0.85
2
4
6
9
5.6
0.69
2
4
6
9
5.6
0.69
0.1
0.8
0.5
0.9
0
1
4
6
3.5
0.5
0
1
4
6
3.5
0.5
0.1
0.8
0.5
0.95
0
1
3
4
2.5
0.5
0
1
3
4
2.5
0.5
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.55
0.65
26
47
72
120
78.1
0.57
35
64
66
109
85.2
0.53
0.1
0.8
0.55
0.7
11
20
33
53
33.7
0.59
13
25
31
49
38
0.46
0.1
0.8
0.55
0.75
6
11
20
31
18.9
0.6
6
12
17
26
19.4
0.47
0.1
0.8
0.55
0.8
4
7
13
20
11.1
0.68
3
6
12
18
11.3
0.56
0.1
0.8
0.55
0.85
3
5
8
12
6.8
0.74
1
3
7
10
7
0.43
0.1
0.8
0.55
0.9
2
4
6
8
5.6
0.61
2
4
6
8
5.6
0.61
0.1
0.8
0.55
0.95
1
2
3
4
2.6
0.7
1
2
3
4
2.6
0.7
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.6
0.7
28
46
76
117
74.2
0.6
48
79
67
102
88.3
0.6
0.1
0.8
0.6
0.75
10
17
33
49
31.3
0.55
14
24
30
44
33.8
0.51
0.1
0.8
0.6
0.8
7
11
21
31
16.9
0.7
6
11
17
24
17.9
0.47
0.1
0.8
0.6
0.85
3
5
14
20
10.1
0.66
4
7
11
15
10.4
0.58
0.1
0.8
0.6
0.9
4
6
8
11
7.2
0.77
4
6
8
11
7.2
0.77
0.1
0.8
0.6
0.95
2
3
6
8
4.1
0.78
3
4
5
7
4.4
0.87
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.65
0.75
29
44
75
107
68.8
0.61
43
64
69
98
74.6
0.69
0.1
0.8
0.65
0.8
13
19
37
52
28.8
0.7
12
19
30
41
29.6
0.52
0.1
0.8
0.65
0.85
5
8
18
24
14.8
0.57
5
8
18
24
14.8
0.57
0.1
0.8
0.65
0.9
4
6
10
13
8.2
0.68
4
6
10
13
8.2
0.68
0.1
0.8
0.65
0.95
2
3
7
9
4.6
0.73
2
3
7
9
4.6
0.73
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.7
0.8
27
38
72
96
60.2
0.62
29
42
65
86
63.8
0.5
0.1
0.8
0.7
0.85
10
14
33
43
24.3
0.64
9
13
31
40
24.4
0.58
0.1
0.8
0.7
0.9
4
6
16
20
11.9
0.58
4
6
16
20
11.9
0.58
0.1
0.8
0.7
0.95
2
3
9
11
5.7
0.66
2
3
9
11
5.7
0.66
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.75
0.85
25
33
65
81
51.9
0.61
34
46
59
73
59.8
0.49
0.1
0.8
0.75
0.9
10
13
28
34
20
0.67
7
10
25
30
20.5
0.47
0.1
0.8
0.75
0.95
2
3
15
18
9.3
0.58
0
1
12
14
10.8
0.25
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.8
0.9
23
28
56
66
40
0.69
28
34
52
61
42.1
0.7
0.1
0.8
0.8
0.95
9
11
21
24
15.2
0.68
12
14
20
23
15.8
0.8
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.8
0.85
0.95
12
14
43
48
26.1
0.64
23
26
37
41
29.4
0.77
α=0.1, Power = 0.8 π0 = proportion of success at or below which treatment is abandoned π1 = proportion of success at or above which treatment will be accepted for further development and trials r1 = number of successes above which stage 2 can be entered, and at or below which the trial terminates after n1 cases, and the treatment rejected n1 = maximum sample size in stage 1 rTot = the total number of successes (stage 1 and 2 combined) above which the study can be terminated and the treatment accepted for further development and trials, and at or below which after nTot cases the treatment is rejected. nTot = the total maximum sample size (stage 1 and 2 combined) EN = expected sample size, an estimated average sample size that will be used before the trial terminates. This is a measure of efficiency in detecting PET = probability of early termination (when treatment is abandoned), a measure of efficiency in rejecting ineffective treatments.
α=0.05 pw=0.8
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.05
0.15
1
23
5
56
33.6
0.68
1
30
5
52
39.8
0.55
0.05
0.8
0.05
0.2
0
10
3
29
17.6
0.6
0
13
3
27
19.8
0.51
0.05
0.8
0.05
0.25
0
9
2
17
12
0.63
0
12
2
16
13.8
0.54
0.05
0.8
0.05
0.3
0
5
2
18
7.9
0.77
0
7
2
14
9.1
0.7
0.05
0.8
0.05
0.35
0
4
2
16
6.2
0.81
0
6
2
12
7.6
0.74
0.05
0.8
0.05
0.4
0
4
1
8
4.7
0.81
0
5
1
7
5.5
0.77
0.05
0.8
0.05
0.45
0
3
1
8
3.7
0.86
0
4
1
6
4.4
0.81
0.05
0.8
0.05
0.5
0
3
1
6
3.4
0.86
0
4
1
5
4.2
0.81
0.05
0.8
0.05
0.55
0
3
1
5
3.3
0.86
0
3
1
5
3.3
0.86
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.1
0.2
3
30
13
89
50.8
0.65
4
45
12
78
60.6
0.53
0.05
0.8
0.1
0.25
2
18
7
43
24.7
0.73
2
22
7
40
28.8
0.62
0.05
0.8
0.1
0.3
1
10
5
29
15
0.74
1
15
5
25
19.5
0.55
0.05
0.8
0.1
0.35
1
8
4
22
10.6
0.81
1
11
4
18
13.1
0.7
0.05
0.8
0.1
0.4
0
4
3
15
7.8
0.66
1
8
3
13
8.9
0.81
0.05
0.8
0.1
0.45
0
4
2
9
5.7
0.66
0
4
2
9
5.7
0.66
0.05
0.8
0.1
0.5
0
3
2
9
4.6
0.73
0
4
2
8
5.4
0.66
0.05
0.8
0.1
0.55
0
3
2
7
4.1
0.73
0
3
2
7
4.1
0.73
0.05
0.8
0.1
0.6
0
2
2
8
3.1
0.81
0
3
2
6
3.8
0.73
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.15
0.25
6
38
23
116
64.6
0.66
8
55
20
97
73.7
0.56
0.05
0.8
0.15
0.3
3
19
12
55
30.4
0.68
3
23
11
48
34.5
0.54
0.05
0.8
0.15
0.35
1
9
8
34
19
0.6
2
15
7
28
20.1
0.6
0.05
0.8
0.15
0.4
1
7
6
25
12.1
0.72
1
9
5
19
13
0.6
0.05
0.8
0.15
0.45
1
6
5
19
8.9
0.78
0
5
4
14
10
0.44
0.05
0.8
0.15
0.5
1
5
4
16
6.8
0.84
0
4
3
10
6.9
0.52
0.05
0.8
0.15
0.55
0
3
3
10
5.7
0.61
0
4
3
9
6.4
0.52
0.05
0.8
0.15
0.6
0
3
2
6
4.2
0.61
0
3
2
6
4.2
0.61
0.05
0.8
0.15
0.65
0
2
2
6
3.1
0.72
0
2
2
6
3.1
0.72
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.2
0.3
10
46
35
141
75.1
0.69
13
66
30
116
88.6
0.55
0.05
0.8
0.2
0.35
5
22
19
72
35.4
0.73
6
31
15
53
40.4
0.57
0.05
0.8
0.2
0.4
3
13
12
43
20.6
0.75
4
18
10
33
22.3
0.72
0.05
0.8
0.2
0.45
2
10
7
22
13.9
0.68
2
13
7
21
17
0.5
0.05
0.8
0.2
0.5
2
8
6
18
10
0.8
2
9
6
17
11.1
0.74
0.05
0.8
0.2
0.55
1
5
5
14
7.4
0.74
1
6
4
11
7.7
0.66
0.05
0.8
0.2
0.6
1
4
4
12
5.4
0.82
1
5
4
10
6.3
0.74
0.05
0.8
0.2
0.65
0
2
3
8
4.2
0.64
1
5
3
7
5.5
0.74
0.05
0.8
0.2
0.7
0
2
3
7
3.8
0.64
0
2
3
7
3.8
0.64
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.25
0.35
15
56
45
149
85.3
0.69
20
80
40
129
101.6
0.56
0.05
0.8
0.25
0.4
5
20
23
71
39.5
0.62
16
51
20
60
52
0.89
0.05
0.8
0.25
0.45
5
17
14
41
22.6
0.77
4
17
13
36
25.1
0.57
0.05
0.8
0.25
0.5
2
9
9
24
15
0.6
2
9
9
24
15
0.6
0.05
0.8
0.25
0.55
2
7
8
21
10.4
0.76
2
9
7
17
12.2
0.6
0.05
0.8
0.25
0.6
1
5
5
12
7.6
0.63
1
5
5
12
7.6
0.63
0.05
0.8
0.25
0.65
1
4
5
12
6.1
0.74
0
3
4
9
6.5
0.42
0.05
0.8
0.25
0.7
0
2
4
9
5.1
0.56
0
3
4
8
5.9
0.42
0.05
0.8
0.25
0.75
1
3
3
7
3.6
0.84
0
2
3
6
3.8
0.56
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.3
0.4
19
59
59
168
91.7
0.7
36
107
51
142
113.2
0.82
0.05
0.8
0.3
0.45
9
27
30
81
41.7
0.73
16
46
25
65
49.6
0.81
0.05
0.8
0.3
0.5
5
15
18
46
23.6
0.72
6
19
16
39
25.7
0.67
0.05
0.8
0.3
0.55
3
9
14
35
16
0.73
2
9
11
25
17.6
0.46
0.05
0.8
0.3
0.6
3
8
10
24
11.1
0.81
2
10
8
17
14.3
0.38
0.05
0.8
0.3
0.65
1
4
7
15
7.8
0.65
2
7
7
14
9.5
0.65
0.05
0.8
0.3
0.7
0
2
5
10
6.1
0.49
0
2
5
10
6.1
0.49
0.05
0.8
0.3
0.75
1
3
5
10
4.5
0.78
2
5
4
8
5.5
0.84
0.05
0.8
0.3
0.8
1
3
4
7
3.9
0.78
2
4
3
6
4.2
0.92
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.35
0.45
23
62
71
176
97.6
0.69
38
113
61
148
133.2
0.42
0.05
0.8
0.35
0.5
10
27
33
77
43.5
0.67
22
55
29
66
57
0.82
0.05
0.8
0.35
0.55
5
14
20
44
24.8
0.64
8
21
18
39
26.3
0.71
0.05
0.8
0.35
0.6
3
9
13
27
16
0.61
6
18
13
26
21.6
0.55
0.05
0.8
0.35
0.65
2
6
10
20
10.9
0.65
8
16
9
18
16.1
0.93
0.05
0.8
0.35
0.7
2
5
8
16
7.6
0.76
2
6
7
13
8.5
0.65
0.05
0.8
0.35
0.75
1
3
7
13
5.8
0.72
2
5
5
9
5.9
0.76
0.05
0.8
0.35
0.8
1
3
4
7
4.1
0.72
1
3
4
7
4.1
0.72
0.05
0.8
0.35
0.85
0
1
4
7
3.1
0.65
0
1
4
7
3.1
0.65
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.4
0.5
25
60
80
176
99.9
0.66
53
121
71
154
126.7
0.83
0.05
0.8
0.4
0.55
11
26
40
84
44.9
0.67
28
59
34
70
60.1
0.9
0.05
0.8
0.4
0.6
7
16
23
46
24.5
0.72
17
34
20
39
34.4
0.91
0.05
0.8
0.4
0.65
5
11
16
31
15.9
0.75
5
12
14
26
16.7
0.67
0.05
0.8
0.4
0.7
3
7
11
20
10.8
0.71
6
12
10
18
12.9
0.84
0.05
0.8
0.4
0.75
2
5
8
14
7.9
0.68
2
5
8
14
7.9
0.68
0.05
0.8
0.4
0.8
1
3
6
10
5.5
0.65
1
3
6
10
5.5
0.65
0.05
0.8
0.4
0.85
0
1
6
10
4.6
0.6
2
4
5
8
4.7
0.82
0.05
0.8
0.4
0.9
0
1
4
6
3
0.6
0
1
4
6
3
0.6
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.45
0.55
31
66
88
174
101.3
0.67
47
106
79
154
130.7
0.49
0.05
0.8
0.45
0.6
12
26
41
77
45.1
0.63
19
42
38
70
53.9
0.58
0.05
0.8
0.45
0.65
7
15
24
43
24.7
0.65
16
30
22
39
31.2
0.86
0.05
0.8
0.45
0.7
5
10
19
33
16
0.74
5
12
15
25
18.1
0.53
0.05
0.8
0.45
0.75
2
5
11
18
10.3
0.59
2
6
10
16
11.6
0.44
0.05
0.8
0.45
0.8
1
3
9
14
7.7
0.57
1
3
9
14
7.7
0.57
0.05
0.8
0.45
0.85
2
4
6
9
5.2
0.76
2
4
6
9
5.2
0.76
0.05
0.8
0.45
0.9
1
2
7
11
3.8
0.8
1
3
5
7
4.7
0.57
0.05
0.8
0.45
0.95
0
1
3
4
2.4
0.55
0
1
3
4
2.4
0.55
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.5
0.6
30
58
101
182
101
0.65
68
125
87
155
129.2
0.86
0.05
0.8
0.5
0.65
15
28
48
83
43.7
0.71
39
66
40
68
66.1
0.95
0.05
0.8
0.5
0.7
8
15
26
43
23.5
0.7
12
23
23
37
27.7
0.66
0.05
0.8
0.5
0.75
6
11
16
25
14.8
0.73
7
14
15
23
17.6
0.6
0.05
0.8
0.5
0.8
4
7
13
20
9.9
0.77
3
6
12
18
10.1
0.66
0.05
0.8
0.5
0.85
2
4
9
13
6.8
0.69
7
10
8
12
10.1
0.95
0.05
0.8
0.5
0.9
1
2
9
14
5
0.75
2
4
6
8
5.3
0.69
0.05
0.8
0.5
0.95
1
2
5
7
3.3
0.75
3
4
4
6
4.1
0.94
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.55
0.65
35
61
107
177
96.9
0.69
50
90
92
150
115
0.58
0.05
0.8
0.55
0.7
15
26
48
76
42
0.68
20
35
43
67
45.8
0.66
0.05
0.8
0.55
0.75
9
15
28
43
22.3
0.74
15
24
24
36
26.1
0.83
0.05
0.8
0.55
0.8
4
7
19
28
13.6
0.68
5
9
16
23
14.1
0.64
0.05
0.8
0.55
0.85
3
5
14
20
8.8
0.74
4
7
11
15
9.5
0.68
0.05
0.8
0.55
0.9
3
5
9
12
6.8
0.74
5
7
8
11
7.4
0.9
0.05
0.8
0.55
0.95
1
2
6
8
3.8
0.7
1
2
6
8
3.8
0.7
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.6
0.7
34
55
107
163
92
0.66
92
139
94
142
139.2
0.94
0.05
0.8
0.6
0.75
17
27
46
67
39.3
0.69
18
30
43
62
43.8
0.57
0.05
0.8
0.6
0.8
7
11
30
43
20.5
0.7
8
13
25
35
20.8
0.65
0.05
0.8
0.6
0.85
6
9
17
23
12.2
0.77
4
7
15
20
12.5
0.58
0.05
0.8
0.6
0.9
4
6
10
13
7.6
0.77
4
6
10
13
7.6
0.77
0.05
0.8
0.6
0.95
2
3
8
10
4.5
0.78
2
3
8
10
4.5
0.78
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.65
0.75
34
51
104
147
84.7
0.65
80
114
94
132
115.9
0.9
0.05
0.8
0.65
0.8
12
18
49
67
35.4
0.65
20
31
41
55
41.9
0.54
0.05
0.8
0.65
0.85
10
14
25
33
18.2
0.78
19
25
23
30
25.4
0.92
0.05
0.8
0.65
0.9
4
6
15
19
10.1
0.68
12
15
14
18
15.2
0.94
0.05
0.8
0.65
0.95
2
3
9
11
5.2
0.73
2
3
9
11
5.2
0.73
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.7
0.8
34
47
105
139
75.6
0.69
66
89
90
118
93.8
0.83
0.05
0.8
0.7
0.85
14
19
46
59
30.3
0.72
16
23
39
49
34.4
0.56
0.05
0.8
0.7
0.9
4
6
22
27
14.8
0.58
19
23
21
26
23.2
0.95
0.05
0.8
0.7
0.95
3
4
17
21
8.1
0.76
5
7
12
14
9.3
0.67
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.75
0.85
36
46
95
118
64.2
0.75
77
95
82
101
95.4
0.93
0.05
0.8
0.75
0.9
10
13
40
48
24.6
0.67
17
22
33
39
27.5
0.68
0.05
0.8
0.75
0.95
2
3
19
22
11
0.58
14
16
17
20
16.3
0.94
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.8
0.9
23
28
83
97
49.7
0.69
60
70
69
80
70.8
0.92
0.05
0.8
0.8
0.95
7
9
26
29
17.7
0.56
7
9
26
29
17.7
0.56
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.8
0.85
0.95
11
13
59
65
33.7
0.6
35
39
52
57
41.6
0.86
α=0.1, Power = 0.8 π0 = proportion of success at or below which treatment is abandoned π1 = proportion of success at or above which treatment will be accepted for further development and trials r1 = number of successes above which stage 2 can be entered, and at or below which the trial terminates after n1 cases, and the treatment rejected n1 = maximum sample size in stage 1 rTot = the total number of successes (stage 1 and 2 combined) above which the study can be terminated and the treatment accepted for further development and trials, and at or below which after nTot cases the treatment is rejected. nTot = the total maximum sample size (stage 1 and 2 combined) EN = expected sample size, an estimated average sample size that will be used before the trial terminates. This is a measure of efficiency in detecting PET = probability of early termination (when treatment is abandoned), a measure of efficiency in rejecting ineffective treatments.
α=0.01 pw=0.8
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.05
0.15
2
30
10
102
43.5
0.81
2
45
9
82
59.5
0.61
0.01
0.8
0.05
0.2
1
16
6
50
22.4
0.81
1
25
6
44
31.8
0.64
0.01
0.8
0.05
0.25
1
12
5
36
14.8
0.88
0
11
4
26
17.5
0.57
0.01
0.8
0.05
0.3
0
5
4
27
10
0.77
0
7
3
18
10.3
0.7
0.01
0.8
0.05
0.35
0
4
4
24
7.7
0.81
0
7
3
15
9.4
0.7
0.01
0.8
0.05
0.4
0
4
3
15
6
0.81
0
6
3
13
7.9
0.74
0.01
0.8
0.05
0.45
0
3
3
14
4.6
0.86
0
4
2
9
4.9
0.81
0.01
0.8
0.05
0.5
0
3
2
9
3.9
0.86
0
4
2
8
4.7
0.81
0.01
0.8
0.05
0.55
0
3
2
7
3.6
0.86
0
3
2
7
3.6
0.86
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.1
0.2
5
41
23
149
65.1
0.78
6
58
20
122
81
0.64
0.01
0.8
0.1
0.25
3
22
14
80
32
0.83
3
30
12
62
41.3
0.65
0.01
0.8
0.1
0.3
2
14
9
46
19.1
0.84
2
18
8
37
23.1
0.73
0.01
0.8
0.1
0.35
1
8
8
37
13.4
0.81
1
11
6
25
15.2
0.7
0.01
0.8
0.1
0.4
1
7
6
25
9.7
0.85
1
9
5
19
11.3
0.77
0.01
0.8
0.1
0.45
1
6
5
19
7.5
0.89
0
5
4
14
8.7
0.59
0.01
0.8
0.1
0.5
1
5
4
16
5.9
0.92
1
8
4
12
8.7
0.81
0.01
0.8
0.1
0.55
1
5
3
10
5.4
0.92
0
4
3
9
5.7
0.66
0.01
0.8
0.1
0.6
0
2
3
10
3.5
0.81
0
3
3
8
4.4
0.73
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.15
0.25
10
56
39
188
83.8
0.79
12
76
34
157
104.3
0.65
0.01
0.8
0.15
0.3
5
26
23
101
39.9
0.82
7
42
18
73
51
0.71
0.01
0.8
0.15
0.35
3
15
16
65
23.9
0.82
3
19
12
44
26.9
0.68
0.01
0.8
0.15
0.4
2
10
11
41
15.6
0.82
2
13
9
30
18.2
0.69
0.01
0.8
0.15
0.45
2
9
8
27
11.5
0.86
2
13
7
21
15.5
0.69
0.01
0.8
0.15
0.5
1
5
8
27
8.6
0.84
1
7
6
17
9.8
0.72
0.01
0.8
0.15
0.55
1
5
5
14
6.5
0.84
1
6
5
13
7.6
0.78
0.01
0.8
0.15
0.6
1
4
5
15
5.2
0.89
0
3
4
10
5.7
0.61
0.01
0.8
0.15
0.65
0
2
4
10
4.2
0.72
0
3
4
9
5.3
0.61
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.2
0.3
14
62
56
215
99.8
0.75
45
173
49
183
173.2
0.98
0.01
0.8
0.2
0.35
7
29
31
109
45.8
0.79
9
42
26
87
56.6
0.68
0.01
0.8
0.2
0.4
4
16
21
69
26.7
0.8
6
25
17
52
30.9
0.78
0.01
0.8
0.2
0.45
2
9
14
42
17.6
0.74
4
16
12
34
19.6
0.8
0.01
0.8
0.2
0.5
2
8
11
31
12.7
0.8
3
12
10
26
14.9
0.79
0.01
0.8
0.2
0.55
2
7
9
24
9.5
0.85
2
9
8
19
11.6
0.74
0.01
0.8
0.2
0.6
1
4
8
21
7.1
0.82
5
12
6
14
12
0.98
0.01
0.8
0.2
0.65
1
4
6
14
5.8
0.82
2
6
5
11
6.5
0.9
0.01
0.8
0.2
0.7
0
2
5
10
4.9
0.64
0
2
5
10
4.9
0.64
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.25
0.4
11
36
41
123
50.5
0.83
11
42
34
97
61.3
0.65
0.01
0.8
0.25
0.45
5
17
25
69
29.2
0.77
6
24
21
55
36.2
0.61
0.01
0.8
0.25
0.5
4
13
17
43
19.2
0.79
5
17
16
39
22.2
0.77
0.01
0.8
0.25
0.55
2
7
14
34
13.6
0.76
2
9
12
27
16.2
0.6
0.01
0.8
0.25
0.6
2
6
12
29
9.9
0.83
2
8
9
19
11.5
0.68
0.01
0.8
0.25
0.65
1
4
8
17
7.4
0.74
2
7
7
14
8.7
0.76
0.01
0.8
0.25
0.7
2
5
8
16
6.1
0.9
2
6
7
13
7.2
0.83
0.01
0.8
0.25
0.75
1
3
6
12
4.4
0.84
0
2
5
9
5.1
0.56
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.3
0.45
13
37
50
129
54.7
0.81
18
59
42
104
77.1
0.6
0.01
0.8
0.3
0.5
7
20
29
69
31.2
0.77
8
25
26
60
36.3
0.68
0.01
0.8
0.3
0.55
5
13
24
56
20.1
0.83
3
11
18
39
23.1
0.57
0.01
0.8
0.3
0.6
4
10
17
37
14.1
0.85
6
18
13
26
20.2
0.72
0.01
0.8
0.3
0.65
2
6
12
24
10.6
0.74
7
15
10
19
15.2
0.95
0.01
0.8
0.3
0.7
2
5
11
21
7.6
0.84
2
6
9
16
8.6
0.74
0.01
0.8
0.3
0.75
1
3
9
17
6
0.78
2
5
7
12
6.1
0.84
0.01
0.8
0.3
0.8
1
3
6
10
4.5
0.78
1
3
6
10
4.5
0.78
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.35
0.5
13
33
60
137
57.5
0.76
19
52
49
108
71.5
0.65
0.01
0.8
0.35
0.55
8
20
34
72
32.4
0.76
12
30
30
62
37
0.78
0.01
0.8
0.35
0.6
5
12
25
51
20.3
0.79
7
17
21
41
22.1
0.79
0.01
0.8
0.35
0.65
4
9
20
39
14.2
0.83
3
9
16
29
16.8
0.61
0.01
0.8
0.35
0.7
3
7
13
23
10.2
0.8
4
9
12
21
11.1
0.83
0.01
0.8
0.35
0.75
3
6
12
21
7.8
0.88
7
12
9
15
12.1
0.97
0.01
0.8
0.35
0.8
2
4
10
17
5.6
0.87
4
7
8
13
7.3
0.94
0.01
0.8
0.35
0.85
0
1
7
11
4.5
0.65
2
4
6
9
4.6
0.87
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.4
0.55
15
34
65
132
58.6
0.75
20
49
56
111
73.4
0.61
0.01
0.8
0.4
0.6
11
23
43
84
33
0.84
13
32
34
63
44.3
0.6
0.01
0.8
0.4
0.65
6
13
26
47
20.8
0.77
8
18
24
42
24.3
0.74
0.01
0.8
0.4
0.7
5
10
19
33
13.8
0.83
5
12
17
28
17.4
0.67
0.01
0.8
0.4
0.75
3
6
16
27
9.8
0.82
9
15
13
21
15.2
0.97
0.01
0.8
0.4
0.8
2
4
14
23
7.4
0.82
2
5
10
15
8.2
0.68
0.01
0.8
0.4
0.85
2
4
9
13
5.6
0.82
7
10
8
12
10
0.99
0.01
0.8
0.4
0.9
1
2
9
14
3.9
0.84
2
4
6
8
4.7
0.82
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.45
0.6
22
43
74
137
58.7
0.83
48
90
62
112
91
0.95
0.01
0.8
0.45
0.65
11
21
47
83
32.4
0.82
16
34
36
61
43.1
0.66
0.01
0.8
0.45
0.7
7
13
32
54
20.3
0.82
9
19
25
40
25.9
0.67
0.01
0.8
0.45
0.75
5
9
23
37
13.6
0.83
11
18
18
28
18.5
0.95
0.01
0.8
0.45
0.8
4
7
16
24
9.6
0.85
5
9
13
19
10.7
0.83
0.01
0.8
0.45
0.85
3
5
13
19
6.8
0.87
6
9
10
14
9.2
0.95
0.01
0.8
0.45
0.9
1
2
11
16
4.8
0.8
5
7
8
11
7.1
0.96
0.01
0.8
0.45
0.95
1
2
7
9
3.4
0.8
3
4
6
8
4.2
0.96
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.5
0.65
23
41
81
137
57.7
0.83
60
101
66
109
101.2
0.98
0.01
0.8
0.5
0.7
12
21
46
74
31.2
0.81
13
25
39
61
37.4
0.65
0.01
0.8
0.5
0.75
7
12
33
51
19.6
0.81
11
20
25
37
24.3
0.75
0.01
0.8
0.5
0.8
4
7
22
32
12.7
0.77
17
24
18
26
24
0.99
0.01
0.8
0.5
0.85
3
5
17
24
8.6
0.81
5
8
14
19
9.6
0.86
0.01
0.8
0.5
0.9
3
5
11
14
6.7
0.81
4
6
10
13
6.8
0.89
0.01
0.8
0.5
0.95
2
3
8
10
3.9
0.88
2
3
8
10
3.9
0.88
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.55
0.7
21
35
80
124
54.9
0.78
28
50
69
105
71.4
0.61
0.01
0.8
0.55
0.75
12
19
53
80
29.5
0.83
12
21
40
58
33.6
0.66
0.01
0.8
0.55
0.8
7
11
33
47
17.9
0.81
7
12
26
36
19.3
0.7
0.01
0.8
0.55
0.85
6
9
20
27
11.7
0.85
15
20
18
24
20.1
0.98
0.01
0.8
0.55
0.9
4
6
14
18
8
0.84
12
15
13
17
15
0.99
0.01
0.8
0.55
0.95
3
4
11
14
4.9
0.91
1
2
10
12
5
0.7
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.6
0.75
23
35
82
118
51.2
0.8
31
49
70
99
62.6
0.73
0.01
0.8
0.6
0.8
13
19
49
68
27
0.84
15
23
40
54
30.4
0.76
0.01
0.8
0.6
0.85
7
10
35
47
16.2
0.83
7
12
25
32
20.8
0.56
0.01
0.8
0.6
0.9
5
7
19
24
9.7
0.84
3
5
17
21
10.4
0.66
0.01
0.8
0.6
0.95
2
3
14
17
6
0.78
4
6
12
14
7.9
0.77
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.65
0.8
19
27
86
116
46.4
0.78
30
44
69
91
57.1
0.72
0.01
0.8
0.65
0.85
10
14
45
58
23.7
0.78
25
33
38
48
34
0.93
0.01
0.8
0.65
0.9
5
7
29
36
13.8
0.77
8
11
24
29
14.6
0.8
0.01
0.8
0.65
0.95
3
4
20
24
7.6
0.82
5
7
14
16
9.1
0.77
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.7
0.8
47
64
167
218
99.9
0.77
62
86
149
193
117.9
0.7
0.01
0.8
0.7
0.85
19
25
80
101
39.7
0.81
31
43
64
79
54.8
0.67
0.01
0.8
0.7
0.9
10
13
40
48
20.1
0.8
33
39
35
42
39
0.99
0.01
0.8
0.7
0.95
3
4
28
33
11
0.76
9
11
21
24
12.5
0.89
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.75
0.85
35
45
155
190
85.6
0.72
58
76
135
164
107
0.65
0.01
0.8
0.75
0.9
18
22
74
88
32.7
0.84
30
38
56
65
44.2
0.77
0.01
0.8
0.75
0.95
10
12
32
36
15.8
0.84
27
30
29
33
30
0.99
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.8
0.9
37
44
135
156
65.9
0.8
109
125
115
132
125.1
0.99
0.01
0.8
0.8
0.95
12
14
53
59
22.9
0.8
42
46
45
50
46
0.99
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.8
0.85
0.95
24
27
99
108
43.8
0.79
27
31
85
92
49.1
0.7
α=0.1, Power = 0.8 π0 = proportion of success at or below which treatment is abandoned π1 = proportion of success at or above which treatment will be accepted for further development and trials r1 = number of successes above which stage 2 can be entered, and at or below which the trial terminates after n1 cases, and the treatment rejected n1 = maximum sample size in stage 1 rTot = the total number of successes (stage 1 and 2 combined) above which the study can be terminated and the treatment accepted for further development and trials, and at or below which after nTot cases the treatment is rejected. nTot = the total maximum sample size (stage 1 and 2 combined) EN = expected sample size, an estimated average sample size that will be used before the trial terminates. This is a measure of efficiency in detecting PET = probability of early termination (when treatment is abandoned), a measure of efficiency in rejecting ineffective treatments.
α=0.1 pw=0.9
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.05
0.15
1
28
5
66
43.6
0.59
1
39
5
60
51.3
0.41
0.1
0.9
0.05
0.2
0
12
3
37
23.5
0.54
0
18
3
32
26.4
0.4
0.1
0.9
0.05
0.25
0
9
2
24
14.5
0.63
0
13
2
20
16.4
0.51
0.1
0.9
0.05
0.3
0
7
2
21
11.2
0.7
0
13
2
16
14.5
0.51
0.1
0.9
0.05
0.35
0
6
1
12
7.6
0.74
0
7
1
10
7.9
0.7
0.1
0.9
0.05
0.4
0
5
1
10
6.1
0.77
0
6
1
9
6.8
0.74
0.1
0.9
0.05
0.45
0
4
1
10
5.1
0.81
0
5
1
8
5.7
0.77
0.1
0.9
0.05
0.5
0
4
1
7
4.6
0.81
0
4
1
7
4.6
0.81
0.1
0.9
0.05
0.55
0
3
1
8
3.7
0.86
0
4
1
6
4.4
0.81
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.1
0.2
4
41
13
99
63.7
0.61
5
61
12
86
75.5
0.42
0.1
0.9
0.1
0.25
2
21
7
50
31.2
0.65
2
27
6
40
33.7
0.48
0.1
0.9
0.1
0.3
1
12
5
35
19.8
0.66
1
16
4
25
20.4
0.51
0.1
0.9
0.1
0.35
1
11
3
19
13.4
0.7
0
8
3
18
13.7
0.43
0.1
0.9
0.1
0.4
0
5
3
18
10.3
0.59
0
8
3
15
12
0.43
0.1
0.9
0.1
0.45
0
5
2
11
7.5
0.59
0
8
2
10
9.1
0.43
0.1
0.9
0.1
0.5
0
4
2
10
6.1
0.66
0
5
2
9
6.6
0.59
0.1
0.9
0.1
0.55
0
3
2
11
5.2
0.73
0
5
2
8
6.2
0.59
0.1
0.9
0.1
0.6
0
3
1
6
3.8
0.73
0
4
1
5
4.3
0.66
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.15
0.25
8
52
23
125
79.5
0.62
9
65
20
105
85.7
0.48
0.1
0.9
0.15
0.3
3
23
11
55
37.7
0.54
5
34
11
53
41.7
0.6
0.1
0.9
0.15
0.35
3
19
7
33
23.4
0.68
2
17
7
32
24.2
0.52
0.1
0.9
0.15
0.4
1
10
5
22
15.5
0.54
2
15
5
21
17.4
0.6
0.1
0.9
0.15
0.45
1
8
4
17
11.1
0.66
1
9
4
16
11.8
0.6
0.1
0.9
0.15
0.5
1
7
3
13
8.7
0.72
0
5
3
12
8.9
0.44
0.1
0.9
0.15
0.55
1
6
3
11
7.1
0.78
1
6
3
11
7.1
0.78
0.1
0.9
0.15
0.6
0
3
2
8
4.9
0.61
0
5
2
7
6.1
0.44
0.1
0.9
0.15
0.65
0
3
2
7
4.5
0.61
0
3
2
7
4.5
0.61
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.2
0.3
12
59
34
144
92.8
0.6
14
73
30
124
98.5
0.5
0.1
0.9
0.2
0.35
5
27
16
63
43.6
0.54
6
33
15
58
45.5
0.5
0.1
0.9
0.2
0.4
3
17
10
37
26
0.55
3
19
10
36
28.3
0.46
0.1
0.9
0.2
0.45
3
14
7
25
17.3
0.7
3
15
7
24
18.2
0.65
0.1
0.9
0.2
0.5
2
10
5
17
12.3
0.68
2
10
5
17
12.3
0.68
0.1
0.9
0.2
0.55
0
4
4
13
9.3
0.41
0
4
4
13
9.3
0.41
0.1
0.9
0.2
0.6
1
5
4
14
7.4
0.74
1
7
3
9
7.8
0.58
0.1
0.9
0.2
0.65
0
3
3
9
5.9
0.51
0
3
3
9
5.9
0.51
0.1
0.9
0.2
0.7
0
3
2
6
4.5
0.51
0
3
2
6
4.5
0.51
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.25
0.35
19
74
44
152
104.1
0.61
18
75
41
140
108.6
0.48
0.1
0.9
0.25
0.4
7
29
22
72
48.1
0.56
9
39
20
64
52.1
0.48
0.1
0.9
0.25
0.45
3
14
14
44
28.4
0.52
5
23
13
39
31.5
0.47
0.1
0.9
0.25
0.5
2
10
9
27
18.1
0.53
2
11
9
26
19.2
0.46
0.1
0.9
0.25
0.55
2
9
7
20
13.4
0.6
5
15
6
17
15.3
0.85
0.1
0.9
0.25
0.6
2
8
5
14
9.9
0.68
1
8
5
13
11.2
0.37
0.1
0.9
0.25
0.65
1
5
4
11
7.2
0.63
1
6
4
10
7.9
0.53
0.1
0.9
0.25
0.7
1
4
4
11
5.8
0.74
2
6
3
8
6.3
0.83
0.1
0.9
0.25
0.75
0
2
3
8
4.6
0.56
1
4
3
7
4.8
0.74
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.3
0.4
23
75
58
171
112.9
0.61
22
78
52
151
120.5
0.42
0.1
0.9
0.3
0.45
9
30
29
82
51.4
0.59
16
50
25
69
56
0.68
0.1
0.9
0.3
0.5
7
22
17
46
29.9
0.67
7
28
15
39
35
0.36
0.1
0.9
0.3
0.55
4
13
12
31
19.2
0.65
4
16
10
25
21
0.45
0.1
0.9
0.3
0.6
2
8
8
20
13.4
0.55
2
9
8
19
14.4
0.46
0.1
0.9
0.3
0.65
2
7
7
16
10.2
0.65
3
9
6
14
10.4
0.73
0.1
0.9
0.3
0.7
2
6
5
12
7.5
0.74
2
7
4
9
7.7
0.65
0.1
0.9
0.3
0.75
0
2
4
9
5.6
0.49
0
2
4
9
5.6
0.49
0.1
0.9
0.3
0.8
1
4
3
6
4.7
0.65
1
4
3
6
4.7
0.65
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.35
0.45
27
77
67
171
118.7
0.56
61
155
62
157
155.2
0.89
0.1
0.9
0.35
0.5
12
34
33
81
53.2
0.59
14
43
30
72
59.3
0.44
0.1
0.9
0.35
0.55
7
20
20
47
30.8
0.6
15
36
18
42
36.9
0.84
0.1
0.9
0.35
0.6
6
16
12
27
19.4
0.69
6
16
12
27
19.4
0.69
0.1
0.9
0.35
0.65
2
7
10
22
14
0.53
4
13
8
17
15
0.5
0.1
0.9
0.35
0.7
2
6
7
15
9.2
0.65
3
9
7
14
11
0.61
0.1
0.9
0.35
0.75
2
6
5
10
7.4
0.65
2
6
5
10
7.4
0.65
0.1
0.9
0.35
0.8
0
2
4
8
5.5
0.42
0
2
4
8
5.5
0.42
0.1
0.9
0.35
0.85
1
3
3
6
3.8
0.72
1
3
3
6
3.8
0.72
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.4
0.5
30
74
84
191
122.3
0.59
57
134
73
164
141.3
0.76
0.1
0.9
0.4
0.55
16
38
40
88
54.5
0.67
18
45
34
73
57.2
0.56
0.1
0.9
0.4
0.6
7
18
22
46
30.2
0.56
11
28
20
41
33.8
0.55
0.1
0.9
0.4
0.65
5
13
14
28
19.4
0.57
5
13
14
28
19.4
0.57
0.1
0.9
0.4
0.7
5
11
10
20
13.2
0.75
2
7
10
19
14
0.42
0.1
0.9
0.4
0.75
2
6
7
13
9.2
0.54
2
6
7
13
9.2
0.54
0.1
0.9
0.4
0.8
1
4
5
9
6.6
0.48
1
4
5
9
6.6
0.48
0.1
0.9
0.4
0.85
1
3
5
9
5.1
0.65
0
2
4
7
5.2
0.36
0.1
0.9
0.4
0.9
1
3
3
5
3.7
0.65
1
3
3
5
3.7
0.65
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.45
0.55
38
83
90
183
122.9
0.6
56
128
81
163
148.2
0.42
0.1
0.9
0.45
0.6
14
32
40
78
54.2
0.52
34
67
38
74
68
0.86
0.1
0.9
0.45
0.65
9
20
24
45
30.2
0.59
9
21
22
41
30.8
0.51
0.1
0.9
0.45
0.7
5
12
15
27
19.1
0.53
6
15
14
25
20.5
0.45
0.1
0.9
0.45
0.75
5
10
11
20
12.6
0.74
2
6
11
19
13.3
0.44
0.1
0.9
0.45
0.8
2
5
9
15
9.1
0.59
4
9
8
13
10.5
0.62
0.1
0.9
0.45
0.85
1
3
6
10
6
0.57
1
3
6
10
6
0.57
0.1
0.9
0.45
0.9
2
4
5
8
5
0.76
3
5
4
7
5.3
0.87
0.1
0.9
0.45
0.95
1
2
4
7
3
0.8
0
1
4
6
3.3
0.55
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.5
0.6
45
87
103
191
121.8
0.67
78
146
89
163
149.1
0.82
0.1
0.9
0.5
0.65
18
35
47
84
53
0.63
19
40
41
72
58
0.44
0.1
0.9
0.5
0.7
11
21
26
45
29
0.67
11
23
23
39
31
0.5
0.1
0.9
0.5
0.75
6
12
17
28
18.2
0.61
13
22
15
25
22.4
0.86
0.1
0.9
0.5
0.8
3
7
10
16
11.5
0.5
3
7
10
16
11.5
0.5
0.1
0.9
0.5
0.85
4
7
8
13
8.4
0.77
4
8
8
12
9.5
0.64
0.1
0.9
0.5
0.9
2
4
6
9
5.6
0.69
2
4
6
9
5.6
0.69
0.1
0.9
0.5
0.95
0
1
4
6
3.5
0.5
0
1
4
6
3.5
0.5
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.55
0.65
44
79
102
172
116.9
0.59
52
96
95
159
129.1
0.47
0.1
0.9
0.55
0.7
19
34
46
75
50.1
0.61
35
58
43
70
60.1
0.83
0.1
0.9
0.55
0.75
10
18
26
41
27
0.61
20
32
24
38
32.9
0.85
0.1
0.9
0.55
0.8
4
8
17
26
16.6
0.52
13
20
15
23
20.4
0.87
0.1
0.9
0.55
0.85
4
7
12
18
10.5
0.68
2
5
11
16
11.5
0.41
0.1
0.9
0.55
0.9
2
4
7
10
6.3
0.61
2
4
7
10
6.3
0.61
0.1
0.9
0.55
0.95
0
1
5
7
4.3
0.45
0
1
5
7
4.3
0.45
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.6
0.7
43
71
108
168
111.4
0.58
83
131
97
150
134.6
0.81
0.1
0.9
0.6
0.75
21
34
47
71
47.1
0.65
25
43
43
64
54.4
0.46
0.1
0.9
0.6
0.8
6
11
26
38
25.4
0.47
18
27
24
35
28.5
0.82
0.1
0.9
0.6
0.85
7
11
16
23
14.6
0.7
5
9
15
21
14.8
0.52
0.1
0.9
0.6
0.9
3
5
12
17
9
0.66
7
10
10
14
10.7
0.83
0.1
0.9
0.6
0.95
0
1
6
8
5.2
0.4
0
1
6
8
5.2
0.4
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.65
0.75
47
71
106
153
101.6
0.63
45
71
97
139
109.7
0.43
0.1
0.9
0.65
0.8
20
30
45
63
41.8
0.64
22
33
43
60
42.6
0.64
0.1
0.9
0.65
0.85
10
15
25
34
21.7
0.65
8
13
23
31
22
0.5
0.1
0.9
0.65
0.9
5
8
13
17
11.9
0.57
5
8
13
17
11.9
0.57
0.1
0.9
0.65
0.95
3
5
8
10
7.1
0.57
3
5
8
10
7.1
0.57
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.7
0.8
37
53
99
133
89.8
0.54
44
64
92
123
95.8
0.46
0.1
0.9
0.7
0.85
14
20
45
59
36.2
0.58
15
22
40
52
36.8
0.51
0.1
0.9
0.7
0.9
6
9
22
28
17.8
0.54
11
16
20
25
20
0.55
0.1
0.9
0.7
0.95
5
7
13
16
10
0.67
7
9
12
15
10.2
0.8
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.75
0.85
39
51
99
125
77
0.65
68
87
83
104
90.6
0.79
0.1
0.9
0.75
0.9
12
16
39
48
29
0.6
20
27
33
40
33.1
0.53
0.1
0.9
0.75
0.95
6
8
16
19
12
0.63
6
8
16
19
12
0.63
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.8
0.9
30
37
80
95
58.5
0.63
66
78
71
84
78.7
0.88
0.1
0.9
0.8
0.95
5
7
27
31
20.8
0.42
5
7
27
31
20.8
0.42
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.1
0.9
0.85
0.95
25
29
53
59
39.5
0.65
40
45
52
58
47.3
0.83
α=0.1, Power = 0.8 π0 = proportion of success at or below which treatment is abandoned π1 = proportion of success at or above which treatment will be accepted for further development and trials r1 = number of successes above which stage 2 can be entered, and at or below which the trial terminates after n1 cases, and the treatment rejected n1 = maximum sample size in stage 1 rTot = the total number of successes (stage 1 and 2 combined) above which the study can be terminated and the treatment accepted for further development and trials, and at or below which after nTot cases the treatment is rejected. nTot = the total maximum sample size (stage 1 and 2 combined) EN = expected sample size, an estimated average sample size that will be used before the trial terminates. This is a measure of efficiency in detecting PET = probability of early termination (when treatment is abandoned), a measure of efficiency in rejecting ineffective treatments.
α=0.05 pw=0.9
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.05
0.15
2
37
7
84
50.2
0.72
2
46
7
77
58.6
0.59
0.05
0.9
0.05
0.2
1
21
4
41
26.7
0.72
1
29
4
38
32.9
0.57
0.05
0.9
0.05
0.25
0
9
3
30
16.8
0.63
0
15
3
25
20.4
0.46
0.05
0.9
0.05
0.3
0
9
2
17
12
0.63
0
13
2
16
14.5
0.51
0.05
0.9
0.05
0.35
0
6
2
17
8.9
0.74
0
8
2
14
10
0.66
0.05
0.9
0.05
0.4
0
5
2
14
7
0.77
0
7
2
12
8.5
0.7
0.05
0.9
0.05
0.45
0
4
2
14
5.9
0.81
0
8
2
10
8.7
0.66
0.05
0.9
0.05
0.5
0
4
1
7
4.6
0.81
0
4
1
7
4.6
0.81
0.05
0.9
0.05
0.55
0
3
1
8
3.7
0.86
0
4
1
6
4.4
0.81
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.1
0.2
5
47
18
130
74.3
0.67
6
70
16
109
91.8
0.44
0.05
0.9
0.1
0.25
2
21
10
66
36.8
0.65
3
31
9
55
40
0.62
0.05
0.9
0.1
0.3
2
18
6
35
22.5
0.73
2
22
6
33
26.2
0.62
0.05
0.9
0.1
0.35
1
11
5
27
15.8
0.7
1
13
5
25
17.5
0.62
0.05
0.9
0.1
0.4
1
9
4
20
11.5
0.77
1
12
4
18
14
0.66
0.05
0.9
0.1
0.45
0
5
3
14
8.7
0.59
0
7
3
13
10.1
0.48
0.05
0.9
0.1
0.5
0
4
3
13
7.1
0.66
0
5
3
12
7.9
0.59
0.05
0.9
0.1
0.55
0
4
2
9
5.7
0.66
0
5
2
8
6.2
0.59
0.05
0.9
0.1
0.6
0
3
2
8
4.4
0.73
0
5
2
7
5.8
0.59
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.15
0.25
9
57
31
162
93.6
0.65
11
78
27
136
107.6
0.49
0.05
0.9
0.15
0.3
5
30
17
82
45.1
0.71
6
42
14
64
51.8
0.55
0.05
0.9
0.15
0.35
3
19
10
44
26.9
0.68
3
23
9
38
29.9
0.54
0.05
0.9
0.15
0.4
2
13
7
29
17.9
0.69
2
16
7
27
20.8
0.56
0.05
0.9
0.15
0.45
1
9
5
19
13
0.6
1
9
5
19
13
0.6
0.05
0.9
0.15
0.5
1
7
5
18
10.1
0.72
0
6
4
14
11
0.38
0.05
0.9
0.15
0.55
1
6
4
14
7.8
0.78
0
4
4
13
8.3
0.52
0.05
0.9
0.15
0.6
1
5
3
11
6
0.84
1
7
3
9
7.6
0.72
0.05
0.9
0.15
0.65
0
3
3
9
5.3
0.61
0
3
3
9
5.3
0.61
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.2
0.3
15
71
45
184
109.5
0.66
18
92
40
160
124.6
0.52
0.05
0.9
0.2
0.35
8
37
22
83
51.4
0.69
8
42
21
77
58.4
0.53
0.05
0.9
0.2
0.4
4
19
15
54
30.4
0.67
5
24
13
45
31.2
0.66
0.05
0.9
0.2
0.45
4
16
11
38
20.4
0.8
4
19
9
29
22.3
0.67
0.05
0.9
0.2
0.5
2
10
7
22
13.9
0.68
2
12
7
21
16
0.56
0.05
0.9
0.2
0.55
2
8
7
22
10.8
0.8
1
7
6
17
11.2
0.58
0.05
0.9
0.2
0.6
1
5
6
18
8.4
0.74
1
8
5
13
10.5
0.5
0.05
0.9
0.2
0.65
1
5
4
11
6.6
0.74
1
6
4
10
7.4
0.66
0.05
0.9
0.2
0.7
1
4
4
11
5.3
0.82
2
6
3
8
6.2
0.9
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.25
0.35
24
89
61
207
122.8
0.71
30
125
54
179
155
0.45
0.05
0.9
0.25
0.4
10
37
31
99
56.2
0.69
13
57
27
83
72.1
0.42
0.05
0.9
0.25
0.45
6
22
19
57
32.5
0.7
6
26
17
49
37.1
0.52
0.05
0.9
0.25
0.5
5
17
13
37
21.7
0.77
4
16
12
33
22.3
0.63
0.05
0.9
0.25
0.55
2
9
9
24
15
0.6
3
13
9
23
17.2
0.58
0.05
0.9
0.25
0.6
2
8
7
18
11.2
0.68
2
9
7
17
12.2
0.6
0.05
0.9
0.25
0.65
1
5
6
15
8.7
0.63
1
6
6
14
9.7
0.53
0.05
0.9
0.25
0.7
1
4
6
15
6.9
0.74
1
6
4
9
7.4
0.53
0.05
0.9
0.25
0.75
0
2
4
9
5.1
0.56
0
2
4
9
5.1
0.56
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.3
0.4
28
89
77
222
133.4
0.67
41
142
68
193
171.3
0.42
0.05
0.9
0.3
0.45
13
40
40
110
60.8
0.7
27
77
33
88
78.5
0.86
0.05
0.9
0.3
0.5
8
24
24
63
34.7
0.73
7
24
21
53
36.6
0.56
0.05
0.9
0.3
0.55
5
15
16
40
22
0.72
8
22
14
34
24.2
0.81
0.05
0.9
0.3
0.6
3
10
12
28
16.3
0.65
7
18
10
23
18.7
0.86
0.05
0.9
0.3
0.65
2
7
9
20
11.6
0.65
2
10
8
17
14.3
0.38
0.05
0.9
0.3
0.7
2
6
7
15
8.3
0.74
3
9
7
14
10.4
0.73
0.05
0.9
0.3
0.75
1
4
6
12
6.8
0.65
2
6
5
10
7
0.74
0.05
0.9
0.3
0.8
2
5
4
8
5.5
0.84
2
5
4
8
5.5
0.84
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.35
0.5
16
43
44
105
62.7
0.68
16
46
40
94
67.4
0.56
0.05
0.9
0.35
0.55
7
20
26
59
35.6
0.6
12
37
24
53
45.9
0.45
0.05
0.9
0.35
0.6
5
14
18
39
23
0.64
7
19
16
34
24
0.67
0.05
0.9
0.35
0.65
5
12
14
30
15.8
0.79
5
14
12
24
17.6
0.64
0.05
0.9
0.35
0.7
3
8
10
20
11.5
0.71
8
16
9
18
16.1
0.93
0.05
0.9
0.35
0.75
2
6
7
13
8.5
0.65
2
6
7
13
8.5
0.65
0.05
0.9
0.35
0.8
2
5
6
11
6.4
0.76
2
5
6
11
6.4
0.76
0.05
0.9
0.35
0.85
1
3
5
9
4.7
0.72
1
3
5
9
4.7
0.72
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.4
0.55
19
45
49
104
64
0.68
24
62
45
94
78.9
0.47
0.05
0.9
0.4
0.6
11
25
32
66
36
0.73
12
29
27
54
38.1
0.64
0.05
0.9
0.4
0.65
7
16
20
39
22.5
0.72
9
23
18
34
27.9
0.56
0.05
0.9
0.4
0.7
4
10
13
24
15.1
0.63
4
10
13
24
15.1
0.63
0.05
0.9
0.4
0.75
3
7
11
20
10.8
0.71
6
12
10
18
12.9
0.84
0.05
0.9
0.4
0.8
2
5
8
14
7.9
0.68
3
8
8
13
10
0.59
0.05
0.9
0.4
0.85
1
3
6
10
5.5
0.65
1
3
6
10
5.5
0.65
0.05
0.9
0.4
0.9
2
4
5
8
4.7
0.82
2
4
5
8
4.7
0.82
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.45
0.6
19
40
60
116
64
0.68
49
93
50
95
93.1
0.94
0.05
0.9
0.45
0.65
11
23
33
61
34.9
0.69
14
31
30
54
40.6
0.58
0.05
0.9
0.45
0.7
7
15
20
35
21.9
0.65
9
19
19
33
23.6
0.67
0.05
0.9
0.45
0.75
5
10
17
29
15
0.74
5
12
14
23
17.2
0.53
0.05
0.9
0.45
0.8
2
5
11
18
10.3
0.59
3
7
10
16
10.5
0.61
0.05
0.9
0.45
0.85
1
3
9
14
7.7
0.57
1
4
8
12
8.9
0.39
0.05
0.9
0.45
0.9
2
4
6
9
5.2
0.76
2
4
6
9
5.2
0.76
0.05
0.9
0.45
0.95
0
1
5
7
3.7
0.55
0
1
5
7
3.7
0.55
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.5
0.65
22
42
60
105
62.3
0.68
28
57
54
93
75
0.5
0.05
0.9
0.5
0.7
13
24
36
61
34
0.73
14
27
32
53
36.1
0.65
0.05
0.9
0.5
0.75
7
13
25
41
21.1
0.71
12
21
20
32
23.1
0.81
0.05
0.9
0.5
0.8
5
9
18
29
14.1
0.75
13
20
14
22
20.1
0.94
0.05
0.9
0.5
0.85
4
7
12
18
9.5
0.77
7
11
10
15
11.5
0.89
0.05
0.9
0.5
0.9
2
4
7
10
5.9
0.69
2
4
7
10
5.9
0.69
0.05
0.9
0.5
0.95
0
1
6
8
4.5
0.5
0
1
6
8
4.5
0.5
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.55
0.7
22
38
68
110
59.8
0.7
50
81
56
89
81.7
0.91
0.05
0.9
0.55
0.75
10
18
35
54
32.1
0.61
20
33
32
49
36.3
0.79
0.05
0.9
0.55
0.8
7
12
24
36
19.3
0.7
8
14
21
31
19.7
0.66
0.05
0.9
0.55
0.85
3
6
14
20
12.2
0.56
3
6
14
20
12.2
0.56
0.05
0.9
0.55
0.9
3
5
12
17
8.1
0.74
7
10
10
14
10.4
0.9
0.05
0.9
0.55
0.95
1
2
10
14
5.6
0.7
2
4
7
9
6
0.61
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.6
0.7
54
88
138
212
132.4
0.64
121
186
128
196
186.7
0.93
0.05
0.9
0.6
0.75
21
34
64
95
55.6
0.65
48
72
57
84
73.2
0.9
0.05
0.9
0.6
0.8
12
19
37
53
29.5
0.69
15
26
32
45
35.9
0.48
0.05
0.9
0.6
0.85
7
11
23
32
17.2
0.7
8
14
20
27
20.3
0.51
0.05
0.9
0.6
0.9
5
8
13
17
10.8
0.68
5
8
13
17
10.8
0.68
0.05
0.9
0.6
0.95
3
5
8
10
6.7
0.66
3
5
8
10
6.7
0.66
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.65
0.75
52
78
142
203
120.3
0.66
84
129
127
180
152.3
0.54
0.05
0.9
0.65
0.8
21
31
67
93
50.3
0.69
34
52
55
75
61.8
0.57
0.05
0.9
0.65
0.85
10
15
33
44
25.2
0.65
28
37
30
40
37.2
0.94
0.05
0.9
0.65
0.9
7
10
21
27
14.4
0.74
11
15
18
23
16.4
0.83
0.05
0.9
0.65
0.95
5
7
13
16
9.1
0.77
6
8
12
15
9.2
0.83
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.7
0.8
51
71
135
180
106.5
0.67
120
159
122
162
159.2
0.95
0.05
0.9
0.7
0.85
18
25
61
79
43.4
0.66
33
44
53
68
48.5
0.81
0.05
0.9
0.7
0.9
11
15
29
36
21.2
0.7
13
18
26
32
22.7
0.67
0.05
0.9
0.7
0.95
7
9
15
18
10.8
0.8
7
9
15
18
10.8
0.8
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.75
0.85
47
61
124
155
90.1
0.69
93
117
111
138
119.3
0.89
0.05
0.9
0.75
0.9
18
23
52
63
34.3
0.72
19
25
45
54
36
0.62
0.05
0.9
0.75
0.95
7
9
24
28
14.7
0.7
19
22
22
26
22.2
0.94
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.8
0.9
36
44
108
127
70.8
0.68
69
82
94
110
85.9
0.86
0.05
0.9
0.8
0.95
16
19
37
42
24.4
0.76
31
35
35
40
35.3
0.94
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.05
0.9
0.85
0.95
26
30
75
83
47
0.68
35
40
68
75
49.2
0.74
α=0.1, Power = 0.8 π0 = proportion of success at or below which treatment is abandoned π1 = proportion of success at or above which treatment will be accepted for further development and trials r1 = number of successes above which stage 2 can be entered, and at or below which the trial terminates after n1 cases, and the treatment rejected n1 = maximum sample size in stage 1 rTot = the total number of successes (stage 1 and 2 combined) above which the study can be terminated and the treatment accepted for further development and trials, and at or below which after nTot cases the treatment is rejected. nTot = the total maximum sample size (stage 1 and 2 combined) EN = expected sample size, an estimated average sample size that will be used before the trial terminates. This is a measure of efficiency in detecting PET = probability of early termination (when treatment is abandoned), a measure of efficiency in rejecting ineffective treatments.
α=0.01 pw=0.9
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.05
0.15
3
44
13
144
61.7
0.82
3
65
11
108
82.6
0.59
0.01
0.9
0.05
0.2
1
21
7
61
32.3
0.72
2
34
7
57
39.5
0.76
0.01
0.9
0.05
0.25
1
15
6
47
20.5
0.83
1
23
5
35
26.8
0.68
0.01
0.9
0.05
0.3
1
12
5
35
14.7
0.88
0
11
4
25
17
0.57
0.01
0.9
0.05
0.35
0
6
4
25
11
0.74
2
15
3
18
15.1
0.96
0.01
0.9
0.05
0.4
0
5
3
18
7.9
0.77
0
8
3
15
10.4
0.66
0.01
0.9
0.05
0.45
0
4
3
18
6.6
0.81
0
7
3
13
8.8
0.7
0.01
0.9
0.05
0.5
0
4
2
10
5.1
0.81
0
5
2
9
5.9
0.77
0.01
0.9
0.05
0.55
0
3
3
14
4.6
0.86
0
5
2
8
5.7
0.77
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.1
0.2
8
66
28
187
91.6
0.79
9
87
25
160
114.1
0.63
0.01
0.9
0.1
0.25
4
31
17
102
44.7
0.81
4
39
14
78
52.7
0.65
0.01
0.9
0.1
0.3
2
17
11
59
27
0.76
3
27
10
49
33.2
0.72
0.01
0.9
0.1
0.35
2
14
9
44
18.8
0.84
2
18
8
35
22.5
0.73
0.01
0.9
0.1
0.4
1
9
7
30
13.7
0.77
1
13
6
24
17.2
0.62
0.01
0.9
0.1
0.45
1
8
5
20
10.2
0.81
1
9
5
19
11.3
0.77
0.01
0.9
0.1
0.5
1
7
5
18
8.6
0.85
0
6
4
14
9.7
0.53
0.01
0.9
0.1
0.55
1
6
4
14
6.9
0.89
0
4
4
13
7.1
0.66
0.01
0.9
0.1
0.6
0
3
4
13
5.7
0.73
1
7
3
9
7.3
0.85
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.15
0.3
7
38
27
122
54.9
0.8
9
54
23
98
66.6
0.71
0.01
0.9
0.15
0.35
4
22
17
69
32.6
0.77
5
30
15
58
38.1
0.71
0.01
0.9
0.15
0.4
3
15
14
55
22.1
0.82
3
19
11
39
25.3
0.68
0.01
0.9
0.15
0.45
2
11
10
34
16.1
0.78
2
14
9
29
19.3
0.65
0.01
0.9
0.15
0.5
2
9
9
31
12.1
0.86
2
12
7
21
14.4
0.74
0.01
0.9
0.15
0.55
1
6
7
21
9.4
0.78
1
7
6
17
9.8
0.72
0.01
0.9
0.15
0.6
1
5
6
18
7.1
0.84
1
8
5
13
9.7
0.66
0.01
0.9
0.15
0.65
1
5
5
13
6.3
0.84
0
4
4
10
6.9
0.52
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.2
0.35
11
46
37
134
63.2
0.8
27
99
32
112
99.4
0.97
0.01
0.9
0.2
0.4
6
25
24
80
37.1
0.78
9
38
21
67
44.3
0.78
0.01
0.9
0.2
0.45
5
19
17
53
24.5
0.84
4
21
15
44
30.5
0.59
0.01
0.9
0.2
0.5
3
12
13
38
17.3
0.79
2
11
12
33
19.4
0.62
0.01
0.9
0.2
0.55
2
8
12
34
13.3
0.8
3
13
9
23
15.5
0.75
0.01
0.9
0.2
0.6
1
5
9
24
10
0.74
2
9
8
19
11.6
0.74
0.01
0.9
0.2
0.65
1
5
7
17
8.2
0.74
2
7
7
16
8.3
0.85
0.01
0.9
0.2
0.7
1
4
7
16
6.2
0.82
1
5
6
13
7.1
0.74
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.25
0.4
15
51
51
156
70.5
0.81
19
70
42
124
85.4
0.71
0.01
0.9
0.25
0.45
8
27
32
91
40.7
0.79
23
66
26
71
66.1
0.97
0.01
0.9
0.25
0.5
6
19
23
62
26.5
0.83
7
26
19
48
32.9
0.69
0.01
0.9
0.25
0.55
3
11
15
37
18.5
0.71
8
22
14
34
22.9
0.93
0.01
0.9
0.25
0.6
3
10
12
28
14
0.78
7
18
11
25
18.4
0.94
0.01
0.9
0.25
0.65
2
7
10
22
10.7
0.76
2
8
9
19
11.5
0.68
0.01
0.9
0.25
0.7
2
6
8
17
7.9
0.83
1
5
8
16
9
0.63
0.01
0.9
0.25
0.75
1
4
7
14
6.6
0.74
2
6
7
13
7.2
0.83
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.3
0.45
19
55
62
164
75.4
0.81
24
80
52
133
103.6
0.55
0.01
0.9
0.3
0.5
11
31
38
94
43.1
0.81
27
67
32
77
67.3
0.97
0.01
0.9
0.3
0.55
7
20
24
55
28
0.77
7
21
23
52
29.6
0.72
0.01
0.9
0.3
0.6
4
12
18
39
19.5
0.72
5
15
17
36
20.8
0.72
0.01
0.9
0.3
0.65
5
12
15
32
14.4
0.88
3
10
13
26
15.6
0.65
0.01
0.9
0.3
0.7
2
6
12
24
10.6
0.74
2
7
10
19
11.2
0.65
0.01
0.9
0.3
0.75
3
7
9
17
8.3
0.87
2
6
9
16
8.6
0.74
0.01
0.9
0.3
0.8
2
5
8
14
6.5
0.84
1
4
7
12
6.8
0.65
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.35
0.5
24
60
73
170
78.8
0.83
59
135
62
141
135.1
0.99
0.01
0.9
0.35
0.55
12
30
43
94
44.1
0.78
36
77
37
79
77
0.99
0.01
0.9
0.35
0.6
8
19
33
70
28.5
0.81
11
29
25
50
35.2
0.7
0.01
0.9
0.35
0.65
5
12
23
46
19.2
0.79
5
14
19
36
21.9
0.64
0.01
0.9
0.35
0.7
5
11
16
30
13.8
0.85
4
12
14
25
17.4
0.58
0.01
0.9
0.35
0.75
3
7
13
23
10.2
0.8
9
16
11
19
16.1
0.98
0.01
0.9
0.35
0.8
2
5
10
17
7.8
0.76
7
12
9
15
12.1
0.97
0.01
0.9
0.35
0.85
1
3
9
14
6.1
0.72
1
4
8
12
7.5
0.56
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.4
0.55
23
53
79
163
81.4
0.74
36
86
71
144
104.6
0.68
0.01
0.9
0.4
0.6
14
31
48
94
44.8
0.78
26
54
43
83
56.5
0.91
0.01
0.9
0.4
0.65
10
21
33
62
28.1
0.83
24
45
28
51
45.2
0.97
0.01
0.9
0.4
0.7
5
11
24
43
18.9
0.75
8
18
21
36
22.7
0.74
0.01
0.9
0.4
0.75
5
10
18
31
13.5
0.83
4
10
16
26
15.9
0.63
0.01
0.9
0.4
0.8
4
8
12
19
9.9
0.83
4
8
12
19
9.9
0.83
0.01
0.9
0.4
0.85
3
6
10
15
7.6
0.82
3
6
10
15
7.6
0.82
0.01
0.9
0.4
0.9
2
4
9
13
5.6
0.82
3
6
8
11
6.9
0.82
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.45
0.6
30
60
91
171
80.2
0.82
75
139
78
144
139.1
0.99
0.01
0.9
0.45
0.65
16
31
58
105
44.2
0.82
23
50
46
80
61.6
0.61
0.01
0.9
0.45
0.7
10
19
39
67
27.8
0.82
26
44
31
52
44.2
0.98
0.01
0.9
0.45
0.75
7
13
27
44
18.5
0.82
7
16
21
33
23.4
0.56
0.01
0.9
0.45
0.8
5
9
19
30
12.5
0.83
4
9
16
24
14.7
0.62
0.01
0.9
0.45
0.85
4
7
14
21
9.1
0.85
5
10
12
17
11.8
0.74
0.01
0.9
0.45
0.9
2
4
10
14
6.4
0.76
2
4
10
14
6.4
0.76
0.01
0.9
0.45
0.95
1
2
10
14
4.4
0.8
2
4
7
9
5.2
0.76
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.5
0.65
30
55
97
166
78.2
0.79
81
137
84
142
137.1
0.99
0.01
0.9
0.5
0.7
15
27
59
97
42.5
0.78
43
70
49
79
70.2
0.98
0.01
0.9
0.5
0.75
11
19
37
58
26
0.82
15
26
32
49
29.8
0.84
0.01
0.9
0.5
0.8
7
12
26
39
17.2
0.81
9
16
22
32
19.6
0.77
0.01
0.9
0.5
0.85
4
7
19
27
11.5
0.77
7
13
16
22
15.6
0.71
0.01
0.9
0.5
0.9
3
5
15
21
8
0.81
5
8
12
16
9.2
0.86
0.01
0.9
0.5
0.95
1
2
13
18
6
0.75
8
10
9
12
10
0.99
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.55
0.7
29
49
101
159
74.6
0.77
83
130
87
135
130.1
0.98
0.01
0.9
0.55
0.75
18
29
63
96
40.4
0.83
19
33
50
74
46.1
0.68
0.01
0.9
0.55
0.8
10
16
40
58
24.3
0.8
15
26
32
45
32.1
0.68
0.01
0.9
0.55
0.85
7
11
24
33
15.2
0.81
19
26
22
30
26.1
0.98
0.01
0.9
0.55
0.9
3
5
18
24
9.9
0.74
7
10
16
21
11.1
0.9
0.01
0.9
0.55
0.95
4
6
12
15
7.5
0.84
4
6
12
15
7.5
0.84
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.6
0.75
29
45
106
155
69.7
0.78
38
61
89
128
81.9
0.69
0.01
0.9
0.6
0.8
15
23
58
81
36.8
0.76
38
54
50
69
54.6
0.96
0.01
0.9
0.6
0.85
11
16
37
50
21.7
0.83
27
36
31
41
36.1
0.98
0.01
0.9
0.6
0.9
6
9
22
28
13.4
0.77
5
9
20
25
16.7
0.52
0.01
0.9
0.6
0.95
5
7
15
18
8.7
0.84
7
9
14
17
9.6
0.93
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.65
0.8
27
39
103
140
63
0.76
76
103
87
117
103.3
0.98
0.01
0.9
0.65
0.85
18
25
53
69
32.6
0.83
46
59
47
61
59
0.99
0.01
0.9
0.65
0.9
7
10
32
40
17.8
0.74
25
31
29
36
31.1
0.98
0.01
0.9
0.65
0.95
4
6
18
21
10.8
0.68
4
6
18
21
10.8
0.68
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.7
0.85
27
36
99
126
54.3
0.8
79
99
80
101
99
0.99
0.01
0.9
0.7
0.9
13
17
53
65
26.7
0.8
38
46
43
52
46.1
0.98
0.01
0.9
0.7
0.95
8
10
29
34
13.6
0.85
16
19
25
29
19.5
0.95
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.75
0.9
25
31
87
104
43.9
0.82
66
78
71
84
78.1
0.99
0.01
0.9
0.75
0.95
8
10
45
52
20.2
0.76
31
35
35
40
35.1
0.99
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.8
0.9
49
59
166
193
90
0.77
142
164
145
168
164
0.99
0.01
0.9
0.8
0.95
16
19
60
67
30.4
0.76
22
26
55
61
33.2
0.79
Optimal Model
Minimax Model
α
Pw
π0
π1
r1
n1
rTot
nTot
EN
PET
r1
n1
rTot
nTot
EN
PET
0.01
0.9
0.85
0.95
41
46
123
135
60.3
0.84
99
108
107
117
108.1
0.99
α=0.1, Power = 0.8 π0 = proportion of success at or below which treatment is abandoned π1 = proportion of success at or above which treatment will be accepted for further development and trials r1 = number of successes above which stage 2 can be entered, and at or below which the trial terminates after n1 cases, and the treatment rejected n1 = maximum sample size in stage 1 rTot = the total number of successes (stage 1 and 2 combined) above which the study can be terminated and the treatment accepted for further development and trials, and at or below which after nTot cases the treatment is rejected. nTot = the total maximum sample size (stage 1 and 2 combined) EN = expected sample size, an estimated average sample size that will be used before the trial terminates. This is a measure of efficiency in detecting PET = probability of early termination (when treatment is abandoned), a measure of efficiency in rejecting ineffective treatments.
Javascript Program
Data required
Type I Error (alpha;). The common values used are 0.05 or 0.1
Power (1-β). The common values used are 0.8 or 0.9
π0. The proportion of success at or below which treatment is abandoned
π1, The proportion of success at or above which treatment will be accepted for further development and trials
Please Note; If the difference between π0 and π1 is too narrow, the sample size required will exceed 120, and the calculations may become too long so the program will abort
User should replace the default example data with his/her choices
Type I Error (α)
Power (1-β)
Success rate too low, reject treatment (π0)
Success rate high enough, accept treatment for further evaluation (π1)
R Codes
See Technical considerations section for commentary
Simon's Procedure for Phase II
Simon R (1989) Optimal two-stage designs for phase II clinical trials. Control Clin Trials 10:1-10
Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed.
Blackwell Science IBSN 0-86542-870-0 p. 256-257
Algorithm adapted from text book by Machin et.al.
# Data
# a = alpha, probability to Type I Error
# pw = power 1 - beta where beta is probability of Type II Error
#p0 = success rate below which treatment being trialed is rejected as inadequate
#p1 = success rate above which treatment being trialed is accepted as worthy of further evaluation
a = 0.05
pw = 0.8
p0 = 0.1
p1 = 0.3
# Program Starts
#calculate start and finish equation 10.12 p.257
za = qnorm(a)
b = 1 - pw # beta Type II error
zb = qnorm(b)
cp = (p0 + p1) / 2
cx = (za + zb) / (p1 - p0)
start = floor(cp * (1.0 - cp) * cx^2)
finish = start + 30
# Reference matrices
# precalculated values for formulae 10.10 and 10.11, p. 256
mxbP0 <- matrix(1,finish+1,finish+1)
mxbP1 <- matrix(1,finish+1,finish+1)
for(i in 0 : finish)
{
for(j in 0 : finish)
{
bincoef = choose(i,j)
mxbP0[i+1,j+1] = bincoef * p0^j * (1.0-p0)^(i-j)
mxbP1[i+1,j+1] = bincoef * p1^j * (1.0-p1)^(i-j)
}
}
mxBP0 <- matrix(0,finish+1,finish+1)
mxBP1 <- matrix(0,finish+1,finish+1)
for(i in 0 : finish)
{
for(j in 0 : finish)
{
for(x in 0 : j)
{
mxBP0[i+1, j+1] = mxBP0[i+1 , j+1] + mxbP0[i+1 , x+1]
mxBP1[i+1, j+1] = mxBP1[i+1 , j+1] + mxbP1[i+1 , x+1]
}
}
}
# Optimal and Minimax are vectors for results in each loop
#OptimalFinal and MinimaxFinal are the best results when the calculations are
# repeated from start-10 to start. Best is having the lowest total sample size nTot
#initialize the 4 vectors and set the minimum nTot to impossibly high value
Minimax <-c(0,0,0,0,0,0)
Optimal <- Minimax
OptimalFinal <- Optimal
MinimaxFinal <- Minimax
minNTot = 1e10
startF = start # startF is start value inotially calculated
start = startF - 10 # start is set back by 10 for iteration
if(start<2) # making sure is is not zero
{
start = 2
}
while(start<=startF) #loop begins from startF-10 to startF
{
start = start + 1
Minimax <-c(0,0,0,0,0,0) # clear working vectors for each loop
Optimal <- Minimax
MinEN = start # calculation begins
Pr = 1
rs = 0
for(ntot in start : finish)
{
for(n1 in 1 : ntot-1)
{
n2 = ntot - n1
rr = ceiling(Pr * ntot)
rx = rs
for(rtot in rx : rr)
{
minv = n1
if(rtot<minv)
{
minv = rtot
}
for(r1 in 0 : minv)
{
if((r1<rtot) & (r1>=(n1 + rtot - ntot)))
{
PET = mxBP1[n1+1,r1+1]
testbeta = 0; # test beta begins
for(x in (r1+1) : minv)
{
if(rtot-x>=0)
{
lb = mxbP1[n1+1,x+1]
bb = mxBP1[n2+1,rtot-x+1]
testbeta = testbeta + lb*bb
}
}
testbeta = testbeta + PET
y = b - testbeta
if(y>0)
{
PET = mxBP0[n1+1,r1+1]
testalpha = 0;
for(x in r1+1 : minv)
{
if(rtot-x>0)
{
lb = mxbP0[n1+1,x+1]
bb = mxBP0[n2+1,rtot-x+1]
testalpha = testalpha + lb*bb
}
}
testalpha = testalpha + PET
EN = n1 + (1 - PET) * n2
z = testalpha - (1 - a)
if((z>0) & (EN<MinEN))
{
rs = rtot;
Pr = 1 * rtot / ntot
Optimal <- c(r1,n1,rtot,ntot,round(EN,1),round(PET,1))
if((Minimax[4]<1) | (Minimax[4]==ntot))
{
Minimax <- Optimal
}
MinEN = EN
}
}
}
}
}
}
}
# ckeck whether current Minimax nTot is less than the minimum
# if so then replace the result vectors with the current and reset the minimum value
if(Minimax[4]>0 & Minimax[4]<minNTot)
{
OptimalFinal <- Optimal
MinimaxFinal <- Minimax
minNTot = (Minimax[4])
}
} # end of loop
Results <- c("r1","n1","rTotal","nTotal","EN","PET")
Optimal <- OptimalFinal
Minimax <- MinimaxFinal
data.frame(Results,Optimal,Minimax)
The results output are
r1 = number of successes in phase. When surpassed stage 2 can begin
n1 = maximum number in phase 1 to reject new treatment if r1 is not surpassed
rTotal = number of successes in total (stage 1 and 2). New treatment can be accepted for further studies in phase III trial when rTotal is surpassed
nTotal = maximum number (stage 1 and 2) for rejecting new treatment if r2 is not surpassed
EN = The average expected number of cases for a decision
PET = The probability of early termination of the study (rejection of new treatment) if success rate is below requirements
