|
Related Links:
Phase II Studies Explained Page
Sample Size for Phase II Study (Gehan's Procedure) Program Page
Introduction
Sample Size Tables
References
Gehan's procedure, and its comparison against other procedures available for the Phase II trial, are discussed in
the Phase II Studies Explained Page
. This page provides support for the use of the program in the
Sample Size for Phase II Study (Gehan's Procedure) Program Page
and the tables in this page.
Gehan's is a preliminary study of a new treatment, to test whether the proportion of success from the treatment
satisfies the level required to warrant further detailed study in a Phase II trial.
- Stage 1 :
- Sample size for stage 1 (s1) is the number of cases to be used in stage 1. This depends on
- The proportion of success required (p) is defined, a value between 0 (0%) and 1 (100%)
- The power required, the ability to detect that proportion if it is there. Mathematically power = 1-β,
where β is the probability of error in rejecting the new treatment
- At the end of stage 1, after s1 cases are included, k = the number of successes found
- If k=0 or k< a critical value where reaching the required proportion of success is unlikely, the treatment
is rejected without proceeding to stage 2
- If k> a critical value where reaching the required proportion of success is likely, the treatment
is accepted without proceeding to stage 2
- Otherwise, stage 2 of the procedure is required for decision making.
- Stage 2 :
- The sample size in stage 2 is the number of additional cases to be used in stage 2 This depends on
- The number of cases alr3eady used in stage 1 (s1)
- The number of successes (k) found in stage 1
- The acceptable probability of error for a decision to acceot the treatment (ε)
- The total number of successes is the number of successes in stage 1 (k) + number of successes in stage 2 .
- The required total number of successes for a decision to accept the treatment is n = p(s1+s2)
- if the total number of successes reached n, then the trial can be terminated, with a decision to accept the treatment
- If at the end of stage 2, the total number of successes had not reached n, then a decision to reject the treatment is made.
The program in the Sample Size for Phase II Study (Gehan's Procedure) Program Page
produces the results of Gehan's 2 stage procedures.
- The entry data is a single column of 3 rows, where
- Row 1 = ε, Probability of error for accepting the treatment for further study
- Row 2 = Power(1-β), where β is the probability of error for rejecting the treatment for further study
- Row 3 = p, the probability or proportion of successes required for accepting the treatment for further study
- The result are :
- The sample size for stage 1
- A table where each row is the parameters related to each value of k
- k = the number of successes in stage 1
- s2 = the sample size for stage 2
- n = the number of successes required (combine stage 1 and 2) for the treatment to be accepted.
The tables on this page provides the results of calculations for Gehan's 2 stage prod=cdures
- Three levels of power (1-β) at 0.8, 0.9, and 0.95
- Two level of error (ε) at 0.05 and 0.1
- Required proportion of successes :
- From 0.03 (3%) to 0.2 (20%) at 0.01 (1%) intervals
- From 0.2 (20%) to 0.3 (30%) at 0.02 (2%) intervals
- From 0.3 (30%) to 0.5 (50%) at 0.05 (5%) intervals
- From 0.5 (50%) to 0.9 (90%) at 0.1 (10%) intervals
- The cells contains results
- Blanks for impossible situations, where k>s1
- R is the combination of s1 and k that allows the treatment to be rejected at the end of stage 1
- A is the combination of s1 and k that allows the treatment to be accepted at the end of stage 1
- Cells with numbers are combinations of s1 and k that From 0.3 (30%) to 0.5 (50%) at 0.05 (5%) intervals
- From 0.5 (50%) to 0.9 (90%) at 0.1 (10%) intervals
Power=0.8
Power=0.9
Power=0.95
Power=0.8, π=proportion of success required, ε=Error of accepting treatment Stage 1 : s1=sample size for stage 1 r=numbers success in stage 1 Stagte 2 [cells] : sample size for stage 2 + (total number of successes required to accept treatment, inclusive of stage 1 and 2) R = Reject and A = Accept treatment for further studies at the end of stage 1
| | Error (ε) = 0.1 | Error (ε) = 0.05 |
|---|
| π | s1 | r=1 | r=2 | r=3 | r=4 | r=5 | r=1 | r=2 | r=3 | r=4 | r=5 | r=6 | r=7 | r=8 | r=9 |
|---|
| 0.03 | 53 | R | A | A | A | A | R | A | A | A | A | A | A | A | A |
|---|
| 0.04 | 40 | R | A | A | A | A | R | A | A | A | A | A | A | A | A |
|---|
| 0.05 | 32 | R | A | A | A | A | R | 10(3) | A | A | A | A | A | A | A |
|---|
| 0.06 | 27 | R | A | A | A | A | 8(3) | 22(3) | 33(4) | 43(5) | A | A | A | A | A |
|---|
| 0.07 | 23 | R | A | A | A | A | 18(3) | 32(4) | 44(5) | 54(6) | 62(6) | 69(7) | A | A | A |
|---|
| 0.08 | 20 | R | A | A | A | A | 25(4) | 41(5) | 54(6) | 64(7) | 71(8) | 76(8) | 80(8) | A | A |
|---|
| 0.09 | 18 | R | A | A | A | A | 31(5) | 48(6) | 61(8) | 70(8) | 77(9) | 81(9) | 82(9) | 81(9) | A |
|---|
| 0.10 | 16 | R | A | A | A | A | 38(6) | 55(8) | 68(9) | 77(10) | 82(10) | 84(10) | 83(10) | 79(10) | A |
|---|
| 0.11 | 14 | 1(2) | 6(3) | A | A | A | 46(7) | 64(9) | 76(10) | 83(11) | 86(11) | 85(11) | 81(11) | 72(10) | A |
|---|
| 0.12 | 13 | 3(2) | 8(3) | A | A | A | 50(8) | 68(10) | 80(12) | 86(12) | 87(12) | 84(12) | 77(11) | 66(10) | A |
|---|
| 0.13 | 12 | 5(3) | 10(3) | 12(4) | A | A | 55(9) | 73(12) | 84(13) | 88(13) | 87(13) | 81(13) | 71(11) | 56(9) | A |
|---|
| 0.14 | 11 | 7(3) | 11(4) | 14(4) | A | A | 60(10) | 77(13) | 87(14) | 89(14) | 86(14) | 76(13) | 62(11) | A | A |
|---|
| 0.15 | 10 | 9(3) | 13(4) | 15(4) | A | A | 65(12) | 82(14) | 90(15) | 89(15) | 82(14) | 68(12) | 48(9) | A | A |
|---|
| 0.16 | 10 | 9(4) | 13(4) | 15(4) | A | A | 65(12) | 82(15) | 90(16) | 89(16) | 82(15) | 68(13) | 48(10) | A | A |
|---|
| 0.17 | 9 | 11(4) | 15(5) | 16(5) | 15(5) | A | 71(14) | 87(17) | 91(17) | 87(17) | 74(15) | 54(11) | A | A | A |
|---|
| 0.18 | 9 | 11(4) | 15(5) | 16(5) | 15(5) | A | 71(15) | 87(18) | 91(19) | 87(18) | 74(15) | 54(12) | 30(8) | A | A |
|---|
| 0.19 | 8 | 14(5) | 17(5) | 17(5) | 15(5) | A | 77(17) | 91(19) | 91(19) | 81(17) | 61(14) | 35(9) | A | A | |
|---|
| 0.20 | 8 | 14(5) | 17(5) | 17(5) | 15(5) | A | 77(17) | 91(20) | 91(20) | 81(18) | 61(14) | 35(9) | A | A | |
|---|
| 0.22 | 7 | 16(6) | 18(6) | 17(6) | 12(5) | A | 83(20) | 93(22) | 88(21) | 69(17) | 41(11) | A | A | | |
|---|
| 0.24 | 6 | 18(6) | 19(6) | 15(6) | A | A | 90(24) | 93(24) | 78(21) | 49(14) | A | A | | | |
|---|
| 0.26 | 6 | 18(7) | 19(7) | 15(6) | A | A | 90(25) | 93(26) | 78(22) | 49(15) | A | A | | | |
|---|
| 0.28 | 5 | 20(8) | 19(7) | 11(5) | A | A | 95(29) | 88(27) | 58(18) | 17(7) | A | | | | |
|---|
| 0.30 | 5 | 20(8) | 19(8) | 11(5) | A | A | 95(30) | 88(28) | 58(19) | 17(7) | A | | | | |
|---|
| 0.35 | 4 | 21(9) | 15(7) | A | A | | 96(35) | 70(26) | 22(10) | A | | | | | |
|---|
| 0.40 | 4 | 21(10) | 15(8) | A | A | | 96(40) | 70(30) | 22(11) | A | | | | | |
|---|
| 0.45 | 3 | 19(10) | 6(5) | A | | | 85(40) | 31(16) | A | | | | | | |
|---|
| 0.50 | 3 | 19(11) | 6(5) | A | | | 85(44) | 31(17) | A | | | | | | |
|---|
| 0.50 | 3 | 19(11) | 6(5) | A | | | 85(44) | 31(17) | A | | | | | | |
|---|
| 0.60 | 2 | 10(8) | A | | | | 45(29) | A | | | | | | | |
|---|
| 0.70 | 2 | 10(9) | A | | | | 45(33) | A | | | | | | | |
|---|
| 0.80 | 2 | 10(10) | A | | | | 45(38) | A | | | | | | | |
|---|
| 0.90 | 1 | A | | | | | A | | | | | | | | |
|---|
Power=0.9, π=proportion of success required, ε=Error of accepting treatment Stage 1 : s1=sample size for stage 1 r=numbers success in stage 1 Stagte 2 [cells] : sample size for stage 2 + (total number of successes required to accept treatment, inclusive of stage 1 and 2) R = Reject and A = Accept treatment for further studies at the end of stage 1
| | Error (ε) = 0.1 | Error (ε) = 0.05 |
|---|
| π | s1 | r=1 | r=2 | r=3 | r=4 | r=5 | r=6 | r=1 | r=2 | r=3 | r=4 | r=5 | r=6 | r=7 | r=8 | r=9 | r=10 | r=11 |
|---|
| 0.03 | 76 | R | R | A | A | A | A | R | R | A | A | A | A | A | A | A | A | A |
|---|
| 0.04 | 57 | R | R | A | A | A | A | R | R | A | A | A | A | A | A | A | A | A |
|---|
| 0.05 | 45 | R | R | A | A | A | A | R | R | A | A | A | A | A | A | A | A | A |
|---|
| 0.06 | 38 | R | R | A | A | A | A | R | R | A | A | A | A | A | A | A | A | A |
|---|
| 0.07 | 32 | R | R | A | A | A | A | R | 10(3) | 21(4) | 30(5) | A | A | A | A | A | A | A |
|---|
| 0.08 | 28 | R | R | A | A | A | A | 6(3) | 19(4) | 30(5) | 40(6) | 48(7) | 55(7) | 61(8) | A | A | A | A |
|---|
| 0.09 | 25 | R | R | A | A | A | A | 13(4) | 27(5) | 38(6) | 48(7) | 57(8) | 63(8) | 68(9) | 72(9) | A | A | A |
|---|
| 0.10 | 22 | R | R | A | A | A | A | 20(5) | 35(6) | 47(7) | 57(8) | 65(9) | 71(10) | 75(10) | 78(10) | 78(10) | A | A |
|---|
| 0.11 | 20 | R | R | A | A | A | A | 25(5) | 41(7) | 54(9) | 64(10) | 71(11) | 76(11) | 80(11) | 80(11) | 80(11) | 77(11) | A |
|---|
| 0.12 | 19 | R | R | A | A | A | A | 28(6) | 45(8) | 57(10) | 67(11) | 74(12) | 79(12) | 81(12) | 81(12) | 79(12) | 75(12) | A |
|---|
| 0.13 | 17 | R | 1(3) | A | A | A | A | 35(7) | 52(9) | 65(11) | 74(12) | 80(13) | 83(13) | 83(13) | 81(13) | 76(13) | 69(12) | A |
|---|
| 0.14 | 16 | R | 2(3) | A | A | A | A | 38(8) | 55(10) | 68(12) | 77(14) | 82(14) | 84(14) | 83(14) | 79(14) | 73(13) | 64(12) | A |
|---|
| 0.15 | 15 | R | 4(3) | 7(4) | A | A | A | 42(9) | 59(12) | 72(14) | 80(15) | 85(15) | 85(15) | 83(15) | 77(14) | 68(13) | 56(11) | A |
|---|
| 0.16 | 14 | R | 6(4) | 9(4) | A | A | A | 46(10) | 64(13) | 76(15) | 83(16) | 86(16) | 85(16) | 81(16) | 72(14) | 61(12) | A | A |
|---|
| 0.17 | 13 | 3(3) | 8(4) | 11(5) | 12(5) | A | A | 50(11) | 68(14) | 80(16) | 86(17) | 87(17) | 84(17) | 77(16) | 66(14) | 51(11) | A | A |
|---|
| 0.18 | 12 | 5(4) | 10(4) | 12(5) | 13(5) | A | A | 55(13) | 73(16) | 84(18) | 88(19) | 87(18) | 81(17) | 71(15) | 56(13) | 38(10) | A | A |
|---|
| 0.19 | 11 | 7(4) | 11(5) | 14(5) | 14(5) | A | A | 60(14) | 77(17) | 87(19) | 89(20) | 86(19) | 76(17) | 62(14) | 43(11) | A | A | A |
|---|
| 0.20 | 11 | 7(4) | 11(5) | 14(5) | 14(5) | A | A | 60(15) | 77(18) | 87(20) | 89(20) | 86(20) | 76(18) | 62(15) | 43(11) | A | A | A |
|---|
| 0.22 | 10 | 9(5) | 13(6) | 15(6) | 15(6) | 13(6) | A | 65(17) | 82(21) | 90(22) | 89(22) | 82(21) | 68(18) | 48(13) | A | A | A | |
|---|
| 0.24 | 9 | 11(5) | 15(6) | 16(6) | 15(6) | 12(6) | A | 71(20) | 87(24) | 91(24) | 87(24) | 74(20) | 54(16) | 30(10) | A | A | | |
|---|
| 0.26 | 8 | 14(6) | 17(7) | 17(7) | 15(6) | A | A | 77(23) | 91(26) | 91(26) | 81(24) | 61(18) | 35(12) | A | A | | | |
|---|
| 0.28 | 8 | 14(7) | 17(8) | 17(8) | 15(7) | 10(6) | A | 77(24) | 91(28) | 91(28) | 81(25) | 61(20) | 35(13) | A | A | | | |
|---|
| 0.30 | 7 | 16(7) | 18(8) | 17(8) | 12(6) | A | A | 83(27) | 93(30) | 88(29) | 69(23) | 41(15) | A | A | | | | |
|---|
| 0.35 | 6 | 18(9) | 19(9) | 15(8) | 8(5) | A | A | 90(34) | 93(35) | 78(30) | 49(20) | 12(7) | A | | | | | |
|---|
| 0.40 | 5 | 20(10) | 19(10) | 11(7) | A | A | | 95(40) | 88(38) | 58(26) | 17(9) | A | | | | | | |
|---|
| 0.45 | 4 | 21(12) | 15(9) | 3(4) | A | | | 96(45) | 70(34) | 22(12) | A | | | | | | | |
|---|
| 0.50 | 4 | 21(13) | 15(10) | 3(4) | A | | | 96(50) | 70(37) | 22(13) | A | | | | | | | |
|---|
| 0.50 | 4 | 21(13) | 15(10) | 3(4) | A | | | 96(50) | 70(37) | 22(13) | A | | | | | | | |
|---|
| 0.60 | 3 | 19(14) | 6(6) | A | | | | 85(53) | 31(21) | A | | | | | | | | |
|---|
| 0.70 | 2 | 10(9) | A | | | | | 45(33) | A | | | | | | | | | |
|---|
| 0.80 | 2 | 10(10) | A | | | | | 45(38) | A | | | | | | | | | |
|---|
| 0.90 | 2 | 10(11) | A | | | | | 45(43) | A | | | | | | | | | |
|---|
Power=0.95, π=proportion of success required, ε=Error of accepting treatment Stage 1 : s1=sample size for stage 1 r=numbers success in stage 1 Stagte 2 [cells] : sample size for stage 2 + (total number of successes required to accept treatment, inclusive of stage 1 and 2) R = Reject and A = Accept treatment for further studies at the end of stage 1
| | Error (ε) = 0.1 | Error (ε) = 0.05 |
|---|
| π | s1 | r=1 | r=2 | r=3 | r=4 | r=5 | r=6 | r=1 | r=2 | r=3 | r=4 | r=5 | r=6 | r=7 | r=8 | r=9 | r=10 | r=11 | r=12 | r=13 |
|---|
| 0.03 | 99 | R | R | A | A | A | A | R | R | A | A | A | A | A | A | A | A | A | A | A |
|---|
| 0.04 | 74 | R | R | A | A | A | A | R | R | A | A | A | A | A | A | A | A | A | A | A |
|---|
| 0.05 | 59 | R | R | A | A | A | A | R | R | A | A | A | A | A | A | A | A | A | A | A |
|---|
| 0.06 | 49 | R | R | A | A | A | A | R | R | A | A | A | A | A | A | A | A | A | A | A |
|---|
| 0.07 | 42 | R | R | A | A | A | A | R | R | A | A | A | A | A | A | A | A | A | A | A |
|---|
| 0.08 | 36 | R | R | A | A | A | A | R | 2(4) | 12(4) | 21(5) | 28(6) | A | A | A | A | A | A | A | A |
|---|
| 0.09 | 32 | R | R | A | A | A | A | R | 10(4) | 21(5) | 30(6) | 38(7) | 45(7) | 51(8) | A | A | A | A | A | A |
|---|
| 0.10 | 29 | R | R | A | A | A | A | 4(4) | 17(5) | 28(6) | 37(7) | 46(8) | 52(9) | 58(9) | 63(10) | 67(10) | A | A | A | A |
|---|
| 0.11 | 26 | R | R | A | A | A | A | 11(5) | 24(6) | 36(7) | 46(8) | 54(9) | 60(10) | 66(11) | 70(11) | 73(11) | 74(11) | A | A | A |
|---|
| 0.12 | 24 | R | R | A | A | A | A | 15(5) | 29(7) | 41(8) | 51(9) | 59(10) | 66(11) | 71(12) | 74(12) | 76(12) | 76(12) | 76(12) | A | A |
|---|
| 0.13 | 22 | R | R | A | A | A | A | 20(6) | 35(8) | 47(9) | 57(11) | 65(12) | 71(13) | 75(13) | 78(13) | 78(13) | 78(13) | 75(13) | 71(13) | A |
|---|
| 0.14 | 20 | R | R | A | A | A | A | 25(7) | 41(9) | 54(11) | 64(12) | 71(13) | 76(14) | 80(14) | 80(14) | 80(14) | 77(14) | 72(13) | 66(13) | A |
|---|
| 0.15 | 19 | R | R | A | A | A | A | 28(8) | 45(10) | 57(12) | 67(13) | 74(14) | 79(15) | 81(15) | 81(15) | 79(15) | 75(15) | 69(14) | A | A |
|---|
| 0.16 | 18 | R | R | 2(4) | A | A | A | 31(8) | 48(11) | 61(13) | 70(15) | 77(16) | 81(16) | 82(16) | 81(16) | 78(16) | 73(15) | 65(14) | A | A |
|---|
| 0.17 | 17 | R | R | 4(4) | A | A | A | 35(9) | 52(12) | 65(14) | 74(16) | 80(17) | 83(17) | 83(17) | 81(17) | 76(16) | 69(15) | 59(13) | A | A |
|---|
| 0.18 | 16 | R | 2(4) | 5(4) | 8(5) | A | A | 38(10) | 55(13) | 68(16) | 77(17) | 82(18) | 84(19) | 83(18) | 79(18) | 73(17) | 64(15) | 52(13) | A | A |
|---|
| 0.19 | 15 | R | 4(4) | 7(5) | 9(5) | A | A | 42(11) | 59(15) | 72(17) | 80(19) | 85(20) | 85(20) | 83(19) | 77(18) | 68(16) | 56(14) | A | A | A |
|---|
| 0.20 | 14 | R | 6(4) | 9(5) | 11(5) | A | A | 46(12) | 64(16) | 76(18) | 83(20) | 86(20) | 85(20) | 81(19) | 72(18) | 61(15) | 47(13) | A | A | A |
|---|
| 0.22 | 13 | 3(4) | 8(5) | 11(6) | 12(6) | 12(6) | A | 50(14) | 68(18) | 80(21) | 86(22) | 87(22) | 84(22) | 77(20) | 66(18) | 51(15) | 34(11) | A | A | A |
|---|
| 0.24 | 11 | 7(5) | 11(6) | 14(6) | 14(6) | 14(6) | A | 60(18) | 77(22) | 87(24) | 89(24) | 86(24) | 76(21) | 62(18) | 43(13) | A | A | A | | |
|---|
| 0.26 | 10 | 9(5) | 13(6) | 15(7) | 15(7) | 13(6) | A | 65(20) | 82(24) | 90(26) | 89(26) | 82(24) | 68(21) | 48(16) | 25(10) | A | A | | | |
|---|
| 0.28 | 10 | 9(6) | 13(7) | 15(8) | 15(8) | 13(7) | A | 65(22) | 82(26) | 90(29) | 89(28) | 82(26) | 68(22) | 48(17) | 25(10) | A | A | | | |
|---|
| 0.30 | 9 | 11(6) | 15(8) | 16(8) | 15(8) | 12(7) | A | 71(24) | 87(29) | 91(30) | 87(29) | 74(25) | 54(19) | 30(12) | A | A | | | | |
|---|
| 0.35 | 7 | 16(9) | 18(9) | 17(9) | 12(7) | A | A | 83(32) | 93(35) | 88(34) | 69(27) | 41(17) | A | A | | | | | | |
|---|
| 0.40 | 6 | 18(10) | 19(10) | 15(9) | 8(6) | A | A | 90(39) | 93(40) | 78(34) | 49(22) | 12(8) | A | | | | | | | |
|---|
| 0.45 | 6 | 18(11) | 19(12) | 15(10) | 8(7) | A | A | 90(44) | 93(45) | 78(38) | 49(25) | 12(9) | A | | | | | | | |
|---|
| 0.50 | 5 | 20(13) | 19(12) | 11(8) | A | A | | 95(50) | 88(47) | 58(32) | 17(11) | A | | | | | | | | |
|---|
| 0.50 | 5 | 20(13) | 19(12) | 11(8) | A | A | | 95(50) | 88(47) | 58(32) | 17(11) | A | | | | | | | | |
|---|
| 0.60 | 4 | 21(15) | 15(12) | 3(5) | A | | | 96(60) | 70(45) | 22(16) | A | | | | | | | | | |
|---|
| 0.70 | 3 | 19(16) | 6(7) | A | | | | 85(62) | 31(24) | A | | | | | | | | | | |
|---|
| 0.80 | 2 | 10(10) | A | | | | | 45(38) | A | | | | | | | | | | | |
|---|
| 0.90 | 2 | 10(11) | A | | | | | 45(43) | A | | | | | | | | | | | |
|---|
Gehan E A (1961) The determination of the number of patients required in a preliminary and a follow-up trial of a
new chemotherapeutic agent Journal of Chronic Diseases. 13:4, Pages 346-353
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. 255-256
|
