This example requests three 5-stage group sequential designs for normally distributed statistics. Each design uses a triangular method with the specified one-sided upper alternative reference . The resulting boundary values are displayed with the score scale. Note that these unified family triangular designs are different from Whitehead’s triangular designs.
The following statements request three designs with different stopping criterion:
ods graphics on; proc seqdesign altref=0.2 bscale=score errspend plots=(combinedboundary errspend(hscale=info)) ; StopToRejectAccept: design nstages=5 method=tri alt=upper stop=both; StopToReject: design nstages=5 method=tri alt=upper stop=reject; StopToAccept: design nstages=5 method=tri alt=upper stop=accept; run; ods graphics off;
The first design has early stopping to reject or accept the null hypothesis .
The “Design Information” table in Output 87.6.1 displays design specifications and derived statistics. With the specified alternative reference, the maximum information is derived.
Output 87.6.1: Triangular Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Score |
Alternative Hypothesis | Upper |
Early Stop | Accept/Reject Null |
Method | Triangular |
Boundary Key | Both |
Alternative Reference | 0.2 |
Number of Stages | 5 |
Alpha | 0.05 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 140.0293 |
Max Information | 299.797 |
Null Ref ASN (Percent of Fixed Sample) | 59.11973 |
Alt Ref ASN (Percent of Fixed Sample) | 66.94909 |
The “Method Information” table in Output 87.6.2 displays the and errors and the derived drift parameter, which is the standardized alternative reference at the final stage. The table also shows the corresponding parameters for a triangular method as a unified family method.
Output 87.6.2: Method Information
Method Information | ||||||||
---|---|---|---|---|---|---|---|---|
Boundary | Method | Alpha | Beta | Unified Family | Alternative Reference |
Drift | ||
Rho | Tau | C | ||||||
Upper Alpha | Triangular | 0.05000 | . | 0.5 | 1 | 0.94394 | 0.2 | 3.46293 |
Upper Beta | Triangular | . | 0.10000 | 0.5 | 1 | 0.78753 | 0.2 | 3.46293 |
The “Boundary Information” table in Output 87.6.3 displays information level, alternative reference, and boundary values. With the specified BOUNDARYSCALE=SCORE option, the alternative reference and boundary values are displayed in the score statistic scale. With a score scale, the alternative reference is , where is the specified alternative reference and is the information level at stage k, .
Output 87.6.3: Boundary Information
Boundary Information (Score Scale) Null Reference = 0 |
|||||
---|---|---|---|---|---|
_Stage_ | Alternative | Boundary Values | |||
Information Level | Reference | Upper | |||
Proportion | Actual | Upper | Beta | Alpha | |
1 | 0.2000 | 59.9594 | 11.99188 | -4.37102 | 19.61274 |
2 | 0.4000 | 119.9188 | 23.98376 | 4.89371 | 22.88154 |
3 | 0.6000 | 179.8782 | 35.97564 | 14.15845 | 26.15033 |
4 | 0.8000 | 239.8376 | 47.96752 | 23.42318 | 29.41912 |
5 | 1.0000 | 299.797 | 59.95940 | 32.68791 | 32.68791 |
The “Error Spending Information” table in Output 87.6.4 displays cumulative error spending at each stage for each boundary.
Output 87.6.4: Error Spending Information
Error Spending Information | |||
---|---|---|---|
_Stage_ | Information Level |
Cumulative Error Spending | |
Upper | |||
Proportion | Beta | Alpha | |
1 | 0.2000 | 0.01729 | 0.00566 |
2 | 0.4000 | 0.04927 | 0.02138 |
3 | 0.6000 | 0.07611 | 0.03643 |
4 | 0.8000 | 0.09357 | 0.04641 |
5 | 1.0000 | 0.10000 | 0.05000 |
With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 87.6.5. With the STOP=BOTH option, both the acceptance and rejection boundaries at interim stages are displayed. With the score scale, the acceptance and rejection boundaries are straight lines and form a triangular-shape continuation region.
The second design has early stopping only to reject the null hypothesis .
The “Design Information” table in Output 87.6.6 displays design specifications and derived statistics. With the specified alternative reference, the maximum information is derived.
Output 87.6.6: Triangular Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Score |
Alternative Hypothesis | Upper |
Early Stop | Reject Null |
Method | Triangular |
Boundary Key | Both |
Alternative Reference | 0.2 |
Number of Stages | 5 |
Alpha | 0.05 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 113.4443 |
Max Information | 242.8799 |
Null Ref ASN (Percent of Fixed Sample) | 111.3399 |
Alt Ref ASN (Percent of Fixed Sample) | 67.41968 |
The “Method Information” table in Output 87.6.7 displays the and errors and the derived drift parameter. The table also shows the corresponding parameters for a triangular method as a unified family method.
Output 87.6.7: Method Information
Method Information | ||||||||
---|---|---|---|---|---|---|---|---|
Boundary | Method | Alpha | Beta | Unified Family | Alternative Reference |
Drift | ||
Rho | Tau | C | ||||||
Upper Alpha | Triangular | 0.05000 | 0.10000 | 0.5 | 1 | 0.9833 | 0.2 | 3.116921 |
The “Boundary Information” table in Output 87.6.8 displays information level, alternative reference, and boundary values. With the specified BOUNDARYSCALE=SCORE option, the alternative reference and boundary values are displayed in the score statistic scale.
Output 87.6.8: Boundary Information
Boundary Information (Score Scale) Null Reference = 0 |
||||
---|---|---|---|---|
_Stage_ | Alternative | Boundary Values | ||
Information Level | Reference | Upper | ||
Proportion | Actual | Upper | Alpha | |
1 | 0.2000 | 48.57597 | 9.71519 | 18.38919 |
2 | 0.4000 | 97.15194 | 19.43039 | 21.45405 |
3 | 0.6000 | 145.7279 | 29.14558 | 24.51891 |
4 | 0.8000 | 194.3039 | 38.86078 | 27.58378 |
5 | 1.0000 | 242.8799 | 48.57597 | 30.64864 |
The “Error Spending Information” table in Output 87.6.9 displays cumulative error spending at each stage for each boundary.
Output 87.6.9: Error Spending Information
Error Spending Information | |||
---|---|---|---|
_Stage_ | Information Level |
Cumulative Error Spending | |
Upper | |||
Proportion | Beta | Alpha | |
1 | 0.2000 | 0.00000 | 0.00416 |
2 | 0.4000 | 0.00000 | 0.01705 |
3 | 0.6000 | 0.00000 | 0.03027 |
4 | 0.8000 | 0.00000 | 0.04127 |
5 | 1.0000 | 0.10000 | 0.05000 |
With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 87.6.10. For a triangular design, these rejection boundaries form a straight line with the score scale.
The third design has early stopping to accept the null hypothesis .
The “Design Information” table in Output 87.6.11 displays design specifications and derived statistics. With the specified alternative reference, the maximum information is derived.
Output 87.6.11: Triangular Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Score |
Alternative Hypothesis | Upper |
Early Stop | Accept Null |
Method | Triangular |
Boundary Key | Both |
Alternative Reference | 0.2 |
Number of Stages | 5 |
Alpha | 0.05 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 114.9925 |
Max Information | 246.1945 |
Null Ref ASN (Percent of Fixed Sample) | 57.83208 |
Alt Ref ASN (Percent of Fixed Sample) | 110.2477 |
The “Method Information” table in Output 87.6.12 displays the and errors and the derived drift parameter. The table also shows the corresponding parameters for a triangular method as a unified family method.
Output 87.6.12: Method Information
Method Information | ||||||||
---|---|---|---|---|---|---|---|---|
Boundary | Method | Alpha | Beta | Unified Family | Alternative Reference |
Drift | ||
Rho | Tau | C | ||||||
Upper Beta | Triangular | 0.05000 | 0.10000 | 0.5 | 1 | 0.82154 | 0.2 | 3.138117 |
The “Boundary Information” table in Output 87.6.13 displays information level, alternative reference, and boundary values. With the specified BOUNDARYSCALE=SCORE option, the alternative reference and boundary values are displayed in the score statistic scale.
Output 87.6.13: Boundary Information
Boundary Information (Score Scale) Null Reference = 0 |
||||
---|---|---|---|---|
_Stage_ | Alternative | Boundary Values | ||
Information Level | Reference | Upper | ||
Proportion | Actual | Upper | Beta | |
1 | 0.2000 | 49.2389 | 9.84778 | -5.62074 |
2 | 0.4000 | 98.4778 | 19.69556 | 1.64895 |
3 | 0.6000 | 147.7167 | 29.54334 | 8.91865 |
4 | 0.8000 | 196.9556 | 39.39112 | 16.18834 |
5 | 1.0000 | 246.1945 | 49.23890 | 23.45803 |
The “Error Spending Information” table in Output 87.6.14 displays cumulative error spending at each stage for each boundary.
Output 87.6.14: Error Spending Information
Error Spending Information | |||
---|---|---|---|
_Stage_ | Information Level |
Cumulative Error Spending | |
Upper | |||
Proportion | Beta | Alpha | |
1 | 0.2000 | 0.01375 | 0.00000 |
2 | 0.4000 | 0.04149 | 0.00000 |
3 | 0.6000 | 0.06594 | 0.00000 |
4 | 0.8000 | 0.08513 | 0.00000 |
5 | 1.0000 | 0.10000 | 0.05000 |
With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 87.6.15. For a triangular design, these rejection boundaries form a straight line with the score scale.
With the PLOTS=COMBINEDBOUNDARY option, a plot of the resulting sequential boundaries for all designs is displayed, as shown in Output 87.6.16. The plot shows that the design with early stopping to reject and to accept has larger maximum information than the other two designs.
With the PLOTS=ERRSPEND(HSCALE=INFO) option, the error spending plot is displayed with the information level on the horizontal axis, as shown in Output 87.6.17. The design with early stopping to reject or accept the null hypothesis has larger spending and larger spending in early stages than the other two designs.