This example requests two 3-stage group sequential designs for normally distributed statistics. Each design uses a Haybittle-Peto method with a two-sided alternative and early stopping to reject the hypothesis. One design uses the specified interim boundary Z values and derives the final-stage boundary value for the specified and errors. The other design uses the specified boundary Z values and derives the overall and errors.
The following statements specify the interim boundary Z values and derive the final-stage boundary value for the specified and :
ods graphics on; proc seqdesign altref=0.25 errspend stopprob plots=errspend ; OneSidedPeto: design nstages=3 method=peto( z=3) alt=upper stop=reject alpha=0.05 beta=0.10; run; ods graphics off;
The “Design Information” table in Output 87.5.1 displays design specifications and maximum information in percentage of its corresponding fixed-sample design.
Output 87.5.1: Haybittle-Peto Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Standardized Z |
Alternative Hypothesis | Upper |
Early Stop | Reject Null |
Method | Haybittle-Peto |
Boundary Key | Both |
Alternative Reference | 0.25 |
Number of Stages | 3 |
Alpha | 0.05 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 100.2466 |
Max Information | 137.3592 |
Null Ref ASN (Percent of Fixed Sample) | 100.1192 |
Alt Ref ASN (Percent of Fixed Sample) | 87.35 |
The “Method Information” table in Output 87.5.2 displays the and errors and the derived drift parameter, which is the standardized alternative reference at the final stage.
Output 87.5.2: Method Information
Method Information | |||||
---|---|---|---|---|---|
Boundary | Method | Alpha | Beta | Alternative Reference |
Drift |
Upper Alpha | Haybittle-Peto | 0.05000 | 0.10000 | 0.25 | 2.930009 |
With the STOPPROB option, the “Expected Cumulative Stopping Probabilities” table in Output 87.5.3 displays the expected stopping stage and cumulative stopping probability to reject the null hypothesis at each stage under various hypothetical references , where is the alternative reference and are the default values in the CREF= option.
Output 87.5.3: Stopping Probabilities
Expected Cumulative Stopping Probabilities Reference = CRef * (Alt Reference) |
|||||
---|---|---|---|---|---|
CRef | Expected Stopping Stage |
Source | Stopping Probabilities | ||
Stage_1 | Stage_2 | Stage_3 | |||
0.0000 | 2.996 | Reject Null | 0.00135 | 0.00246 | 0.05000 |
0.5000 | 2.941 | Reject Null | 0.01561 | 0.04372 | 0.42762 |
1.0000 | 2.614 | Reject Null | 0.09538 | 0.29057 | 0.90000 |
1.5000 | 1.944 | Reject Null | 0.32185 | 0.73442 | 0.99698 |
The “Boundary Information” table in Output 87.5.4 displays information level, alternative references, and boundary values. By default (or equivalently if you specify BOUNDARYSCALE=STDZ), the standardized Z scale is used to display the alternative references and boundary values. The resulting standardized alternative reference at stage k is given by , where is the alternative reference and is the information level at stage k, .
Output 87.5.4: Boundary Information
Boundary Information (Standardized Z Scale) Null Reference = 0 |
||||
---|---|---|---|---|
_Stage_ | Alternative | Boundary Values | ||
Information Level | Reference | Upper | ||
Proportion | Actual | Upper | Alpha | |
1 | 0.3333 | 45.7864 | 1.69164 | 3.00000 |
2 | 0.6667 | 91.57281 | 2.39234 | 3.00000 |
3 | 1.0000 | 137.3592 | 2.93001 | 1.65042 |
At each interim stage, if the standardized statistic , the trial is stopped and the null hypothesis is rejected. If the statistic , the trial continues to the next stage. At the final stage, the null hypothesis is rejected if the statistic . Otherwise, the hypothesis is accepted. Note that the boundary values at the final stage, 1.65, are close to the critical values 1.645 in the corresponding fixed-sample design.
The “Error Spending Information” in Output 87.5.5 displays cumulative error spending at each stage for each boundary. The stage 1 spending 0.00135 corresponds to the one-sided p-value for a standardized Z statistic, .
Output 87.5.5: Error Spending Information
Error Spending Information | |||
---|---|---|---|
_Stage_ | Information Level |
Cumulative Error Spending | |
Upper | |||
Proportion | Beta | Alpha | |
1 | 0.3333 | 0.00000 | 0.00135 |
2 | 0.6667 | 0.00000 | 0.00246 |
3 | 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 87.5.6. With the STOP=REJECT option, the interim rejection boundaries are displayed.
With the PLOTS=ERRSPEND option, the procedure displays a plot of error spending for each boundary, as shown in Output 87.5.7. The error spending values in the “Error Spending Information” in Output 87.5.4 are displayed in the plot. As expected, the error spending at each of the first two stages is small, with the standardized Z boundary value 3.
The following statements specify the boundary Z values and derive the and errors from these completely specified boundary values:
ods graphics on; proc seqdesign altref=0.25 maxinfo=200 errspend stopprob plots=errspend ; OneSidedPeto: design nstages=3 method=peto(z=3 2.5 2) alt=upper stop=reject boundarykey=none ; run; ods graphics off;
The “Design Information” table in Output 87.5.8 displays design specifications and derived and error levels.
Output 87.5.8: Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Standardized Z |
Alternative Hypothesis | Upper |
Early Stop | Reject Null |
Method | Haybittle-Peto |
Boundary Key | None |
Alternative Reference | 0.25 |
Number of Stages | 3 |
Alpha | 0.02532 |
Beta | 0.06035 |
Power | 0.93965 |
Max Information (Percent of Fixed Sample) | 101.6769 |
Max Information | 200 |
Null Ref ASN (Percent of Fixed Sample) | 101.3933 |
Alt Ref ASN (Percent of Fixed Sample) | 73.74031 |
The “Method Information” table in Output 87.5.9 displays the and errors and the derived drift parameter for each boundary.
Output 87.5.9: Method Information
Method Information | |||||
---|---|---|---|---|---|
Boundary | Method | Alpha | Beta | Alternative Reference |
Drift |
Upper Alpha | Haybittle-Peto | 0.02532 | 0.06035 | 0.25 | 3.535534 |
With the STOPPROB option, the “Expected Cumulative Stopping Probabilities” table in Output 87.5.10 displays the expected stopping stage and cumulative stopping probability to reject the null hypothesis at each stage under various hypothetical references , where is the alternative reference and are the default values in the CREF= option.
Output 87.5.10: Stopping Probabilities
Expected Cumulative Stopping Probabilities Reference = CRef * (Alt Reference) |
|||||
---|---|---|---|---|---|
CRef | Expected Stopping Stage |
Source | Stopping Probabilities | ||
Stage_1 | Stage_2 | Stage_3 | |||
0.0000 | 2.992 | Reject Null | 0.00135 | 0.00702 | 0.02532 |
0.5000 | 2.826 | Reject Null | 0.02389 | 0.15030 | 0.41775 |
1.0000 | 2.176 | Reject Null | 0.16884 | 0.65544 | 0.93965 |
1.5000 | 1.508 | Reject Null | 0.52466 | 0.96708 | 0.99954 |
The “Boundary Information” table in Output 87.5.11 displays information level, alternative references, and boundary values.
Output 87.5.11: Boundary Information
Boundary Information (Standardized Z Scale) Null Reference = 0 |
||||
---|---|---|---|---|
_Stage_ | Alternative | Boundary Values | ||
Information Level | Reference | Upper | ||
Proportion | Actual | Upper | Alpha | |
1 | 0.3333 | 66.66667 | 2.04124 | 3.00000 |
2 | 0.6667 | 133.3333 | 2.88675 | 2.50000 |
3 | 1.0000 | 200 | 3.53553 | 2.00000 |
The “Error Spending Information” in Output 87.5.12 displays cumulative error spending at each stage for each boundary. The first-stage spending 0.00135 corresponds to the one-sided p-value for a standardized Z statistic, .
Output 87.5.12: Error Spending Information
Error Spending Information | |||
---|---|---|---|
_Stage_ | Information Level |
Cumulative Error Spending | |
Upper | |||
Proportion | Beta | Alpha | |
1 | 0.3333 | 0.00000 | 0.00135 |
2 | 0.6667 | 0.00000 | 0.00702 |
3 | 1.0000 | 0.06035 | 0.02532 |
With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 87.5.13. With the STOP=REJECT option, the interim rejection boundaries are displayed.
With the PLOTS=ERRSPEND option, the procedure displays a plot of error spending for each boundary, as shown in Output 87.5.14. The error spending values in the “Error Spending Information” table in Output 87.5.10 are displayed in the plot.