This example requests three group sequential designs for normally distributed statistics. Each design uses the power family error spending function with the default power parameter . The specified error spending method is between the approximated Pocock method () and the approximated O’Brien-Fleming method () (Jennison and Turnbull, 2000, p. 148). The three designs are identical except for the specified number of stages. The following statements request these three group sequential designs:
ods graphics on; proc seqdesign plots=( asn power combinedboundary errspend(hscale=info) ) ; TwoStageDesign: design nstages=2 method=errfuncpow alt=upper stop=reject ; FiveStageDesign: design nstages=5 method=errfuncpow alt=upper stop=reject ; TenStageDesign: design nstages=10 method=errfuncpow alt=upper stop=reject ; run; ods graphics off;
The “Design Information” table in Output 83.9.1 displays design information for the two-stage design.
Output 83.9.1: Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Standardized Z |
Alternative Hypothesis | Upper |
Early Stop | Reject Null |
Method | Error Spending |
Boundary Key | Both |
Number of Stages | 2 |
Alpha | 0.05 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 102.4167 |
Null Ref ASN (Percent of Fixed Sample) | 101.7766 |
Alt Ref ASN (Percent of Fixed Sample) | 79.81021 |
The “Boundary Information” table in Output 83.9.2 displays the information level, alternative reference, and boundary values. By default (or equivalently if you specify BOUNDARYSCALE=STDZ), the alternative reference and boundary values are displayed with the standardized normal Z scale. 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 83.9.2: Boundary Information in Scale
Boundary Information (Standardized Z Scale) Null Reference = 0 |
|||
---|---|---|---|
_Stage_ | Information Level |
Alternative | Boundary Values |
Reference | Upper | ||
Proportion | Upper | Alpha | |
1 | 0.5000 | 2.09414 | 2.24140 |
2 | 1.0000 | 2.96156 | 1.69970 |
The “Design Information” table in Output 83.9.3 displays design information for the five-stage design. Compared with the two-stage design in Output 83.9.1, the maximum information increases from 102.42 to 105.62, and the average sample number under the alternative reference (Alt Ref ASN) decreases from 79.81 to 69.64.
Output 83.9.3: Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Standardized Z |
Alternative Hypothesis | Upper |
Early Stop | Reject Null |
Method | Error Spending |
Boundary Key | Both |
Number of Stages | 5 |
Alpha | 0.05 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 105.6235 |
Null Ref ASN (Percent of Fixed Sample) | 104.356 |
Alt Ref ASN (Percent of Fixed Sample) | 69.64322 |
The “Boundary Information” table in Output 83.9.4 displays the information level, alternative reference, and boundary values with the default standardized normal Z scale.
Output 83.9.4: Boundary Information in Scale
Boundary Information (Standardized Z Scale) Null Reference = 0 |
|||
---|---|---|---|
_Stage_ | Information Level |
Alternative | Boundary Values |
Reference | Upper | ||
Proportion | Upper | Alpha | |
1 | 0.2000 | 1.34502 | 2.87816 |
2 | 0.4000 | 1.90215 | 2.47023 |
3 | 0.6000 | 2.32965 | 2.20095 |
4 | 0.8000 | 2.69005 | 1.98182 |
5 | 1.0000 | 3.00756 | 1.79024 |
The “Design Information” table in Output 83.9.5 displays design information for the ten-stage design. Compared with the five-stage design in Output 83.9.3, the maximum information increases further from 105.62 to 107.26 and under the alternative reference, the average sample number decreases further from 69.64 to 66.36.
Output 83.9.5: Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Standardized Z |
Alternative Hypothesis | Upper |
Early Stop | Reject Null |
Method | Error Spending |
Boundary Key | Both |
Number of Stages | 10 |
Alpha | 0.05 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 107.256 |
Null Ref ASN (Percent of Fixed Sample) | 105.7276 |
Alt Ref ASN (Percent of Fixed Sample) | 66.35565 |
The “Boundary Information” table in Output 83.9.6 displays the information level, alternative reference, and boundary values with the default standardized normal Z scale.
Output 83.9.6: Boundary Information in Scale
Boundary Information (Standardized Z Scale) Null Reference = 0 |
|||
---|---|---|---|
_Stage_ | Information Level |
Alternative | Boundary Values |
Reference | Upper | ||
Proportion | Upper | Alpha | |
1 | 0.1000 | 0.95840 | 3.29053 |
2 | 0.2000 | 1.35538 | 2.94037 |
3 | 0.3000 | 1.65999 | 2.72115 |
4 | 0.4000 | 1.91679 | 2.54808 |
5 | 0.5000 | 2.14304 | 2.40114 |
6 | 0.6000 | 2.34758 | 2.27127 |
7 | 0.7000 | 2.53568 | 2.15359 |
8 | 0.8000 | 2.71076 | 2.04503 |
9 | 0.9000 | 2.87519 | 1.94355 |
10 | 1.0000 | 3.03072 | 1.84765 |
With the PLOTS=ASN option, the procedure displays a plot of average sample numbers under various hypothetical references for all designs simultaneously, as shown in Output 83.9.7. By default, the option CREF= and expected sample sizes under the hypothetical references are displayed, where are values specified in the CREF= option. These CREF= values are displayed on the horizontal axis.
The plot shows that as the number of stages increases, the average sample number as a percentage of the fixed-sample design increases under the null hypothesis () but decreases under the alternative hypothesis ().
With the PLOTS=POWER option, the procedure displays a plot of the power curves under various hypothetical references for all designs simultaneously, as shown in Output 83.9.8. By default, the option CREF= and powers under hypothetical references are displayed, where are values specified in the CREF= option. These CREF= values are displayed on the horizontal axis.
Under the null hypothesis, , the power is 0.05, the upper Type I error probability. Under the alternative hypothesis, , the power is 0.9, one minus the Type II error probability. The plot shows only minor difference among the three designs.
With the PLOTS=COMBINEDBOUNDARY option, the procedure displays a plot of sequential boundaries for all designs simultaneously, as shown in Output 83.9.9. By default (or equivalently if you specify COMBINEDBOUNDARY(HSCALE=INFO)), the information levels are used on the horizontal axis. Since the maximum information is not available for the design, the percent information ratios with respect to the corresponding fixed-sample design are displayed in the plot.
The plot shows that as the number of stages increases, the maximum information increases and the boundary values also increase.
With the PLOTS=ERRSPEND(HSCALE=INFO) option, the procedure displays a plot of cumulative error spends for all boundaries in the designs simultaneously, as shown in Output 83.9.10.
The plot shows similar error spending for these three designs since all three designs are generated from the same power family error spending function.