## Example 80.9 Creating Designs with Various Number of Stages

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 1999, 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 SEQDESIGN Procedure
Design: TwoStageDesign

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 80.9.2 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 scale. The resulting standardized alternative reference at stage is given by , where is the alternative reference and is the information level at stage , .

Output 80.9.2 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 80.9.3 displays design information for the five-stage design. Compared with the two-stage design in Output 80.9.1, the maximum information increases from to , and the average sample number under the alternative reference (Alt Ref ASN) decreases from to .

The SEQDESIGN Procedure
Design: FiveStageDesign

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 80.9.4 displays the information level, alternative reference, and boundary values with the default standardized normal scale.

Output 80.9.4 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 80.9.5 displays design information for the ten-stage design. Compared with the five-stage design in Output 80.9.3, the maximum information increases further from to and under the alternative reference, the average sample number decreases further from to .

The SEQDESIGN Procedure
Design: TenStageDesign

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 80.9.6 displays the information level, alternative reference, and boundary values with the default standardized normal scale.

Output 80.9.6 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 80.9.7. 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.

Output 80.9.7 ASN Plot

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 80.9.8. 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.

Output 80.9.8 Power Plot

Under the null hypothesis, , the power is , the upper Type I error probability. Under the alternative hypothesis, , the power is , 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 80.9.9. 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.

Output 80.9.9 Combined Boundary 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 80.9.10.

Output 80.9.10 Error Spending Plot

The plot shows similar error spending for these three designs since all three designs are generated from the same power family error spending function.