The SEQDESIGN Procedure

Example 89.9 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 $\rho =2$ . The specified error spending method is between the approximated Pocock method ( $\rho =1$ ) and the approximated O’Brien-Fleming method ( $\rho =3$ ) (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 89.9.1 displays design information for the two-stage design.

Output 89.9.1: Design Information

The SEQDESIGN Procedure

Design: TwoStageDesign

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 89.9.2 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 $\theta _1 \sqrt {I_ k}$ , where $\theta _1$ is the alternative reference and $I_ k$ is the information level at stage k, $k= 1, 2$ .

Output 89.9.2: Boundary Information in $\Mathtext{Z}$ Scale

Boundary Information (Standardized Z Scale) Null Reference = 0
_Stage_	Information Level	Alternative	Boundary Values
	Information Level	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 89.9.3 displays design information for the five-stage design. Compared with the two-stage design in Output 89.9.1, 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 89.9.3: Design Information

The SEQDESIGN Procedure

Design: FiveStageDesign

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

Output 89.9.4: Boundary Information in $\Mathtext{Z}$ Scale

Boundary Information (Standardized Z Scale) Null Reference = 0
_Stage_	Information Level	Alternative	Boundary Values
	Information Level	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 89.9.5 displays design information for the ten-stage design. Compared with the five-stage design in Output 89.9.3, 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 89.9.5: Design Information

The SEQDESIGN Procedure

Design: TenStageDesign

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

Output 89.9.6: Boundary Information in $\Mathtext{Z}$ Scale

Boundary Information (Standardized Z Scale) Null Reference = 0
_Stage_	Information Level	Alternative	Boundary Values
	Information Level	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 89.9.7. By default, the option CREF= $0, 0.01, 0.02, \ldots , 1.50$ and expected sample sizes under the hypothetical references $\theta = c_{i} \, \theta _{1}$ are displayed, where $c_{i}$ are values specified in the CREF= option. These CREF= values are displayed on the horizontal axis.

Output 89.9.7: 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 ( $c_{i}=0$ ) but decreases under the alternative hypothesis ( $c_{i}=1$ ).

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 89.9.8. By default, the option CREF= $0, 0.01, 0.02, \ldots , 1.50$ and powers under hypothetical references $\theta = c_{i} \, \theta _{1}$ are displayed, where $c_{i}$ are values specified in the CREF= option. These CREF= values are displayed on the horizontal axis.

Output 89.9.8: Power Plot

Under the null hypothesis, $c_{i}=0$ , the power is 0.05, the upper Type I error probability. Under the alternative hypothesis, $c_{i}=1$ , 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 89.9.9. 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 89.9.9: Combined Boundary Plot

The plot shows that as the number of stages increases, the maximum information increases and the $\alpha$ 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 89.9.10.

Output 89.9.10: 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.