### Example 87.6 Creating Designs with Various Stopping Criteria

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.

The SEQDESIGN Procedure
Design: StopToRejectAccept

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.

Output 87.6.5: Boundary Plot with Score Statistics

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.

The SEQDESIGN Procedure
Design: StopToReject

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.

Output 87.6.10: Boundary Plot with Score Statistics

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.

The SEQDESIGN Procedure
Design: StopToAccept

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.

Output 87.6.15: Boundary Plot with 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.

Output 87.6.16: Combined Boundary Plot with Score Scale

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.

Output 87.6.17: Error Spending Plot