# The SEQDESIGN Procedure

### Example 89.7 Creating Whitehead’s Triangular Designs

This example requests three 4-stage Whitehead’s triangular designs for normally distributed statistics. Each design has a one-sided alternative hypothesis with early stopping to reject or accept the null hypothesis . Note that Whitehead’s triangular designs are different from unified family triangular designs.

The following statements invoke the SEQDESIGN procedure and specify three Whitehead’s triangular designs:

ods graphics on;
proc seqdesign altref=0.693147
bscale=score
plots=combinedboundary
;
BoundaryKeyNone:  design nstages=4
boundarykey=none
alt=upper   stop=both
alpha=0.05  beta=0.20
;
BoundaryKeyAlpha: design nstages=4
boundarykey=alpha
alt=upper   stop=both
alpha=0.05  beta=0.20
;

BoundaryKeyBeta:  design nstages=4
boundarykey=beta
alt=upper   stop=both
alpha=0.05  beta=0.20
;
run;
ods graphics off;


Whitehead methods with early stopping to reject or accept the null hypothesis create boundaries that approximately satisfy the Type I and Type II error probability specification. The BOUNDARYKEY=NONE option specifies no adjustment to the boundary value at the final stage to maintain either a Type I or a Type II error probability level.

The "Design Information" table in Output 89.7.1 displays design specifications and maximum information. Note that with the BOUNDARYKEY=NONE option, the derived errors and are not the same as the specified errors and .

The SEQDESIGN Procedure
Design: BoundaryKeyNone

Statistic Distribution Normal
Boundary Scale Score
Alternative Hypothesis Upper
Early Stop Accept/Reject Null
Boundary Key None
Alternative Reference 0.693147
Number of Stages 4
Alpha 0.05071
Beta 0.19771
Power 0.80229
Max Information (Percent of Fixed Sample) 129.6815
Max Information 16.70639
Null Ref ASN (Percent of Fixed Sample) 62.48184
Alt Ref ASN (Percent of Fixed Sample) 73.82535

The "Method Information" table in Output 89.7.2 displays the derived and errors and the derived drift parameter. The derived errors and are not exactly the same as the specified errors and with the BOUNDARYKEY=NONE option.

Output 89.7.2: Method Information

Method Information
Boundary Method Alpha Beta Whitehead Alternative
Reference
Drift
Tau C
Upper Alpha Whitehead 0.05071 . 0.25 4.60517 0.693147 2.833131
Upper Beta Whitehead . 0.19771 0.25 4.60517 0.693147 2.833131

The "Boundary Information" table in Output 89.7.3 displays information level, alternative reference, and boundary values. With the specified BOUNDARYSCALE=SCORE option, the alternative reference and boundary values are displayed with the score statistics scale.

Output 89.7.3: Boundary Information

Boundary Information (Score Scale)
Null Reference = 0
_Stage_   Alternative Boundary Values
Information Level Reference Upper
Proportion Actual Upper Beta Alpha
1 0.2500 4.176597 2.89500 -0.95755 4.78775
2 0.5000 8.353195 5.78999 1.91510 5.74530
3 0.7500 12.52979 8.68499 4.78775 6.70285
4 1.0000 16.70639 11.57998 7.66039 7.66039

With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 89.7.4.

Output 89.7.4: Boundary Plot

The second design uses the BOUNDARYKEY=ALPHA option to adjust the boundary value at the final stage to maintain the Type I error probability level.

The "Design Information" table in Output 89.7.5 displays design specifications and the derived maximum information. Note that with the BOUNDARYKEY=ALPHA option, the specified Type I error probability is maintained.

The SEQDESIGN Procedure
Design: BoundaryKeyAlpha

Statistic Distribution Normal
Boundary Scale Score
Alternative Hypothesis Upper
Early Stop Accept/Reject Null
Boundary Key Alpha
Alternative Reference 0.693147
Number of Stages 4
Alpha 0.05
Beta 0.20044
Power 0.79956
Max Information (Percent of Fixed Sample) 129.9894
Max Information 16.70639
Null Ref ASN (Percent of Fixed Sample) 62.6302
Alt Ref ASN (Percent of Fixed Sample) 74.00064

The "Method Information" table in Output 89.7.6 displays the specified and derived and errors and the derived drift parameter. The derived Type I error probability is the same as the specified and the derived Type II error probability is not the same as the specified with the BOUNDARYKEY=ALPHA option.

Output 89.7.6: Method Information

Method Information
Boundary Method Alpha Beta Whitehead Alternative
Reference
Drift
Tau C
Upper Alpha Whitehead 0.05000 . 0.25 4.60517 0.693147 2.833131
Upper Beta Whitehead . 0.20044 0.25 4.60517 0.693147 2.833131

The "Boundary Information" table in Output 89.7.7 displays information level, alternative reference, and boundary values.

Output 89.7.7: Boundary Information

Boundary Information (Score Scale)
Null Reference = 0
_Stage_   Alternative Boundary Values
Information Level Reference Upper
Proportion Actual Upper Beta Alpha
1 0.2500 4.176597 2.89500 -0.95755 4.78775
2 0.5000 8.353195 5.78999 1.91510 5.74530
3 0.7500 12.52979 8.68499 4.78775 6.70285
4 1.0000 16.70639 11.57998 7.81300 7.81300

With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 89.7.8.

Output 89.7.8: Boundary Plot

The third design specifies the BOUNDARYKEY=BETA option to derive the boundary values to maintain the Type II error probability level .

The "Design Information" table in Output 89.7.9 displays design specifications and the derived maximum information. Note that with the BOUNDARYKEY=BETA option, the specified Type II error probability is maintained.

The SEQDESIGN Procedure
Design: BoundaryKeyBeta

Statistic Distribution Normal
Boundary Scale Score
Alternative Hypothesis Upper
Early Stop Accept/Reject Null
Boundary Key Beta
Alternative Reference 0.693147
Number of Stages 4
Alpha 0.05011
Beta 0.2
Power 0.8
Max Information (Percent of Fixed Sample) 129.9364
Max Information 16.70639
Null Ref ASN (Percent of Fixed Sample) 62.60462
Alt Ref ASN (Percent of Fixed Sample) 73.97042

The "Method Information" table in Output 89.7.10 displays the and errors and the derived drift parameter. The derived Type II error probability is the same as the specified and the derived Type I error probability is not the same as the specified with the BOUNDARYKEY=BETA option.

Output 89.7.10: Method Information

Method Information
Boundary Method Alpha Beta Whitehead Alternative
Reference
Drift
Tau C
Upper Alpha Whitehead 0.05011 . 0.25 4.60517 0.693147 2.833131
Upper Beta Whitehead . 0.20000 0.25 4.60517 0.693147 2.833131

The "Boundary Information" table in Output 89.7.11 displays information level, alternative reference, and boundary values.

Output 89.7.11: Boundary Information

Boundary Information (Score Scale)
Null Reference = 0
_Stage_   Alternative Boundary Values
Information Level Reference Upper
Proportion Actual Upper Beta Alpha
1 0.2500 4.176597 2.89500 -0.95755 4.78775
2 0.5000 8.353195 5.78999 1.91510 5.74530
3 0.7500 12.52979 8.68499 4.78775 6.70285
4 1.0000 16.70639 11.57998 7.78899 7.78899

With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 89.7.12.

Output 89.7.12: Boundary Plot

With the PLOTS=COMBINEDBOUNDARY option, a combined plot of group sequential boundaries for all designs is displayed, as shown in Output 89.7.13. It shows that three designs are similar, with a slightly smaller boundary value at the final stage for the design with the BOUNDARYKEY=NONE option.

Output 89.7.13: Combined Boundary Plot