### Example 83.7 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.

Suppose that a clinic is conducting a study of the effect of a new cancer treatment. The study consists of exposing mice to a carcinogen and randomly assigning them to either the control group or the treatment group. The event of interest is death from cancer induced by the carcinogen, and the response is the time from randomization to death.

Following the derivations in the section Test for Two Survival Distributions with a Log-Rank Test, the hypothesis with an alternative hypothesis is used, where is the hazard ratio between the treatment group and the control group.

Also suppose that from past experience, the median survival time for the control group is weeks, and the study wants to detect a weeks’ median survival time with a 80% power in the trial. Assuming exponential survival functions for the two groups, the hazard rates can be computed from

where .

Thus, with and , the hazard ratio , and the alternative reference is

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
method=whitehead
boundarykey=none
alt=upper   stop=both
alpha=0.05  beta=0.20
;
BoundaryKeyAlpha: design nstages=4
method=whitehead
boundarykey=alpha
alt=upper   stop=both
alpha=0.05  beta=0.20
;

BoundaryKeyBeta:  design nstages=4
method=whitehead
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 83.7.1 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 .

Output 83.7.1: Whitehead Design Information

The SEQDESIGN Procedure
Design: BoundaryKeyNone

Design Information
Statistic Distribution Normal
Boundary Scale Score
Alternative Hypothesis Upper
Early Stop Accept/Reject Null
Method Whitehead
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 83.7.2 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 83.7.2: 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 83.7.3 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 83.7.3: 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 83.7.4.

Output 83.7.4: 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 83.7.5 displays design specifications and the derived maximum information. Note that with the BOUNDARYKEY=ALPHA option, the specified Type I error probability is maintained.

Output 83.7.5: Whitehead Design Information

The SEQDESIGN Procedure
Design: BoundaryKeyAlpha

Design Information
Statistic Distribution Normal
Boundary Scale Score
Alternative Hypothesis Upper
Early Stop Accept/Reject Null
Method Whitehead
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 83.7.6 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 83.7.6: 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 83.7.7 displays information level, alternative reference, and boundary values.

Output 83.7.7: 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 83.7.8.

Output 83.7.8: 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 83.7.9 displays design specifications and the derived maximum information. Note that with the BOUNDARYKEY=BETA option, the specified Type II error probability is maintained.

Output 83.7.9: Whitehead Design Information

The SEQDESIGN Procedure
Design: BoundaryKeyBeta

Design Information
Statistic Distribution Normal
Boundary Scale Score
Alternative Hypothesis Upper
Early Stop Accept/Reject Null
Method Whitehead
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 83.7.10 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 83.7.10: 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 83.7.11 displays information level, alternative reference, and boundary values.

Output 83.7.11: 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 83.7.12.

Output 83.7.12: Boundary Plot

With the PLOTS=COMBINEDBOUNDARY option, a combined plot of group sequential boundaries for all designs is displayed, as shown in Output 83.7.13. 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 83.7.13: Combined Boundary Plot

The following statements invoke the SEQDESIGN procedure and specify the SAMPLESIZE statement to derive required sample sizes for a log-rank test comparing two survival distributions for the treatment effect (Jennison and Turnbull 2000, pp. 77–79; Whitehead 1997, pp. 36–39):

proc seqdesign altref=0.693147
bscale=score
;
BoundaryKeyAlpha: design nstages=4
method=whitehead
boundarykey=alpha
alt=upper   stop=both
alpha=0.05  beta=0.20
;
samplesize model=twosamplesurvival
( nullhazard=0.03466 accrate=10);
run;


The design is identical to the previous design with the BOUNDARYKEY=ALPHA option except with the addition of the sample size computation.

The Sample Size Summary table in Output 83.7.14 displays parameters for the sample size computation. Since the ACCTIME= option is not specified for the accrual time, the minimum and maximum accrual times are derived for the specified accrual rate.

Output 83.7.14: Sample Size Summary

The SEQDESIGN Procedure
Design: BoundaryKeyAlpha

Sample Size Summary
Test Two-Sample Survival
Null Hazard Rate 0.03466
Hazard Rate (Group A) 0.01733
Hazard Rate (Group B) 0.03466
Hazard Ratio 0.5
log(Hazard Ratio) -0.69315
Reference Hazards Alt Ref
Accrual Uniform
Accrual Rate 10
Min Accrual Time 6.682556
Min Sample Size 66.82556
Max Accrual Time 25.40111
Max Sample Size 254.0111
Max Number of Events 66.82556

If the ACCTIME=20 option is specified in the SAMPLESIZE statement, the Sample Size Summary table in Output 83.7.15 also displays the follow-up time and maximum sample size with the specified accrual time.

Output 83.7.15: Sample Size Summary

The SEQDESIGN Procedure
Design: WhiteheadKeyAlpha

Sample Size Summary
Test Two-Sample Survival
Null Hazard Rate 0.03466
Hazard Rate (Group A) 0.01733
Hazard Rate (Group B) 0.03466
Hazard Ratio 0.5
log(Hazard Ratio) -0.69315
Reference Hazards Alt Ref
Accrual Uniform
Accrual Rate 10
Accrual Time 20
Follow-up Time 6.474376
Total Time 26.47438
Max Number of Events 66.82556
Max Sample Size 200
Expected Sample Size (Null Ref) 161.5941
Expected Sample Size (Alt Ref) 172.4693

The Number of Events (D) and Sample Sizes (N) table in Output 83.7.16 displays the required time at each stage, in both fractional and integer numbers. The derived times under the heading Fractional Time are not integers. These times are rounded up to integers under the heading Ceiling Time. The table also displays the numbers of events and sample sizes at each stage.

Output 83.7.16: Number of Events and Sample Sizes

Numbers of Events (D) and Sample Sizes (N)
Two-Sample Log-Rank Test
_Stage_ Fractional Time Ceiling Time
D D(Grp 1) D(Grp 2) Time N N(Grp 1) N(Grp 2) Information D D(Grp 1) D(Grp 2) Time N N(Grp 1) N(Grp 2) Information
1 16.71 5.82 10.89 11.9867 119.87 59.93 59.93 4.1766 16.74 5.83 10.91 12 120.00 60.00 60.00 4.1854
2 33.41 11.84 21.57 17.3585 173.58 86.79 86.79 8.3532 35.73 12.68 23.04 18 180.00 90.00 90.00 8.9322
3 50.12 18.01 32.11 21.7480 200.00 100.00 100.00 12.5298 51.07 18.37 32.70 22 200.00 100.00 100.00 12.7667
4 66.83 24.46 42.37 26.4744 200.00 100.00 100.00 16.7064 68.55 25.14 43.41 27 200.00 100.00 100.00 17.1378