### Example 83.5 Creating Designs Using Haybittle-Peto Methods

This example requests two 3-stage group sequential designs for normally distributed statistics. Each design uses a Haybittle-Peto method with a two-sided alternative and early stopping to reject the hypothesis. One design uses the specified interim boundary Z values and derives the final-stage boundary value for the specified and errors. The other design uses the specified boundary Z values and derives the overall and errors.

The following statements specify the interim boundary Z values and derive the final-stage boundary value for the specified and :

ods graphics on;
proc seqdesign altref=0.25
errspend
stopprob
plots=errspend
;
OneSidedPeto: design nstages=3
method=peto( z=3)
alt=upper   stop=reject
alpha=0.05  beta=0.10;
run;
ods graphics off;


The Design Information table in Output 83.5.1 displays design specifications and maximum information in percentage of its corresponding fixed-sample design.

The SEQDESIGN Procedure
Design: OneSidedPeto

Statistic Distribution Normal
Boundary Scale Standardized Z
Alternative Hypothesis Upper
Early Stop Reject Null
Method Haybittle-Peto
Boundary Key Both
Alternative Reference 0.25
Number of Stages 3
Alpha 0.05
Beta 0.1
Power 0.9
Max Information (Percent of Fixed Sample) 100.2466
Max Information 137.3592
Null Ref ASN (Percent of Fixed Sample) 100.1192
Alt Ref ASN (Percent of Fixed Sample) 87.35

The Method Information table in Output 83.5.2 displays the and errors and the derived drift parameter, which is the standardized alternative reference at the final stage.

Output 83.5.2: Method Information

Method Information
Boundary Method Alpha Beta Alternative
Reference
Drift
Upper Alpha Haybittle-Peto 0.05000 0.10000 0.25 2.930009

With the STOPPROB option, the Expected Cumulative Stopping Probabilities table in Output 83.5.3 displays the expected stopping stage and cumulative stopping probability to reject the null hypothesis at each stage under various hypothetical references , where is the alternative reference and are the default values in the CREF= option.

Output 83.5.3: Stopping Probabilities

Expected Cumulative Stopping Probabilities
Reference = CRef * (Alt Reference)
CRef Expected
Stopping Stage
Source Stopping Probabilities
Stage_1 Stage_2 Stage_3
0.0000 2.996 Reject Null 0.00135 0.00246 0.05000
0.5000 2.941 Reject Null 0.01561 0.04372 0.42762
1.0000 2.614 Reject Null 0.09538 0.29057 0.90000
1.5000 1.944 Reject Null 0.32185 0.73442 0.99698

The Boundary Information table in Output 83.5.4 displays information level, alternative references, and boundary values. By default (or equivalently if you specify BOUNDARYSCALE=STDZ), the standardized Z scale is used to display the alternative references and boundary values. The resulting standardized alternative reference at stage k is given by , where is the alternative reference and is the information level at stage k, .

Output 83.5.4: Boundary Information

Boundary Information (Standardized Z Scale)
Null Reference = 0
_Stage_   Alternative Boundary Values
Information Level Reference Upper
Proportion Actual Upper Alpha
1 0.3333 45.7864 1.69164 3.00000
2 0.6667 91.57281 2.39234 3.00000
3 1.0000 137.3592 2.93001 1.65042

At each interim stage, if the standardized statistic , the trial is stopped and the null hypothesis is rejected. If the statistic , the trial continues to the next stage. At the final stage, the null hypothesis is rejected if the statistic . Otherwise, the hypothesis is accepted. Note that the boundary values at the final stage, 1.65, are close to the critical values 1.645 in the corresponding fixed-sample design.

The Error Spending Information in Output 83.5.5 displays cumulative error spending at each stage for each boundary. The stage 1 spending 0.00135 corresponds to the one-sided p-value for a standardized Z statistic, .

Output 83.5.5: Error Spending Information

Error Spending Information
_Stage_ Information
Level
Cumulative Error Spending
Upper
Proportion Beta Alpha
1 0.3333 0.00000 0.00135
2 0.6667 0.00000 0.00246
3 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 83.5.6. With the STOP=REJECT option, the interim rejection boundaries are displayed.

Output 83.5.6: Boundary Plot

With the PLOTS=ERRSPEND option, the procedure displays a plot of error spending for each boundary, as shown in Output 83.5.7. The error spending values in the Error Spending Information in Output 83.5.4 are displayed in the plot. As expected, the error spending at each of the first two stages is small, with the standardized Z boundary value 3.

Output 83.5.7: Error Spending Plot

The following statements specify the boundary Z values and derive the and errors from these completely specified boundary values:

ods graphics on;
proc seqdesign altref=0.25
maxinfo=200
errspend
stopprob
plots=errspend
;
OneSidedPeto: design nstages=3
method=peto(z=3 2.5 2)
alt=upper  stop=reject
boundarykey=none
;
run;
ods graphics off;


The Design Information table in Output 83.5.8 displays design specifications and derived and error levels.

The SEQDESIGN Procedure
Design: OneSidedPeto

Statistic Distribution Normal
Boundary Scale Standardized Z
Alternative Hypothesis Upper
Early Stop Reject Null
Method Haybittle-Peto
Boundary Key None
Alternative Reference 0.25
Number of Stages 3
Alpha 0.02532
Beta 0.06035
Power 0.93965
Max Information (Percent of Fixed Sample) 101.6769
Max Information 200
Null Ref ASN (Percent of Fixed Sample) 101.3933
Alt Ref ASN (Percent of Fixed Sample) 73.74031

The Method Information table in Output 83.5.9 displays the and errors and the derived drift parameter for each boundary.

Output 83.5.9: Method Information

Method Information
Boundary Method Alpha Beta Alternative
Reference
Drift
Upper Alpha Haybittle-Peto 0.02532 0.06035 0.25 3.535534

With the STOPPROB option, the Expected Cumulative Stopping Probabilities table in Output 83.5.10 displays the expected stopping stage and cumulative stopping probability to reject the null hypothesis at each stage under various hypothetical references , where is the alternative reference and are the default values in the CREF= option.

Output 83.5.10: Stopping Probabilities

Expected Cumulative Stopping Probabilities
Reference = CRef * (Alt Reference)
CRef Expected
Stopping Stage
Source Stopping Probabilities
Stage_1 Stage_2 Stage_3
0.0000 2.992 Reject Null 0.00135 0.00702 0.02532
0.5000 2.826 Reject Null 0.02389 0.15030 0.41775
1.0000 2.176 Reject Null 0.16884 0.65544 0.93965
1.5000 1.508 Reject Null 0.52466 0.96708 0.99954

The Boundary Information table in Output 83.5.11 displays information level, alternative references, and boundary values.

Output 83.5.11: Boundary Information

Boundary Information (Standardized Z Scale)
Null Reference = 0
_Stage_   Alternative Boundary Values
Information Level Reference Upper
Proportion Actual Upper Alpha
1 0.3333 66.66667 2.04124 3.00000
2 0.6667 133.3333 2.88675 2.50000
3 1.0000 200 3.53553 2.00000

The Error Spending Information in Output 83.5.12 displays cumulative error spending at each stage for each boundary. The first-stage spending 0.00135 corresponds to the one-sided p-value for a standardized Z statistic, .

Output 83.5.12: Error Spending Information

Error Spending Information
_Stage_ Information
Level
Cumulative Error Spending
Upper
Proportion Beta Alpha
1 0.3333 0.00000 0.00135
2 0.6667 0.00000 0.00702
3 1.0000 0.06035 0.02532

With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 83.5.13. With the STOP=REJECT option, the interim rejection boundaries are displayed.

Output 83.5.13: Boundary Plot

With the PLOTS=ERRSPEND option, the procedure displays a plot of error spending for each boundary, as shown in Output 83.5.14. The error spending values in the Error Spending Information table in Output 83.5.10 are displayed in the plot.

Output 83.5.14: Error Spending Plot