The SEQDESIGN Procedure

Example 101.5 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 $\alpha $ and $\beta $ errors. The other design uses the specified boundary Z values and derives the overall $\alpha $ and $\beta $ errors.

The following statements specify the interim boundary Z values and derive the final-stage boundary value for the specified $\alpha =0.05$ and $\beta =0.10$:

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;

The "Design Information" table in Output 101.5.1 displays design specifications and maximum information in percentage of its corresponding fixed-sample design.

Output 101.5.1: Haybittle-Peto Design Information

The SEQDESIGN Procedure
Design: OneSidedPeto

Design Information
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 101.5.2 displays the $\alpha $ and $\beta $ errors and the derived drift parameter, which is the standardized alternative reference at the final stage.

Output 101.5.2: 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 101.5.3 displays the expected stopping stage and cumulative stopping probability to reject the null hypothesis at each stage under various hypothetical references $\theta = c_{i} \theta _{1}$, where $\theta _{1}$ is the alternative reference and $c_{i}=0, 0.5, 1, 1.5$ are the default values in the CREF= option.

Output 101.5.3: 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 101.5.4 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 $\theta _1 \sqrt {I_ k}$, where $\theta _1$ is the alternative reference and $I_ k$ is the information level at stage k, $k= 1, 2, 3$.

Output 101.5.4: 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 $z \geq 3$, the trial is stopped and the null hypothesis is rejected. If the statistic $z < 3$, the trial continues to the next stage. At the final stage, the null hypothesis is rejected if the statistic $z_{3} > 1.65$. 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 101.5.5 displays cumulative error spending at each stage for each boundary. The stage 1 $\alpha $ spending 0.00135 corresponds to the one-sided p-value for a standardized Z statistic, $Z > 3$.

Output 101.5.5: 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 101.5.6. With the STOP=REJECT option, the interim rejection boundaries are displayed.

Output 101.5.6: Boundary Plot

Boundary Plot


With the PLOTS=ERRSPEND option, the procedure displays a plot of error spending for each boundary, as shown in Output 101.5.7. The error spending values in the "Error Spending Information" in Output 101.5.4 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 101.5.7: Error Spending Plot

Error Spending Plot


The following statements specify the boundary Z values and derive the $\alpha $ and $\beta $ 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;

The "Design Information" table in Output 101.5.8 displays design specifications and derived $\alpha $ and $\beta $ error levels.

Output 101.5.8: Design Information

The SEQDESIGN Procedure
Design: OneSidedPeto

Design Information
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 101.5.9 displays the $\alpha $ and $\beta $ errors and the derived drift parameter for each boundary.

Output 101.5.9: 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 101.5.10 displays the expected stopping stage and cumulative stopping probability to reject the null hypothesis at each stage under various hypothetical references $\theta = c_{i} \theta _{1}$, where $\theta _{1}$ is the alternative reference and $c_{i}=0, 0.5, 1, 1.5$ are the default values in the CREF= option.

Output 101.5.10: 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 101.5.11 displays information level, alternative references, and boundary values.

Output 101.5.11: 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 101.5.12 displays cumulative error spending at each stage for each boundary. The first-stage $\alpha $ spending 0.00135 corresponds to the one-sided p-value for a standardized Z statistic, $Z > 3$.

Output 101.5.12: 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 101.5.13. With the STOP=REJECT option, the interim rejection boundaries are displayed.

Output 101.5.13: Boundary Plot

Boundary Plot


With the PLOTS=ERRSPEND option, the procedure displays a plot of error spending for each boundary, as shown in Output 101.5.14. The error spending values in the "Error Spending Information" table in Output 101.5.10 are displayed in the plot.

Output 101.5.14: Error Spending Plot

Error Spending Plot