The SEQDESIGN Procedure

Example 83.10 Creating Two-Sided Error Spending Designs with and without Overlapping Lower and Upper $\bbeta $ Boundaries

This example requests two three-stage group sequential designs for normally distributed statistics. Each design uses a power family error spending function with a specified two-sided alternative hypothesis $H_{1}: \theta _{1}= \pm 0.2$ and early stopping only to accept the null hypothesis $H_{0}$.

The first design uses the BETAOVERLAP=NOADJUST option to derive acceptance boundary values without adjusting for the possible overlapping of the lower and upper $\beta $ boundaries computed from the two corresponding one-sided tests. The second design uses the BETAOVERLAP=ADJUST option to test the overlapping of the $\beta $ boundaries at each interim stage based on the two corresponding one-sided tests and then to set the $\beta $ boundary values at the stage to missing if overlapping occurs at that stage.

The following statements request a two-sided design with the BETAOVERLAP=NOADJUST option:

ods graphics on;
proc seqdesign altref=0.2 errspend;
   design nstages=3
          method=errfuncpow
          alt=twosided  stop=accept
          betaoverlap=noadjust
          beta=0.09
          ;
run;
ods graphics off;

The Design Information table in Output 83.10.1 displays design specifications and the derived statistics for the first design. With the specified alternative reference $\theta _1=0.2$, the maximum information is derived.

Output 83.10.1: Design Information

The SEQDESIGN Procedure
Design: Design_1

Design Information
Statistic Distribution Normal
Boundary Scale Standardized Z
Alternative Hypothesis Two-Sided
Early Stop Accept Null
Method Error Spending
Boundary Key Both
Alternative Reference 0.2
Number of Stages 3
Alpha 0.05
Beta 0.09
Power 0.91
Max Information (Percent of Fixed Sample) 103.8789
Max Information 282.9328
Null Ref ASN (Percent of Fixed Sample) 79.20197
Alt Ref ASN (Percent of Fixed Sample) 102.1476


The Boundary Information table in Output 83.10.2 displays the information level, alternative reference, and boundary values. With a specified alternative reference $\theta _{1}$, the maximum information is derived from the procedure, and the actual information level at each stage is displayed in the table. By default (or equivalently if you specify BOUNDARYSCALE=STDZ), the alternative reference and boundary values are displayed with the standardized Z scale. The alternative reference at stage k is given by $\theta _1 \sqrt {I_ k}$, where $\theta _1$ is the specified alternative reference and $I_ k$ is the information level at stage k, $k= 1, 2, 3$.

Output 83.10.2: Boundary Information

Boundary Information (Standardized Z Scale)
Null Reference = 0
_Stage_   Alternative Boundary Values
Information Level Reference Lower Upper
Proportion Actual Lower Upper Beta Beta
1 0.3333 94.31094 -1.94228 1.94228 -0.08239 0.08239
2 0.6667 188.6219 -2.74679 2.74679 -0.90351 0.90351
3 1.0000 282.9328 -3.36412 3.36412 -1.92519 1.92519


The Error Spending Information table in Output 83.10.3 displays the cumulative error spending at each stage for each boundary.

Output 83.10.3: Error Spending Information

Error Spending Information
_Stage_ Information
Level
Cumulative Error Spending
Lower Upper
Proportion Alpha Beta Beta Alpha
1 0.3333 0.00000 0.01000 0.01000 0.00000
2 0.6667 0.00000 0.04000 0.04000 0.00000
3 1.0000 0.02500 0.09000 0.09000 0.02500


With the STOP=ACCEPT option, the design does not stop at interim stages to reject $H_0$, and the $\alpha $ spending at each interim stage is zero. For the power family error spending function with the default parameter $\rho =2$, the beta spending at stage 1 is $(1/3)^{\rho } \,  \beta = (1/3)^{2} \,  0.09 = 0.01$, and the cumulative beta spending at stage 2 is $(2/3)^{\rho } \,  \beta = (2/3)^{2} \,  0.09 = 0.04$.

With ODS Graphics enabled, a detailed boundary plot with the acceptance and rejection regions is displayed, as shown in Output 83.10.4.

Output 83.10.4: Boundary Plot

Boundary Plot


The following statements request a two-sided design with the BETAOVERLAP=ADJUST option, which is the default:

ods graphics on;
proc seqdesign altref=0.2 errspend;
   design nstages=3
          method=errfuncpow
          alt=twosided
          stop=accept
          betaoverlap=adjust
          beta=0.09
          ;
run;
ods graphics off;

With the BETAOVERLAP=ADJUST option, the procedure first derives the usual $\beta $ boundary values for the two-sided design and then checks for overlapping of the $\beta $ boundaries for the two corresponding one-sided tests at each stage. If this type of overlapping occurs at a particular stage, the $\beta $ boundary values for that stage are set to missing, the $\beta $ spending values at that stage are reset to zero, and the $\beta $ spending values at subsequent stages are adjusted proportionally.

The boundary values without adjusting for the possible overlapping of the two one-sided $\beta $ boundaries are identical to the boundary values derived in the first design (with the BETAOVERLAP=NOADJUST option, as shown in Output 83.10.2). At stage 1, the upper $\beta $ boundary value for the corresponding one-sided test is

\[  \theta _1 \sqrt {I_1} - {\Phi }^{-1} (1 - \beta _1) = 0.2 \sqrt {94.31094} - {\Phi }^{-1} (0.99) = 1.94228 - 2.32635 = -0.38407  \]

where $\theta _1=0.2$ is the upper alternative reference, $I_1=94.31094$ is the information level at stage 1, and $\beta _1=0.01$ is the $\beta $ spending at stage 1 (as shown in Output 83.10.3).

Similarly, the lower $\beta $ boundary value for the corresponding one-sided test is computed as 0.38407. Since the upper $\beta $ boundary value is less than the lower $\beta $ boundary at stage 1, overlapping occurs, and so the $\beta $ boundary values for the two-sided design are set to missing at stage 1.

With the $\beta $ boundary values set to missing at stage 1 and the $\beta $ spending ${\beta }’_1=0$ the $\beta $ spending values at subsequent interim stages are adjusted proportionally. In this example, the adjusted $\beta $ spending at stage 2 is computed as

\[  {\beta }’_2 = {\beta }’_1 + \frac{\beta _2 - \beta _1}{\beta _3 - \beta _1} \;  (\beta _3 - {\beta }’_1) = 0 + \frac{0.04 - 0.01}{0.09 - 0.01} \;  0.09 = 0.03375  \]

where ${\beta _ k}$ is the cumulative $\beta $ spending at stage k before the adjustment, $k=1, 2, 3$.

The Design Information table in Output 83.10.5 displays design specifications and derived statistics for the design.

Output 83.10.5: Design Information

The SEQDESIGN Procedure
Design: Design_1

Design Information
Statistic Distribution Normal
Boundary Scale Standardized Z
Alternative Hypothesis Two-Sided
Early Stop Accept Null
Method Error Spending
Boundary Key Both
Alternative Reference 0.2
Number of Stages 3
Alpha 0.05
Beta 0.09
Power 0.91
Max Information (Percent of Fixed Sample) 101.9388
Max Information 277.649
Null Ref ASN (Percent of Fixed Sample) 80.56408
Alt Ref ASN (Percent of Fixed Sample) 100.792


The Boundary Information table in Output 83.10.6 displays the information levels, alternative references, and boundary values.

Output 83.10.6: Boundary Information

Boundary Information (Standardized Z Scale)
Null Reference = 0
_Stage_   Alternative Boundary Values
Information Level Reference Lower Upper
Proportion Actual Lower Upper Beta Beta
1 0.3333 92.54967 -1.92405 1.92405 . .
2 0.6667 185.0993 -2.72102 2.72102 -0.89469 0.89469
3 1.0000 277.649 -3.33256 3.33256 -1.93494 1.93494


The Error Spending Information table in Output 83.10.7 displays the cumulative error spending at each stage for each boundary.

Output 83.10.7: Error Spending Information

Error Spending Information
_Stage_ Information
Level
Cumulative Error Spending
Lower Upper
Proportion Alpha Beta Beta Alpha
1 0.3333 0.00000 0.00000 0.00000 0.00000
2 0.6667 0.00000 0.03375 0.03375 0.00000
3 1.0000 0.02500 0.09000 0.09000 0.02500


With ODS Graphics enabled, a detailed boundary plot with the acceptance and rejection regions is displayed, as shown in Output 83.10.8.

Output 83.10.8: Boundary Plot

Boundary Plot