The SEQDESIGN Procedure

Example 101.10 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;

The "Design Information" table in Output 101.10.1 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 101.10.1: 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 101.10.2 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 101.10.2: 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 101.10.3 displays the cumulative error spending at each stage for each boundary.

Output 101.10.3: Error Spending Information

Error Spending Information
_Stage_	Information Level	Cumulative Error Spending
	Information Level	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 101.10.4.

Output 101.10.4: 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;

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 101.10.2). 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 101.10.3).

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 101.10.5 displays design specifications and derived statistics for the design.

Output 101.10.5: 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 101.10.6 displays the information levels, alternative references, and boundary values.

Output 101.10.6: 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 101.10.7 displays the cumulative error spending at each stage for each boundary.

Output 101.10.7: Error Spending Information

Error Spending Information
_Stage_	Information Level	Cumulative Error Spending
	Information Level	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 101.10.8.

Output 101.10.8: Boundary Plot

The SEQDESIGN Procedure

Example 101.10 Creating Two-Sided Error Spending Designs with and without Overlapping Lower and Upper Boundaries

Example 101.10 Creating Two-Sided Error Spending Designs with and without Overlapping Lower and Upper $\bbeta$ Boundaries