The SEQDESIGN Procedure

Example 101.8 Creating a One-Sided Error Spending Design

This example requests a five-stage, one-sided group sequential design for normally distributed statistics. The design uses an O’Brien-Fleming-type error spending function for the $\alpha$ boundary and a Pocock-type error spending function for the $\beta$ boundary. The following statements request a one-sided design by using different $\alpha$ and $\beta$ spending functions:

ods graphics on;
proc seqdesign altref=0.2  errspend
               pss(cref=0 0.5 1)
               stopprob(cref=0 0.5 1)
               plots=(asn power errspend)
               ;
   OneSidedErrorSpending: design nstages=5
                          method(alpha)=errfuncobf
                          method(beta)=errfuncpoc
                          alt=upper  stop=both
                          alpha=0.025
                          ;
run;

The "Design Information" table in Output 101.8.1 displays design specifications and the derived statistics. With the specified alternative reference, the maximum information is derived.

Output 101.8.1: Error Spending Method Design Information

The SEQDESIGN Procedure

Design: OneSidedErrorSpending

Design Information
Statistic Distribution	Normal
Boundary Scale	Standardized Z
Alternative Hypothesis	Upper
Early Stop	Accept/Reject Null
Method	Error Spending
Boundary Key	Both
Alternative Reference	0.2
Number of Stages	5
Alpha	0.025
Beta	0.1
Power	0.9
Max Information (Percent of Fixed Sample)	119.4278
Max Information	313.7196
Null Ref ASN (Percent of Fixed Sample)	50.35408
Alt Ref ASN (Percent of Fixed Sample)	78.77223

The "Method Information" table in Output 101.8.2 displays the $\alpha$ and $\beta$ errors, alternative reference, and derived drift parameter, which is the standardized alternative reference at the final stage.

Output 101.8.2: Method Information

Method Information
Boundary	Method	Alpha	Beta	Error Spending	Alternative Reference	Drift
Boundary	Method	Alpha	Beta	Function	Alternative Reference	Drift
Upper Alpha	Error Spending	0.02500	.	Approx O'Brien-Fleming	0.2	3.542426
Upper Beta	Error Spending	.	0.10000	Approx Pocock	0.2	3.542426

With the STOPPROB option, the "Expected Cumulative Stopping Probabilities" table in Output 101.8.3 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 $c_{i}$ are values specified in the CREF= option.

Output 101.8.3: Stopping Probabilities

Expected Cumulative Stopping Probabilities Reference = CRef * (Alt Reference)
CRef	Expected Stopping Stage	Source	Stopping Probabilities
CRef	Expected Stopping Stage	Source	Stage_1	Stage_2	Stage_3	Stage_4	Stage_5
0.0000	2.108	Reject Null	0.00000	0.00039	0.00381	0.01221	0.02500
0.0000	2.108	Accept Null	0.38080	0.69133	0.86162	0.94170	0.97500
0.0000	2.108	Total	0.38080	0.69173	0.86543	0.95391	1.00000
0.5000	3.296	Reject Null	0.00002	0.01265	0.09650	0.24465	0.38724
0.5000	3.296	Accept Null	0.13665	0.28063	0.41080	0.52230	0.61276
0.5000	3.296	Total	0.13667	0.29328	0.50730	0.76695	1.00000
1.0000	3.298	Reject Null	0.00050	0.13209	0.52642	0.80390	0.90000
1.0000	3.298	Accept Null	0.02954	0.05231	0.07085	0.08648	0.10000
1.0000	3.298	Total	0.03004	0.18440	0.59728	0.89039	1.00000

With the PSS option, the "Power and Expected Sample Sizes" table in Output 101.8.4 displays powers and expected sample sizes 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 101.8.4: Power and Expected Sample Size Information

Powers and Expected Sample Sizes Reference = CRef * (Alt Reference)
CRef	Power	Sample Size
CRef	Power	Percent Fixed-Sample
0.0000	0.02500	50.3541
0.5000	0.38724	78.7219
1.0000	0.90000	78.7722

With the PLOTS=ASN option, the procedure displays a plot of expected sample sizes under various hypothetical references, as shown in Output 101.8.5. By default, expected sample sizes under the hypotheses $\theta = c_{i} \, \theta _{1}$ , $c_{i}= 0, 0.01, 0.02, \ldots , 1.50$ , are displayed, where $\theta _{1}$ is the alternative reference.

Output 101.8.5: ASN Plot

With the PLOTS=POWER option, the procedure displays a plot of the power curves under various hypothetical references for all designs simultaneously, as shown in Output 101.8.6. By default, the option CREF= $0, 0.01, 0.02, \ldots , 1.50$ and powers under hypothetical references $\theta = c_{i} \, \theta _{1}$ are displayed, where $c_{i}$ are values specified in the CREF= option. These CREF= values are displayed on the horizontal axis.

Under the null hypothesis, $c_{i}=0$ , the power is 0.025, the upper Type I error probability. Under the alternative hypothesis, $c_{i}=1$ , the power is 0.9, one minus the Type II error probability. The plot shows only minor difference between the two designs.

Output 101.8.6: Power Plot

The "Boundary Information" table in Output 101.8.7 displays information level, alternative reference, and boundary values. By default (or equivalently if you specify BOUNDARYSCALE=STDZ), the alternative reference and boundary values are displayed with the standardized Z scale. That is, the resulting standardized 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, $k= 1, 2, \ldots , 5$ .

Output 101.8.7: Boundary Information

Boundary Information (Standardized Z Scale) Null Reference = 0
_Stage_			Alternative	Boundary Values
	Information Level		Reference	Upper
	Proportion	Actual	Upper	Beta	Alpha
1	0.2000	62.74393	1.58422	-0.30338	4.87688
2	0.4000	125.4879	2.24043	0.41667	3.35706
3	0.6000	188.2318	2.74395	0.97165	2.67766
4	0.8000	250.9757	3.16844	1.43627	2.26535
5	1.0000	313.7196	3.54243	1.87522	1.87522

With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 101.8.8. This plot displays the boundary values in the "Boundary Information" table in Output 101.8.7.

Output 101.8.8: Boundary Plot

The "Error Spending Information" table in Output 101.8.9 displays cumulative error spending at each stage for each boundary.

Output 101.8.9: Error Spending Information

Error Spending Information
_Stage_	Information Level	Cumulative Error Spending
	Information Level	Upper
	Proportion	Beta	Alpha
1	0.2000	0.02954	0.00000
2	0.4000	0.05231	0.00039
3	0.6000	0.07085	0.00381
4	0.8000	0.08648	0.01221
5	1.0000	0.10000	0.02500

With the PLOTS=ERRSPEND option, the procedure displays a plot of error spending for each boundary, as shown in Output 101.8.10. This plot displays the cumulative error spending at each stage in the "Error Spending Information" table in Output 101.8.9. The O’Brien-Fleming-type $\alpha$ spending function is conservative in early stages because it uses much less at early stages than in the later stages. In contrast, the Pocock-type $\beta$ spending function uses more at early stages than in the later stages.

Output 101.8.10: Error Spending Plot