### Example 83.8 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 boundary and a Pocock-type error spending function for the boundary. The following statements request a one-sided design by using different and 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;
ods graphics off;


The Design Information table in Output 83.8.1 displays design specifications and the derived statistics. With the specified alternative reference, the maximum information is derived.

The SEQDESIGN Procedure
Design: OneSidedErrorSpending

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 83.8.2 displays the and errors, alternative reference, and derived drift parameter, which is the standardized alternative reference at the final stage.

Output 83.8.2: Method Information

Method Information
Boundary Method Alpha Beta Error Spending Alternative
Reference
Drift
Function
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 83.8.3 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 values specified in the CREF= option.

Output 83.8.3: Stopping Probabilities

Expected Cumulative Stopping Probabilities
Reference = CRef * (Alt Reference)
CRef Expected
Stopping Stage
Source Stopping Probabilities
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 83.8.4 displays powers and expected sample sizes under various hypothetical references , where is the alternative reference and are the default values in the CREF= option.

Output 83.8.4: Power and Expected Sample Size Information

Powers and Expected Sample Sizes
Reference = CRef * (Alt Reference)
CRef Power Sample Size
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 83.8.5. By default, expected sample sizes under the hypotheses , , are displayed, where is the alternative reference.

Output 83.8.5: 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 83.8.6. By default, the option CREF= and powers under hypothetical references are displayed, where are values specified in the CREF= option. These CREF= values are displayed on the horizontal axis.

Under the null hypothesis, , the power is 0.025, the upper Type I error probability. Under the alternative hypothesis, , the power is 0.9, one minus the Type II error probability. The plot shows only minor difference between the two designs.

Output 83.8.6: Power Plot

The Boundary Information table in Output 83.8.7 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 , where is the specified alternative reference and is the information level at stage k, .

Output 83.8.7: 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 83.8.8. This plot displays the boundary values in the Boundary Information table in Output 83.8.7.

Output 83.8.8: Boundary Plot

The Error Spending Information table in Output 83.8.9 displays cumulative error spending at each stage for each boundary.

Output 83.8.9: Error Spending Information

Error Spending Information
_Stage_ Information
Level
Cumulative Error Spending
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 83.8.10. This plot displays the cumulative error spending at each stage in the Error Spending Information table in Output 83.8.9. The O’Brien-Fleming-type 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 spending function uses more at early stages than in the later stages.

Output 83.8.10: Error Spending Plot