PROC SEQDESIGN: Creating a Two-Sided Asymmetric Error Spending Design with Early Stopping to Reject or Accept :: SAS/STAT(R) 9.2 User's Guide, Second Edition

The SEQDESIGN Procedure

Example 77.12 Creating a Two-Sided Asymmetric Error Spending Design with Early Stopping to Reject or Accept $\text{[math]}$

This example requests a four-stage two-sided asymmetric group sequential design for normally distributed statistics. The O’Brien-Fleming boundary can be approximated by a gamma family error spending function with parameter $\text{[math]}$ or $\text{[math]}$ , and the Pocock boundary can be approximated with parameter $\text{[math]}$ (Hwang, Shih, and DeCani 1990, p. 1440). The following statements use the gamma error spending function with early stopping to reject or accept the null hypothesis $\text{[math]}$ :

   ods graphics on;
   proc seqdesign altref=2
                  pss(cref=0 0.5 1)
                  stopprob(cref=0 1)
                  errspend
                  plots=(asn power errspend)
                  ;
   TwoSidedAsymmetric: design nstages=4
                       method=errfuncgamma(gamma=1)
                       method(beta)=errfuncgamma(gamma=-2)
                       method(upperalpha)=errfuncgamma(gamma=-5)
                       alt=twosided
                       stop=both
                       beta=0.1
                       ;
   run;
   ods graphics off;

The design uses gamma family error spending functions with $\text{[math]}$ for the upper $\text{[math]}$ boundary, $\text{[math]}$ for the lower $\text{[math]}$ boundary, and $\text{[math]}$ for the lower and upper $\text{[math]}$ boundaries.

The "Design Information" table in Output 77.12.1 displays design specifications and the derived maximum information. Note that in order to attain the same information level for the asymmetric lower and upper boundaries, the derived power at the upper alternative $\text{[math]}$ is larger than the specified $\text{[math]}$ .

Output 77.12.1 Design Information

The SEQDESIGN Procedure

Design: TwoSidedAsymmetric

Design Information
Statistic Distribution	Normal
Boundary Scale	Standardized Z
Alternative Hypothesis	Two-Sided
Early Stop	Accept/Reject Null
Method	Error Spending
Boundary Key	Both
Alternative Reference	2
Number of Stages	4
Alpha	0.05
Beta (Lower)	0.1
Beta (Upper)	0.06345
Power (Lower)	0.9
Power (Upper)	0.93655
Max Information (Percent of Fixed Sample)	104.0688
Max Information	3.162386
Null Ref ASN (Percent of Fixed Sample)	74.16654
Lower Alt Ref ASN (Percent of Fixed Sample)	59.10271
Upper Alt Ref ASN (Percent of Fixed Sample)	73.78797

The "Method Information" table in Output 77.11.2 displays the specified $\text{[math]}$ and $\text{[math]}$ error levels and the derived drift parameter. With the same information level used for the asymmetric lower and upper boundaries, only one of the $\text{[math]}$ levels is maintained and the other is derived to have the level less than or equal to the specified level.

Output 77.12.2 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	.	Gamma (Gamma=-5)	2	3.55662
Upper Beta	Error Spending	.	0.06345	Gamma (Gamma=-2)	2	3.55662
Lower Beta	Error Spending	.	0.10000	Gamma (Gamma=-2)	-2	-3.55662
Lower Alpha	Error Spending	0.02500	.	Gamma (Gamma=1)	-2	-3.55662

With the STOPPROB(CREF=0 1) option, the "Expected Cumulative Stopping Probabilities" table in Output 77.12.3 displays the expected stopping stage and cumulative stopping probabilities at each stage under the null reference $\text{[math]}$ and under the alternative reference $\text{[math]}$ .

Output 77.12.3 Stopping Probabilities

Expected Cumulative Stopping Probabilities Reference = CRef * (Alt Reference)
CRef	Ref	Expected Stopping Stage	Source	Stopping Probabilities
CRef	Ref	Expected Stopping Stage	Source	Stage_1	Stage_2	Stage_3	Stage_4
0.0000	Lower Alt	2.851	Rej Null (Lower Alt)	0.00875	0.01556	0.02087	0.02500
0.0000	Lower Alt	2.851	Rej Null (Upper Alt)	0.00042	0.00190	0.00704	0.02500
0.0000	Lower Alt	2.851	Reject Null	0.00917	0.01746	0.02791	0.05000
0.0000	Lower Alt	2.851	Accept Null	0.00000	0.30125	0.79354	0.95000
0.0000	Lower Alt	2.851	Total	0.00917	0.31870	0.82145	1.00000
1.0000	Lower Alt	2.272	Rej Null (Lower Alt)	0.27499	0.58934	0.79601	0.90000
1.0000	Lower Alt	2.272	Rej Null (Upper Alt)	0.00000	0.00000	0.00000	0.00000
1.0000	Lower Alt	2.272	Reject Null	0.27499	0.58934	0.79601	0.90000
1.0000	Lower Alt	2.272	Accept Null	0.00000	0.01863	0.04935	0.10000
1.0000	Lower Alt	2.272	Total	0.27499	0.60797	0.84536	1.00000
0.0000	Upper Alt	2.851	Rej Null (Lower Alt)	0.00875	0.01556	0.02087	0.02500
0.0000	Upper Alt	2.851	Rej Null (Upper Alt)	0.00042	0.00190	0.00704	0.02500
0.0000	Upper Alt	2.851	Reject Null	0.00917	0.01746	0.02791	0.05000
0.0000	Upper Alt	2.851	Accept Null	0.00000	0.30125	0.79354	0.95000
0.0000	Upper Alt	2.851	Total	0.00917	0.31870	0.82145	1.00000
1.0000	Upper Alt	2.836	Rej Null (Lower Alt)	0.00002	0.00002	0.00002	0.00002
1.0000	Upper Alt	2.836	Rej Null (Upper Alt)	0.05945	0.33802	0.72323	0.93655
1.0000	Upper Alt	2.836	Reject Null	0.05947	0.33804	0.72325	0.93657
1.0000	Upper Alt	2.836	Accept Null	0.00000	0.01182	0.03131	0.06343
1.0000	Upper Alt	2.836	Total	0.05947	0.34986	0.75456	1.00000

"Rej Null (Lower Alt)" and "Rej Null (Upper Alt)" under the heading "Source" indicate the probabilities of rejecting the null hypothesis for the lower alternative and for the upper alternative, respectively. "Reject Null" indicates the probability of rejecting the null hypothesis for either the lower or upper alternative, "Accept Null" indicates the probability of accepting the null hypothesis, and "Total" indicates the total probability of stopping the trial.

With the PSS(CREF=0 0.5 1.0) option, the "Power and Expected Sample Sizes" table in Output 77.12.4 displays powers and expected sample sizes under hypothetical references $\text{[math]}$ (null hypothesis $\text{[math]}$ ), $\text{[math]}$ , and $\text{[math]}$ (alternative hypothesis $\text{[math]}$ ), where $\text{[math]}$ is the alternative reference. The expected sample sizes are displayed in a scale that indicates a percentage of its corresponding fixed-sample size design.

Output 77.12.4 Power and Expected Sample Size Information

Powers and Expected Sample Sizes Reference = CRef * (Alt Reference)
CRef	Ref	Power	Sample Size
CRef	Ref	Power	Percent Fixed-Sample
0.0000	Lower Alt	0.02500	74.1665
0.5000	Lower Alt	0.34601	75.8425
1.0000	Lower Alt	0.90000	59.1027
0.0000	Upper Alt	0.02500	74.1665
0.5000	Upper Alt	0.41647	85.3976
1.0000	Upper Alt	0.93655	73.7880

Note that at $\text{[math]}$ , the null reference $\text{[math]}$ , the power with the lower alternative is the lower $\text{[math]}$ error $\text{[math]}$ , and the power with the upper alternative is the upper $\text{[math]}$ error $\text{[math]}$ . At $\text{[math]}$ , the alternative reference $\text{[math]}$ , the power with the lower alternative is the specified power $\text{[math]}$ , and the power with the upper alternative $\text{[math]}$ is greater than the specified power $\text{[math]}$ because the same information level is used for these two asymmetric boundaries.

With the PLOTS=POWER option, the procedure displays a plot of the power curves under various hypothetical references, as shown in Output 77.12.5. By default, powers under the lower hypotheses $\text{[math]}$ and under the upper hypotheses $\text{[math]}$ , are displayed for a two-sided asymmetric design, where $\text{[math]}$ and $\text{[math]}$ and $\text{[math]}$ are the lower and upper alternative references, respectively.

Output 77.12.5 Power Plot

The horizontal axis displays the multiplier of the reference difference. A positive multiplier corresponds to $\text{[math]}$ for the upper alternative hypothesis, and a negative multiplier corresponds to $\text{[math]}$ for the lower alternative hypothesis. For lower reference hypotheses, the power is the lower $\text{[math]}$ error $\text{[math]}$ under the null hypothesis ( $\text{[math]}$ ) and is $\text{[math]}$ under the alternative hypothesis ( $\text{[math]}$ ). For upper reference hypotheses, the power is the upper $\text{[math]}$ error $\text{[math]}$ under the null hypothesis ( $\text{[math]}$ ) and is $\text{[math]}$ under the alternative hypothesis ( $\text{[math]}$ ).

With the PLOTS=ASN option, the procedure displays a plot of expected sample sizes under various hypothetical references, as shown in Output 77.12.6. By default, expected sample sizes under the lower hypotheses $\text{[math]}$ and under the upper hypotheses $\text{[math]}$ are displayed for a two-sided asymmetric design, where $\text{[math]}$ and $\text{[math]}$ and $\text{[math]}$ are the lower and upper alternative references, respectively.

Output 77.12.6 ASN Plot

By default, or when you specify BETAOVERLAP=ADJUST, the SEQDESIGN procedure first derives boundary values without adjusting for the possible overlapping of the two one-sided $\text{[math]}$ boundaries based on two corresponding one-sided tests. Then the procedure checks for overlapping of the $\text{[math]}$ boundaries at the interim stages. Since the two $\text{[math]}$ boundaries overlap at stage $\text{[math]}$ , the $\text{[math]}$ boundary values for stage $\text{[math]}$ are set to missing, the $\text{[math]}$ spending values at stage $\text{[math]}$ are set to zero, and the $\text{[math]}$ spending values at subsequent stages are adjusted proportionally.

The "Boundary Information" table in Output 77.12.7 displays the information levels, alternative references, and boundary values. The default BOUNDARYSCALE=STDZ option specifies that the standardized $\text{[math]}$ scale be used to display the alternative references and boundary values. The resulting standardized alternative references at stage $\text{[math]}$ is given by $\text{[math]}$ , where $\text{[math]}$ is the specified alternative reference and $\text{[math]}$ is the information level at stage $\text{[math]}$ , $\text{[math]}$ .

Output 77.12.7 Boundary Information

Boundary Information (Standardized Z Scale) Null Reference = 0
_Stage_			Alternative		Boundary Values
	Information Level		Reference		Lower		Upper
	Proportion	Actual	Lower	Upper	Alpha	Beta	Beta	Alpha
1	0.2500	0.790597	-1.77831	1.77831	-2.37610	.	.	3.33772
2	0.5000	1.581193	-2.51491	2.51491	-2.35714	-0.48408	0.29400	2.94871
3	0.7500	2.37179	-3.08012	3.08012	-2.34861	-1.36183	1.13898	2.50473
4	1.0000	3.162386	-3.55662	3.55662	-2.32105	-2.32105	1.95675	1.95675

With the specified ODS GRAPHICS ON statement, a detailed boundary plot with the rejection and acceptance regions is displayed by default, as shown in Output 77.12.8.

Output 77.12.8 Boundary Plot

The "Error Spending Information" in Output 77.12.9 displays the cumulative error spending at each stage for each boundary.

Output 77.12.9 Error Spending Information

Error Spending Information
_Stage_	Information Level	Cumulative Error Spending
	Information Level	Lower		Upper
	Proportion	Alpha	Beta	Beta	Alpha
1	0.2500	0.00875	0.00000	0.00002	0.00042
2	0.5000	0.01556	0.01863	0.01184	0.00190
3	0.7500	0.02087	0.04935	0.03132	0.00704
4	1.0000	0.02500	0.10000	0.06345	0.02500

With the $\text{[math]}$ boundary values missing at stage $\text{[math]}$ , there is no early stopping to accept $\text{[math]}$ at stage $\text{[math]}$ , and the corresponding $\text{[math]}$ spending at stage $\text{[math]}$ is computed from the rejection region. For example, the upper $\text{[math]}$ spending at stage $\text{[math]}$ ( $\text{[math]}$ ) is the probability of rejecting $\text{[math]}$ for the lower alternative under the upper alternative reference.

With the PLOTS=ERRSPEND option, the procedure displays a plot of the cumulative error spending on each boundary at each stage, as shown in Output 77.12.10.

Output 77.12.10 Error Spending Plot

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