# The SEQDESIGN Procedure

### Example 101.12 Creating a Two-Sided Asymmetric Error Spending Design with Early Stopping to Reject or Accept

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 or –5, and the Pocock boundary can be approximated with parameter (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 :

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;


The design uses gamma family error spending functions with for the upper boundary, for the lower boundary, and for the lower and upper boundaries.

The "Design Information" table in Output 101.12.1 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 0.93655 is larger than the specified .

The SEQDESIGN Procedure
Design: TwoSidedAsymmetric

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 101.11.2 displays the specified and error levels and the derived drift parameter. With the same information level used for the asymmetric lower and upper boundaries, only one of the levels is maintained and the other is derived to have the level less than or equal to the specified level.

Output 101.12.2: Method Information

Method Information
Boundary Method Alpha Beta Error Spending Alternative
Reference
Drift
Function
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 101.12.3 displays the expected stopping stage and cumulative stopping probabilities at each stage under the null reference and under the alternative reference .

Output 101.12.3: Stopping Probabilities

Expected Cumulative Stopping Probabilities
Reference = CRef * (Alt Reference)
CRef Ref Expected
Stopping Stage
Source Stopping Probabilities
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 101.12.4 displays powers and expected sample sizes under hypothetical references (null hypothesis ), , and (alternative hypothesis ), where 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 101.12.4: Power and Expected Sample Size Information

Powers and Expected Sample Sizes
Reference = CRef * (Alt Reference)
CRef Ref Power Sample Size
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 , the null reference , the power with the lower alternative is the lower error 0.025, and the power with the upper alternative is the upper error 0.025. At , the alternative reference , the power with the lower alternative is the specified power 0.90, and the power with the upper alternative 0.93655 is greater than the specified power 0.90 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 101.12.5. By default, powers under the lower hypotheses and under the upper hypotheses , are displayed for a two-sided asymmetric design, where and and are the lower and upper alternative references, respectively.

Output 101.12.5: Power Plot

The horizontal axis displays the multiplier of the reference difference. A positive multiplier corresponds to for the upper alternative hypothesis, and a negative multiplier corresponds to for the lower alternative hypothesis. For lower reference hypotheses, the power is the lower error 0.025 under the null hypothesis () and is 0.90 under the alternative hypothesis (). For upper reference hypotheses, the power is the upper error 0.025 under the null hypothesis () and is 0.93655 under the alternative hypothesis ().

With the PLOTS=ASN option, the procedure displays a plot of expected sample sizes under various hypothetical references, as shown in Output 101.12.6. By default, expected sample sizes under the lower hypotheses and under the upper hypotheses are displayed for a two-sided asymmetric design, where and and are the lower and upper alternative references, respectively.

Output 101.12.6: ASN Plot

The horizontal axis displays the multiplier of the reference difference. A positive multiplier corresponds to for the upper alternative hypothesis, and a negative multiplier corresponds to for the lower alternative hypothesis.

By default (or equivalently if you specify BETAOVERLAP=ADJUST), the SEQDESIGN procedure first derives boundary values without adjusting for the possible overlapping of the two one-sided boundaries based on two corresponding one-sided tests. Then the procedure checks for overlapping of the boundaries at the interim stages. Since the two boundaries overlap at stage 1, the boundary values for stage 1 are set to missing, the spending values at stage 1 are set to zero, and the spending values at subsequent stages are adjusted proportionally.

The "Boundary Information" table in Output 101.12.7 displays the information levels, alternative references, and boundary values. By default (or equivalently if you specify BOUNDARYSCALE=STDZ), the standardized Z scale is used to display the alternative references and boundary values. The resulting standardized alternative references at stage k is given by , where is the specified alternative reference and is the information level at stage k, .

Output 101.12.7: 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 ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 101.12.8.

Output 101.12.8: Boundary Plot

The "Error Spending Information" in Output 101.12.9 displays the cumulative error spending at each stage for each boundary.

Output 101.12.9: Error Spending Information

Error Spending Information
_Stage_ Information
Level
Cumulative Error Spending
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 boundary values missing at stage 1, there is no early stopping to accept at stage 1, and the corresponding spending at stage 1 is computed from the rejection region. For example, the upper spending at stage 1 (0.00002) is the probability of rejecting 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 101.12.10.

Output 101.12.10: Error Spending Plot