The SEQDESIGN Procedure

Example 89.11 Creating a Two-Sided Asymmetric Error Spending Design with Early Stopping to Reject $H_0$

This example requests a three-stage two-sided asymmetric group sequential design for normally distributed statistics.

The O’Brien-Fleming boundary can be approximated using a power family error spending function with parameter $\rho =3$ , and the Pocock boundary can be approximated using a power family error spending function with parameter $\rho =1$ (Jennison and Turnbull, 2000, p. 148). The following statements use the power family error spending function to creates a two-sided asymmetric design with early stopping to reject the null hypothesis $H_0$ :

ods graphics on;
proc seqdesign altref=1.0
               pss(cref=0 0.5 1)
               stopprob(cref=0 0.5 1)
               errspend
               plots=(asn power errspend)
               ;

   TwoSidedErrorSpending: design nstages=3
                          method(upperalpha)=errfuncpow(rho=3)
                          method(loweralpha)=errfuncpow(rho=1)
                          info=cum(2 3 4)
                          alt=twosided
                          stop=reject
                          alpha=0.075(upper=0.025)
                          ;
run;
ods graphics off;

The design uses power family error spending functions with $\rho =1$ for the lower $\alpha$ boundary and $\rho =3$ for the upper $\alpha$ boundary. Thus, the design is conservative in the early stages and tends to stop the trials early only with a small p-value for the upper $\alpha$ boundary. The upper $\alpha$ level 0.025 is specified explicitly, and the lower $\alpha$ level is computed as 0.075 – 0.025 = 0.05.

The "Design Information" table in Output 89.11.1 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 lower alternative 0.92963 is larger than the default 0.90.

Output 89.11.1: Design Information

The SEQDESIGN Procedure

Design: TwoSidedErrorSpending

Design Information
Statistic Distribution	Normal
Boundary Scale	Standardized Z
Alternative Hypothesis	Two-Sided
Early Stop	Reject Null
Method	Error Spending
Boundary Key	Both
Alternative Reference	1
Number of Stages	3
Alpha	0.075
Alpha (Lower)	0.05
Alpha (Upper)	0.025
Beta (Lower)	0.07037
Beta (Upper)	0.1
Power (Lower)	0.92963
Power (Upper)	0.9
Max Information (Percent of Fixed Sample)	102.4384
Max Information	10.76365
Null Ref ASN (Percent of Fixed Sample)	100.4877
Lower Alt Ref ASN (Percent of Fixed Sample)	64.8288
Upper Alt Ref ASN (Percent of Fixed Sample)	75.98778

The "Method Information" table in Output 89.11.2 displays the specified $\alpha$ and $\beta$ error levels and the derived drift parameter. With the same information level used for the asymmetric lower and upper boundaries, only one of the $\beta$ levels is maintained, and the other is derived to have the level less than or equal to the default level.

Output 89.11.2: 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	0.10000	Power (Rho=3)	1	3.280801
Lower Alpha	Error Spending	0.05000	0.07037	Power (Rho=1)	-1	-3.2808

With the STOPPROB(CREF=0 0.5 1) option, the "Expected Cumulative Stopping Probabilities" table in Output 89.11.3 displays the expected stopping stage and cumulative stopping probability to reject the null hypothesis $H_{0}$ at each stage under hypothetical references $\theta = 0$ (null hypothesis $H_{0}$ ), $\theta = 0.5 \, \theta _{1}$ , and $\theta = \theta _{1}$ (alternative hypothesis $H_{1}$ ), where $\theta _{1}$ is the alternative reference.

Output 89.11.3: 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
0.0000	Lower Alt	2.924	Rej Null (Lower Alt)	0.02500	0.03750	0.05000
0.0000	Lower Alt	2.924	Rej Null (Upper Alt)	0.00313	0.01055	0.02500
0.0000	Lower Alt	2.924	Reject Null	0.02813	0.04805	0.07500
0.5000	Lower Alt	2.456	Rej Null (Lower Alt)	0.21185	0.33190	0.45370
0.5000	Lower Alt	2.456	Rej Null (Upper Alt)	0.00005	0.00012	0.00021
0.5000	Lower Alt	2.456	Reject Null	0.21190	0.33202	0.45391
1.0000	Lower Alt	1.531	Rej Null (Lower Alt)	0.64054	0.82803	0.92963
1.0000	Lower Alt	1.531	Rej Null (Upper Alt)	0.00000	0.00000	0.00000
1.0000	Lower Alt	1.531	Reject Null	0.64054	0.82803	0.92963
0.0000	Upper Alt	2.924	Rej Null (Lower Alt)	0.02500	0.03750	0.05000
0.0000	Upper Alt	2.924	Rej Null (Upper Alt)	0.00313	0.01055	0.02500
0.0000	Upper Alt	2.924	Reject Null	0.02813	0.04805	0.07500
0.5000	Upper Alt	2.758	Rej Null (Lower Alt)	0.00090	0.00110	0.00120
0.5000	Upper Alt	2.758	Rej Null (Upper Alt)	0.05769	0.18269	0.36458
0.5000	Upper Alt	2.758	Reject Null	0.05860	0.18379	0.36578
1.0000	Upper Alt	1.967	Rej Null (Lower Alt)	0.00001	0.00001	0.00001
1.0000	Upper Alt	1.967	Rej Null (Upper Alt)	0.33926	0.69356	0.90000
1.0000	Upper Alt	1.967	Reject Null	0.33927	0.69357	0.90001

"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.

Note that with the STOP=REJECT option, the cumulative stopping probability of accepting the null hypothesis $H_{0}$ at each interim stage is zero and is not displayed.

With the PSS(CREF=0 0.5 1.0) option, the "Power and Expected Sample Sizes" table in Output 89.11.4 displays powers and expected sample sizes under hypothetical references $\theta = 0$ (null hypothesis $H_{0}$ ), $\theta = 0.5 \, \theta _{1}$ , and $\theta = \theta _{1}$ (alternative hypothesis $H_{1}$ ), where $\theta _{1}$ is the alternative reference. The expected sample sizes are displayed in a percentage scale relative to the corresponding fixed-sample size design.

Output 89.11.4: 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	0.05000	100.4877
0.5000	Lower Alt	0.45370	88.5090
1.0000	Lower Alt	0.92963	64.8288
0.0000	Upper Alt	0.02500	100.4877
0.5000	Upper Alt	0.36458	96.2309
1.0000	Upper Alt	0.90000	75.9878

Note that at $c_{i}=0$ , the null reference $\theta =0$ , the power with the lower alternative is the lower $\alpha$ error 0.05, and the power with the upper alternative is the upper $\alpha$ error 0.025. At $c_{i}=1$ , the alternative reference $\theta =\theta _{1}$ , the power with the upper alternative is the specified power 0.90, and the power with the lower alternative 0.92963 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 89.11.5. By default, powers under the lower hypotheses $\theta = c_{i} \, \theta _{1l}$ and under the upper hypotheses $\theta = c_{i} \, \theta _{1u}$ are displayed for a two-sided asymmetric design, where $c_{i}= 0, 0.01, 0.02, \ldots , 1.50$ and $\theta _{1l}=-1$ and $\theta _{1u}=1$ are the lower and upper alternative references, respectively.

Output 89.11.5: Power Plot

The horizontal axis displays the multiplier of the reference difference. A positive multiplier corresponds to $c_{i}$ for the upper alternative hypothesis, and a negative multiplier corresponds to $-c_{i}$ for the lower alternative hypothesis. For lower reference hypotheses, the power is the lower $\alpha$ error 0.05 under the null hypothesis ( $c_{i}=0$ ) and is 0.92963 under the alternative hypothesis ( $c_{i}=1$ ). For upper reference hypotheses, the power is the upper $\alpha$ error 0.025 under the null hypothesis ( $c_{i}=0$ ) and is 0.90 under the alternative hypothesis ( $c_{i}=1$ ).

With the PLOTS=ASN option, the procedure displays a plot of expected sample sizes under various hypothetical references, as shown in Output 89.11.6. By default, expected sample sizes under the lower hypotheses $\theta = c_{i} \, \theta _{1l}$ and under the upper hypotheses $\theta = c_{i} \, \theta _{1u}$ , $c_{i}= 0, 0.01, 0.02, \ldots , 1.50$ , are displayed for a two-sided asymmetric design, where $\theta _{1l}=-1$ and $\theta _{1u}=1$ are the lower and upper alternative references, respectively.

Output 89.11.6: ASN Plot

The horizontal axis displays the multiplier of the reference difference. A positive multiplier corresponds to $c_{i}$ for the upper alternative hypothesis and a negative multiplier corresponds to $-c_{i}$ for the lower alternative hypothesis.

The "Boundary Information" table in Output 89.11.7 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 are given by $\pm \theta _1 \sqrt {I_ k}$ , where $\theta _1$ is the specified alternative reference and $I_ k$ is the information level at stage k, $k= 1, 2, 3$ .

Output 89.11.7: Boundary Information

Boundary Information (Standardized Z Scale) Null Reference = 0
_Stage_			Alternative		Boundary Values
	Information Level		Reference		Lower	Upper
	Proportion	Actual	Lower	Upper	Alpha	Alpha
1	0.5000	5.381827	-2.31988	2.31988	-1.95996	2.73437
2	0.7500	8.07274	-2.84126	2.84126	-1.98394	2.35681
3	1.0000	10.76365	-3.28080	3.28080	-1.90855	2.02853

With ODS Graphics enabled, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Output 89.11.8.

Output 89.11.8: Boundary Plot

The "Error Spending Information" table in Output 89.11.9 displays the cumulative error spending at each stage for each boundary.

Output 89.11.9: Error Spending Information

Error Spending Information
_Stage_	Information Level	Cumulative Error Spending
	Information Level	Lower		Upper
	Proportion	Alpha	Beta	Beta	Alpha
1	0.5000	0.02500	0.00000	0.00001	0.00313
2	0.7500	0.03750	0.00000	0.00001	0.01055
3	1.0000	0.05000	0.07037	0.10000	0.02500

With the STOP=REJECT option, there is no early stopping to accept $H_0$ , and the corresponding $\beta$ spending at an interim stage is computed from the rejection region. For example, the upper $\beta$ spending at stage 1 (0.00001) is the probability of rejecting $H_0$ 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 89.11.10.

Output 89.11.10: Error Spending Plot