Creating a Two-Sided Asymmetric Error Spending Design with Early Stopping to Reject
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The SEQDESIGN Procedure

Example 87.11 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 87.11.1 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 87.11.1: 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 87.11.2 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 87.11.2: Method Information

Method Information
Boundary Method Alpha Beta Error Spending Alternative
Reference 		Drift
Function
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 87.11.3 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 87.11.3: Stopping Probabilities

Expected Cumulative Stopping Probabilities
Reference = CRef * (Alt Reference)
CRef Ref Expected
Stopping Stage 		Source Stopping Probabilities
Stage_1 Stage_2 Stage_3
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 87.11.4 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 87.11.4: 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 0.05000 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 87.11.5. 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 87.11.5: 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 87.11.6. 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 87.11.6: 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 87.11.7 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 87.11.7: 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 87.11.8.

Output 87.11.8: Boundary Plot


The “Error Spending Information” table in Output 87.11.9 displays the cumulative error spending at each stage for each boundary.

Output 87.11.9: Error Spending Information

Error Spending Information
_Stage_ Information
Level 		Cumulative Error Spending
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 87.11.10.

Output 87.11.10: Error Spending Plot


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