This example requests two 4-stage group sequential designs for normally distributed statistics with equally spaced information levels at all stages. One design uses Pocock’s method and the other uses the O’Brien-Fleming method. The following statements invoke the SEQDESIGN procedure and request these two designs:
proc seqdesign altref=0.4 pss stopprob errspend ; TwoSidedPocock: design nstages=4 method=poc; TwoSidedOBrienFleming: design nstages=4 method=obf; samplesize model=twosamplemean(stddev=0.8 weight=2); run;
By default (or equivalently if you specify ALT=TWOSIDED and STOP=REJECT in the DESIGN statement), each design has a null hypothesis with a two-sided alternative with early stopping to reject .
The “Design Information” table in Output 87.3.1 displays design specifications and derived statistics for the Pocock’s design. With the specified ALTREF= option, the maximum information is also derived.
Output 87.3.1: Pocock Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Standardized Z |
Alternative Hypothesis | Two-Sided |
Early Stop | Reject Null |
Method | Pocock |
Boundary Key | Both |
Alternative Reference | 0.4 |
Number of Stages | 4 |
Alpha | 0.05 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 118.3143 |
Max Information | 77.69844 |
Null Ref ASN (Percent of Fixed Sample) | 115.6074 |
Alt Ref ASN (Percent of Fixed Sample) | 69.74805 |
With the corresponding fixed-sample information
the fixed-sample information ratio is .
For a two-sided design with early stopping to reject the null hypothesis, lower and upper boundaries are created. The “Method Information” table in Output 87.3.2 displays the and errors, alternative references, and derived drift parameters, which are the standardized alternative references at the final stage.
Output 87.3.2: Method Information
Method Information | ||||||||
---|---|---|---|---|---|---|---|---|
Boundary | Method | Alpha | Beta | Unified Family | Alternative Reference |
Drift | ||
Rho | Tau | C | ||||||
Upper Alpha | Pocock | 0.02500 | 0.10000 | 0 | 0 | 2.36129 | 0.4 | 3.525869 |
Lower Alpha | Pocock | 0.02500 | 0.10000 | 0 | 0 | 2.36129 | -0.4 | -3.52587 |
With the METHOD=POC option, the Pocock method is used for each boundary. The Pocock method is one of the unified family methods, and the table also displays its corresponding parameters as a unified family method and the derived parameters for the boundary values.
With the PSS option, the “Power and Expected Sample Sizes” table in Output 87.3.3 displays powers and expected sample sizes under various hypothetical references , where is the alternative reference and are values specified in the CREF= option. By default, .
Output 87.3.3: 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 | 115.6074 |
0.5000 | 0.34252 | 104.0615 |
1.0000 | 0.90000 | 69.7480 |
1.5000 | 0.99869 | 43.6600 |
Note that at , the null reference , and the power 0.025 corresponds to the one-sided Type I error probability 0.025. At , , the power 0.9 is the power of the design. The expected sample sizes are displayed in a percentage scale to its corresponding fixed-sample size design. With the specified SAMPLESIZE statement, the expected sample sizes for the specified model in the SAMPLESIZE statement are also displayed.
With the STOPPROB option, the “Expected Cumulative Stopping Probabilities” table in Output 87.3.4 displays the expected cumulative 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. By default, .
Output 87.3.4: 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 | |||
0.0000 | 3.908 | Reject Null | 0.01821 | 0.03155 | 0.04176 | 0.05000 |
0.5000 | 3.518 | Reject Null | 0.07005 | 0.15939 | 0.25242 | 0.34327 |
1.0000 | 2.358 | Reject Null | 0.27482 | 0.58074 | 0.78638 | 0.90002 |
1.5000 | 1.476 | Reject Null | 0.61145 | 0.92348 | 0.98900 | 0.99869 |
Note that at , the cumulative stopping probability to reject at the final stage is the overall Type I error probability 0.05. At , the alternative hypothesis , the cumulative stopping probability to reject includes both the probability in the lower rejection region and the probability in the upper rejection region. This stopping probability to reject at the final stage, 0.90002, is slightly greater than the power , which corresponds to the cumulative stopping probability in the upper rejection region only. See the section Type I and Type II Errors for a detailed description of the Type II error probability .
The “Boundary Information” table in Output 87.3.5 displays the information level, alternative references, and boundary values at each stage. 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 reference at stage k is given by , where is the alternative reference and is the information level at stage k, .
Output 87.3.5: Boundary Information
Boundary Information (Standardized Z Scale) Null Reference = 0 |
|||||||
---|---|---|---|---|---|---|---|
_Stage_ | Alternative | Boundary Values | |||||
Information Level | Reference | Lower | Upper | ||||
Proportion | Actual | N | Lower | Upper | Alpha | Alpha | |
1 | 0.2500 | 19.42461 | 55.94288 | -1.76293 | 1.76293 | -2.36129 | 2.36129 |
2 | 0.5000 | 38.84922 | 111.8858 | -2.49317 | 2.49317 | -2.36129 | 2.36129 |
3 | 0.7500 | 58.27383 | 167.8286 | -3.05349 | 3.05349 | -2.36129 | 2.36129 |
4 | 1.0000 | 77.69844 | 223.7715 | -3.52587 | 3.52587 | -2.36129 | 2.36129 |
By default (or equivalently if you specify INFO=EQUAL in the DESIGN statement), equally spaced information levels are used. With the SAMPLESIZE statement, the required sample size N is also displayed under the heading “Information Level.” With the Pocock method, the standardized Z boundary values are identical at all stages for each boundary.
At each interim stage, the hypothesis of is rejected if the standardized normal test statistic , the lower boundary, or , the upper boundary. Otherwise, the trial continues to the next stage. At the final stage, stage 4, the trial stops and the hypothesis is rejected if the test statistic . Otherwise, the hypothesis is accepted.
The “Error Spending Information” in Output 87.3.6 displays cumulative error spending at each stage for each boundary. It shows that more errors are used in early stages than in later stages.
Output 87.3.6: Error Spending Information
Error Spending Information | |||||
---|---|---|---|---|---|
_Stage_ | Information Level |
Cumulative Error Spending | |||
Lower | Upper | ||||
Proportion | Alpha | Beta | Beta | Alpha | |
1 | 0.2500 | 0.00911 | 0.00002 | 0.00002 | 0.00911 |
2 | 0.5000 | 0.01577 | 0.00002 | 0.00002 | 0.01577 |
3 | 0.7500 | 0.02088 | 0.00002 | 0.00002 | 0.02088 |
4 | 1.0000 | 0.02500 | 0.10000 | 0.10000 | 0.02500 |
The “Sample Size Summary” table in Output 87.3.7 displays the specified parameters for the sample size computation of the two-sample test for mean difference.
Output 87.3.7: Sample Size Summary
Sample Size Summary | |
---|---|
Test | Two-Sample Means |
Mean Difference | 0.4 |
Standard Deviation | 0.8 |
Max Sample Size | 223.7715 |
Expected Sample Size (Null Ref) | 218.652 |
Expected Sample Size (Alt Ref) | 131.9167 |
Weight (Group A) | 2 |
Weight (Group B) | 1 |
The “Sample Sizes (N)” table in Output 87.3.8 displays the derived sample sizes at each stage, in both fractional and integer numbers. With the WEIGHT=2 option, the allocation ratio is 2 for the first group and 1 for the second group. See the section Test for the Difference between Two Normal Means for the derivation of these sample sizes. With the fixed-sample information ratio 1.1831, the derived sample sizes in fractional numbers are derived by multiplying 1.1831 by the corresponding sample sizes in the fixed-sample design.
Output 87.3.8: Sample Sizes
Sample Sizes (N) Two-Sample Z Test for Mean Difference |
||||||||
---|---|---|---|---|---|---|---|---|
_Stage_ | Fractional N | Ceiling N | ||||||
N | N(Grp 1) | N(Grp 2) | Information | N | N(Grp 1) | N(Grp 2) | Information | |
1 | 55.94 | 37.30 | 18.65 | 19.4246 | 57 | 38 | 19 | 19.7917 |
2 | 111.89 | 74.59 | 37.30 | 38.8492 | 113 | 75 | 38 | 39.4082 |
3 | 167.83 | 111.89 | 55.94 | 58.2738 | 168 | 112 | 56 | 58.3333 |
4 | 223.77 | 149.18 | 74.59 | 77.6984 | 225 | 150 | 75 | 78.1250 |
These fractional sample sizes are rounded up to integers under the heading “Ceiling N.” When the resulting integer sample sizes are used, the corresponding information levels are slightly larger than the levels from the design. This can increase the power slightly if a trial uses these integer sample sizes.
Note that compared with other designs, a Pocock design can stop the trial early with a larger p-value. However, this might not be persuasive enough to make a new treatment widely accepted (Pocock and White, 1999).
The “Design Information” table in Output 87.3.9 displays design specifications and the derived statistics for the O’Brien-Fleming design. With the specified ALTREF= option, the maximum information is derived.
Output 87.3.9: O’Brien-Fleming Design Information
Design Information | |
---|---|
Statistic Distribution | Normal |
Boundary Scale | Standardized Z |
Alternative Hypothesis | Two-Sided |
Early Stop | Reject Null |
Method | O'Brien-Fleming |
Boundary Key | Both |
Alternative Reference | 0.4 |
Number of Stages | 4 |
Alpha | 0.05 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 102.2163 |
Max Information | 67.12682 |
Null Ref ASN (Percent of Fixed Sample) | 101.5728 |
Alt Ref ASN (Percent of Fixed Sample) | 76.7397 |
With the corresponding fixed-sample information
the fixed-sample information ratio is . That is, the maximum information for the O’Brien-Fleming design is only 2.2% more than for the corresponding fixed-sample design.
The “Method Information” table in Output 87.3.10 displays the Type I level and Type II level. It also displays the derived drift parameter , which is the standardized alternative reference at the final stage.
Output 87.3.10: Method Information
Method Information | ||||||||
---|---|---|---|---|---|---|---|---|
Boundary | Method | Alpha | Beta | Unified Family | Alternative Reference |
Drift | ||
Rho | Tau | C | ||||||
Upper Alpha | O'Brien-Fleming | 0.02500 | 0.10000 | 0.5 | 0 | 2.02429 | 0.4 | 3.277238 |
Lower Alpha | O'Brien-Fleming | 0.02500 | 0.10000 | 0.5 | 0 | 2.02429 | -0.4 | -3.27724 |
With the METHOD=OBF option, the O’Brien-Fleming method is used for each boundary. The O’Brien-Fleming method is one of the unified family methods, and the table also displays its corresponding parameters as a unified family method and the derived parameter for the boundary values.
With the PSS option, the “Power and Expected Sample Sizes” table in Output 87.3.11 displays powers and expected sample sizes under various hypothetical references , where is the alternative reference and are values specified in the CREF= option.
Output 87.3.11: 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 | 101.5728 |
0.5000 | 0.36495 | 96.3684 |
1.0000 | 0.90000 | 76.7397 |
1.5000 | 0.99821 | 57.2590 |
Compared with the corresponding Pocock design, the O’Brien-Fleming design has a smaller maximum sample size, and smaller expected sample sizes under hypothetical references and , but larger expected sample sizes under hypothetical references and .
With the STOPPROB option, the “Expected Cumulative Stopping Probabilities” table in Output 87.3.12 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 87.3.12: 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 | |||
0.0000 | 3.975 | Reject Null | 0.00005 | 0.00422 | 0.02091 | 0.05000 |
0.5000 | 3.771 | Reject Null | 0.00062 | 0.04430 | 0.18392 | 0.36515 |
1.0000 | 3.003 | Reject Null | 0.00798 | 0.29296 | 0.69603 | 0.90000 |
1.5000 | 2.241 | Reject Null | 0.05584 | 0.73031 | 0.97315 | 0.99821 |
Compared with the corresponding Pocock design, the O’Brien-Fleming design has smaller stopping probabilities in early stages under each hypothetical reference.
The “Boundary Information” table in Output 87.3.13 displays the boundary values for the design that uses the O’Brien-Fleming method. Compared with the Pocock method, the standardized statistics boundary values derived from the O’Brien-Fleming method in absolute values are larger in early stages and smaller in later stages. This makes the O’Brien-Fleming design less likely to reject the null hypothesis in early stages than the Pocock design. With the derived parameter for the boundary, the boundaries at stage j are computed as , .
Output 87.3.13: Boundary Information
Boundary Information (Standardized Z Scale) Null Reference = 0 |
|||||||
---|---|---|---|---|---|---|---|
_Stage_ | Alternative | Boundary Values | |||||
Information Level | Reference | Lower | Upper | ||||
Proportion | Actual | N | Lower | Upper | Alpha | Alpha | |
1 | 0.2500 | 16.7817 | 48.33131 | -1.63862 | 1.63862 | -4.04859 | 4.04859 |
2 | 0.5000 | 33.56341 | 96.66262 | -2.31736 | 2.31736 | -2.86278 | 2.86278 |
3 | 0.7500 | 50.34511 | 144.9939 | -2.83817 | 2.83817 | -2.33745 | 2.33745 |
4 | 1.0000 | 67.12682 | 193.3252 | -3.27724 | 3.27724 | -2.02429 | 2.02429 |
The “Error Spending Information” in Output 87.3.14 displays cumulative error spending at each stage for each boundary. With smaller spending in early stages for the O’Brien-Fleming method, it also indicates that the O’Brien-Fleming design is less likely to reject the null hypothesis in early stages than the Pocock design.
Output 87.3.14: Error Spending Information
Error Spending Information | |||||
---|---|---|---|---|---|
_Stage_ | Information Level |
Cumulative Error Spending | |||
Lower | Upper | ||||
Proportion | Alpha | Beta | Beta | Alpha | |
1 | 0.2500 | 0.00003 | 0.00000 | 0.00000 | 0.00003 |
2 | 0.5000 | 0.00211 | 0.00000 | 0.00000 | 0.00211 |
3 | 0.7500 | 0.01046 | 0.00000 | 0.00000 | 0.01046 |
4 | 1.0000 | 0.02500 | 0.10000 | 0.10000 | 0.02500 |
The “Sample Size Summary” table in Output 87.3.15 displays the specified parameters for the sample size computation of the two-sample test for mean difference.
Output 87.3.15: Sample Size Summary
Sample Size Summary | |
---|---|
Test | Two-Sample Means |
Mean Difference | 0.4 |
Standard Deviation | 0.8 |
Max Sample Size | 193.3252 |
Expected Sample Size (Null Ref) | 192.1081 |
Expected Sample Size (Alt Ref) | 145.1404 |
Weight (Group A) | 2 |
Weight (Group B) | 1 |
The “Sample Sizes (N)” table in Output 87.3.16 displays the derived sample sizes at each stage, in both fractional and integer numbers. With the fixed-sample information ratio 1.0222, the required sample sizes in fractional numbers are derived by multiplying 1.0222 by the corresponding sample sizes in the fixed-sample design.
Output 87.3.16: Derived Sample Sizes
Sample Sizes (N) Two-Sample Z Test for Mean Difference |
||||||||
---|---|---|---|---|---|---|---|---|
_Stage_ | Fractional N | Ceiling N | ||||||
N | N(Grp 1) | N(Grp 2) | Information | N | N(Grp 1) | N(Grp 2) | Information | |
1 | 48.33 | 32.22 | 16.11 | 16.7817 | 50 | 33 | 17 | 17.5313 |
2 | 96.66 | 64.44 | 32.22 | 33.5634 | 98 | 65 | 33 | 34.1996 |
3 | 144.99 | 96.66 | 48.33 | 50.3451 | 146 | 97 | 49 | 50.8669 |
4 | 193.33 | 128.88 | 64.44 | 67.1268 | 194 | 129 | 65 | 67.5338 |