This example illustrates sample size computation for survival data when the accrual is not uniform and when the data contain individual loss to follow-up.
Following the derivations in Example 87.14, the following statements invoke the SEQDESIGN procedure for sample size computation with exponential accrual and individual loss to follow-up:
proc seqdesign; OBrienFleming: design nstages=4 method=obf ; samplesize model=twosamplesurvival ( nullhazard=0.03466 hazard=0.01733 accrual=exp(parm=-0.1) loss=exp(hazard=0.05) acctime=20); run;
The SAMPLESIZE statement derives required sample sizes for the test. The MODEL=TWOSAMPLESURVIVAL option specifies a log-rank test to compare two survival distributions for the treatment effect. The NULLHAZARD=0.03466 option specifies null hazard rates for the two groups under the null hypothesis, and the HAZARD=0.01733 option specifies the hazard rate for the first group under the alternative hypothesis.
The ACCTIME= option specifies the accrual time , and the ACCRUAL=EXP(POWER=–0.1) option specifies that the individual accrual be truncated exponential with a scaled power parameter . That is, the truncated exponential accrual has a power parameter in the accrual time . The LOSS=EXP(HAZARD=0.05) option specifies an exponential loss function with a hazard rate . For a detailed derivation of these required sample sizes, see the section Test for Two Survival Distributions with a Log-Rank Test.
The “Design Information” table in Output 87.15.1 displays design specifications and the derived maximum information.
Output 87.15.1: 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 | 0.693147 |
Number of Stages | 4 |
Alpha | 0.05 |
Beta | 0.1 |
Power | 0.9 |
Max Information (Percent of Fixed Sample) | 102.2163 |
Max Information | 22.35451 |
Null Ref ASN (Percent of Fixed Sample) | 101.5728 |
Alt Ref ASN (Percent of Fixed Sample) | 76.7397 |
The “Boundary Information” table in Output 87.15.2 displays information level, alternative reference, and boundary values.
Output 87.15.2: Boundary Information
Boundary Information (Standardized Z Scale) Null Reference = 0 |
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---|---|---|---|---|---|---|---|
_Stage_ | Alternative | Boundary Values | |||||
Information Level | Reference | Lower | Upper | ||||
Proportion | Actual | Events | Lower | Upper | Alpha | Alpha | |
1 | 0.2500 | 5.588627 | 22.35451 | -1.63862 | 1.63862 | -4.04859 | 4.04859 |
2 | 0.5000 | 11.17725 | 44.70902 | -2.31736 | 2.31736 | -2.86278 | 2.86278 |
3 | 0.7500 | 16.76588 | 67.06352 | -2.83817 | 2.83817 | -2.33745 | 2.33745 |
4 | 1.0000 | 22.35451 | 89.41803 | -3.27724 | 3.27724 | -2.02429 | 2.02429 |
The “Sample Size Summary” table in Output 87.15.3 displays parameters for the sample size computation. Because the ACCNOBS= option is not specified along with the ACCTIME= option, the minimum and maximum sample sizes are derived.
Output 87.15.3: Sample Size Summary
Sample Size Summary | |
---|---|
Test | Two-Sample Survival |
Null Hazard Rate | 0.03466 |
Hazard Rate (Group A) | 0.01733 |
Hazard Rate (Group B) | 0.03466 |
Hazard Ratio | 0.5 |
log(Hazard Ratio) | -0.69315 |
Reference Hazards | Alt Ref |
Accrual | Truncated Exponential with Losses |
Scaled Exponential Parameter | -0.1 |
Loss Hazard Rate | 0.05 |
Accrual Time | 20 |
Min Accrual Sample Size | 269 |
Max Accrual Sample Size | 553 |
Max Number of Events | 89.41803 |
With the minimum and maximum sample sizes of 269 and 553, respectively, the ACCNOBS=360 option specifies an accrual sample size of 360 for the trial.
proc seqdesign; OBrienFleming: design nstages=4 method=obf ; samplesize model=twosamplesurvival ( nullhazard=0.03466 hazard=0.01733 accrual=exp(parm=-0.1) loss=exp(hazard=0.05) acctime=20 accnobs=360 ); run;
With the specified accrual sample size and accrual time, the “Sample Size Summary” table in Output 87.15.4 displays the follow-up time and maximum sample size with the specified accrual time. When you specify the CEILING=TIME option (which is the default), the required times at the stages are rounded up to integers for additional statistics, and the table also displays the follow-up time and total time that correspond to these ceiling times at the stages.
Output 87.15.4: Sample Size Summary
Sample Size Summary | |
---|---|
Test | Two-Sample Survival |
Null Hazard Rate | 0.03466 |
Hazard Rate (Group A) | 0.01733 |
Hazard Rate (Group B) | 0.03466 |
Hazard Ratio | 0.5 |
log(Hazard Ratio) | -0.69315 |
Reference Hazards | Alt Ref |
Accrual | Truncated Exponential with Losses |
Scaled Exponential Parameter | -0.1 |
Loss Hazard Rate | 0.05 |
Accrual Time | 20 |
Follow-up Time | 9.100306 |
Total Time | 29.10031 |
Max Number of Events | 89.41803 |
Max Sample Size | 360 |
Expected Sample Size (Null Ref) | 359.7605 |
Expected Sample Size (Alt Ref) | 342.9204 |
Follow-up Time (Ceiling Time) | 10 |
Total Time (Ceiling Time) | 30 |
The “Number of Events (D) and Sample Sizes (N)” table in Output 87.15.5 displays the required time at each stage, in both fractional and integer numbers. The derived times under the heading “Fractional Time” are not integers. These times are rounded up to integers under the heading “Ceiling Time.” The table also displays the numbers of events and sample sizes at each stage.
Output 87.15.5: Number of Events and Sample Sizes
Numbers of Events (D) and Sample Sizes (N) Two-Sample Log-Rank Test |
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_Stage_ | Fractional Time | Ceiling Time | ||||||||||||||
D | D(Grp 1) | D(Grp 2) | Time | N | N(Grp 1) | N(Grp 2) | Information | D | D(Grp 1) | D(Grp 2) | Time | N | N(Grp 1) | N(Grp 2) | Information | |
1 | 22.35 | 7.74 | 14.62 | 11.4005 | 200.79 | 100.39 | 100.39 | 5.5886 | 24.46 | 8.48 | 15.98 | 12 | 211.67 | 105.83 | 105.83 | 6.1162 |
2 | 44.71 | 15.69 | 29.01 | 17.0454 | 304.52 | 152.26 | 152.26 | 11.1773 | 48.97 | 17.23 | 31.75 | 18 | 322.36 | 161.18 | 161.18 | 12.2436 |
3 | 67.06 | 23.81 | 43.25 | 21.9812 | 360.00 | 180.00 | 180.00 | 16.7659 | 67.14 | 23.84 | 43.30 | 22 | 360.00 | 180.00 | 180.00 | 16.7851 |
4 | 89.42 | 32.39 | 57.03 | 29.1003 | 360.00 | 180.00 | 180.00 | 22.3545 | 91.46 | 33.21 | 58.25 | 30 | 360.00 | 180.00 | 180.00 | 22.8649 |
Alternatively, you can specify the CEILING=N option in the SAMPLESIZE statement as follows to derive additional sample size information with ceiling sample sizes at each stage:
proc seqdesign; OBrienFleming: design nstages=4 method=obf ; samplesize model=twosamplesurvival ( nullhazard=0.03466 hazard=0.01733 accrual=exp(parm=-0.1) loss=exp(hazard=0.05) acctime=20 accnobs=360 ceiling=n); run;
When you specify CEILING=N, the required sample sizes at the stages are rounded up to integers for additional statistics, and the “Sample Size Summary” table in Output 87.15.6 also displays the accrual time, follow-up time, and maximum sample size that correspond to these ceiling sample sizes.
Output 87.15.6: Sample Size Summary
Sample Size Summary | |
---|---|
Test | Two-Sample Survival |
Null Hazard Rate | 0.03466 |
Hazard Rate (Group A) | 0.01733 |
Hazard Rate (Group B) | 0.03466 |
Hazard Ratio | 0.5 |
log(Hazard Ratio) | -0.69315 |
Reference Hazards | Alt Ref |
Accrual | Truncated Exponential with Losses |
Scaled Exponential Parameter | -0.1 |
Loss Hazard Rate | 0.05 |
Accrual Time | 20 |
Follow-up Time | 9.100306 |
Total Time | 29.10031 |
Max Number of Events | 89.41803 |
Max Sample Size | 360 |
Expected Sample Size (Null Ref) | 359.7605 |
Expected Sample Size (Alt Ref) | 342.9204 |
Accrual Time (Ceiling N) | 20 |
Follow-up Time (Ceiling N) | 9.100306 |
Max Sample Size (Ceiling N) | 360 |
The “Number of Events (D) and Sample Sizes (N)” table in Output 87.15.7 displays the required sample sizes at each stage, in both fractional and integer numbers. The derived numbers under the heading “Fractional N” are not integers. These numbers are rounded up to integers to match the sample size allocation under the heading “Ceiling N (Matched).” The table also displays the numbers of events and time at each stage.
Output 87.15.7: Number of Events and Sample Sizes
Numbers of Events (D) and Sample Sizes (N) Two-Sample Log-Rank Test |
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---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
_Stage_ | Fractional N | Ceiling N (Matched) | ||||||||||||||
D | D(Grp 1) | D(Grp 2) | Time | N | N(Grp 1) | N(Grp 2) | Information | D | D(Grp 1) | D(Grp 2) | Time | N | N(Grp 1) | N(Grp 2) | Information | |
1 | 22.35 | 7.74 | 14.62 | 11.4005 | 200.79 | 100.39 | 100.39 | 5.5886 | 22.59 | 7.82 | 14.77 | 11.4674 | 202 | 101 | 101 | 5.6466 |
2 | 44.71 | 15.69 | 29.01 | 17.0454 | 304.52 | 152.26 | 152.26 | 11.1773 | 45.06 | 15.82 | 29.24 | 17.1246 | 306 | 153 | 153 | 11.2645 |
3 | 67.06 | 23.81 | 43.25 | 21.9812 | 360.00 | 180.00 | 180.00 | 16.7659 | 67.06 | 23.81 | 43.25 | 21.9812 | 360 | 180 | 180 | 16.7659 |
4 | 89.42 | 32.39 | 57.03 | 29.1003 | 360.00 | 180.00 | 180.00 | 22.3545 | 89.42 | 32.39 | 57.03 | 29.1003 | 360 | 180 | 180 | 22.3545 |