### Example 87.14 Computing Sample Size for Survival Data That Have Uniform Accrual

This example illustrates sample size computation for survival data when the accrual is uniform and when the data do not contain individual loss to follow-up.

Suppose that a clinic is conducting a study of the effect of a new cancer treatment. The study consists of exposing mice to a carcinogen and randomly assigning them to either the control group or the treatment group. The event of interest is death from cancer induced by the carcinogen, and the response is the time from randomization to death.

Following the derivations in the section Test for Two Survival Distributions with a Log-Rank Test, the hypothesis with the alternative hypothesis is used, where is the hazard ratio between the treatment group and the control group.

Also suppose that from past experience, the median survival time for the control group is weeks, and the study wants to detect a median survival time, weeks, with an 80% power in the trial. Assuming exponential survival functions for the two groups, the hazard rates can be computed from

where j = 0, 1.

Thus, the hazard rates are and under the null and alternative hypotheses, respectively.

The following statements invoke the SEQDESIGN procedure and specify the SAMPLESIZE statement to derive required sample sizes for a log-rank test that compares two survival distributions for the treatment effect (Jennison and Turnbull 2000, pp. 77–79; Whitehead 1997, pp. 36–39):

proc seqdesign;
ErrorSpend: design nstages=4 method=errfuncobf
;
samplesize model=twosamplesurvival
( nullhazard=0.03466 hazard=0.01733
accrual=uniform accrate=15);
run;


In the SAMPLESIZE statement, 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 ACCRUAL= option specifies the method for individual accrual. The ACCRUAL=UNIFORM option (which is the default) specifies that the individual accrual is uniform with a constant accrual rate, and the ACCRATE= option specifies the accrual rate.

You do not have to specify the alternative reference explicitly for the sample size computation in the SEQDESIGN procedure. For the specified null and alternative hazards, and , the hazard ratio , and the alternative reference is

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 87.14.1 displays design specifications and four derived statistics: the actual maximum information, the maximum information, the average sample number under the null hypothesis (Null Ref ASN), and the average sample number under the alternative hypothesis (Alt Ref ASN).

The SEQDESIGN Procedure
Design: ErrorSpend

Statistic Distribution Normal
Boundary Scale Standardized Z
Alternative Hypothesis Two-Sided
Early Stop Reject Null
Method Error Spending
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) 101.8279
Max Information 22.26962
Null Ref ASN (Percent of Fixed Sample) 101.2586
Alt Ref ASN (Percent of Fixed Sample) 77.73131

The Boundary Information table in Output 87.14.2 displays the information level, including the proportion, actual level, and corresponding number of events at each stage. The table also displays the lower and upper alternative references, and the lower and upper boundary values at each stage.

Output 87.14.2: Boundary Information

Boundary Information (Standardized Z Scale)
Null Reference = 0
_Stage_   Alternative Boundary Values
Information Level Reference Lower Upper
Proportion Actual Events Lower Upper Alpha Alpha
1 0.2500 5.567405 22.26962 -1.63550 1.63550 -4.33263 4.33263
2 0.5000 11.13481 44.53924 -2.31295 2.31295 -2.96333 2.96333
3 0.7500 16.70221 66.80886 -2.83278 2.83278 -2.35902 2.35902
4 1.0000 22.26962 89.07847 -3.27101 3.27101 -2.01409 2.01409

The Sample Size Summary table in Output 87.14.3 displays parameters for the sample size computation. Because the ACCTIME= option is not specified along with the ACCRATE= option, the minimum and maximum accrual times are derived.

Output 87.14.3: 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 Uniform
Accrual Rate 15
Min Accrual Time 5.938565
Min Sample Size 89.07847
Max Accrual Time 23.78469
Max Sample Size 356.7704
Max Number of Events 89.07847

With the minimum and maximum accrual times of 5.9386 and 23.7847, respectively, the ACCTIME=18 option specifies an accrual time of 18 for the trial.

The following statements invoke the SEQDESIGN procedure and specify the ACCTIME=18 option in the SAMPLESIZE statement to derive required sample sizes:

proc seqdesign;
ErrorSpend: design nstages=4 method=errfuncobf
;
samplesize model=twosamplesurvival
( nullhazard=0.03466 hazard=0.01733
accrual=uniform accrate=15 acctime=18
ceiling=time);
run;


The Sample Size Summary table in Output 87.14.4 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 87.14.4: Sample Size Summary

The SEQDESIGN Procedure
Design: ErrorSpend

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 Uniform
Accrual Rate 15
Accrual Time 18
Follow-up Time 7.133226
Total Time 25.13323
Max Number of Events 89.07847
Max Sample Size 270
Expected Sample Size (Null Ref) 269.9206
Expected Sample Size (Alt Ref) 263.1141
Follow-up Time (Ceiling Time) 8
Total Time (Ceiling Time) 26

The Number of Events (D) and Sample Sizes (N) table in Output 87.14.5 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 87.14.5: Number of Events and Sample Sizes

Numbers of Events (D) and Sample Sizes (N)
Two-Sample Log-Rank Test
_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 22.27 7.73 14.54 11.2631 168.95 84.47 84.47 5.5674 25.11 8.74 16.37 12 180.00 90.00 90.00 6.2781
2 44.54 15.73 28.81 16.2875 244.31 122.16 122.16 11.1348 48.22 17.07 31.16 17 255.00 127.50 127.50 12.0552
3 66.81 23.93 42.88 20.4926 270.00 135.00 135.00 16.7022 69.39 24.90 44.48 21 270.00 135.00 135.00 17.3468
4 89.08 32.51 56.57 25.1332 270.00 135.00 135.00 22.2696 92.93 34.04 58.89 26 270.00 135.00 135.00 23.2319

Alternatively, you can specify the CEILING=N option in the SAMPLESIZE statement as follows to derive additional sample size information that includes ceiling sample sizes at each stage:

proc seqdesign;
ErrorSpend: design nstages=4 method=errfuncobf
;
samplesize model=twosamplesurvival
( nullhazard=0.03466 hazard=0.01733
accrual=uniform accrate=15 acctime=18
ceiling=n);
run;


When 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 87.14.6 also displays the accrual time, follow-up time, and maximum sample size that correspond to these ceiling sample sizes.

Output 87.14.6: Sample Size Summary

The SEQDESIGN Procedure
Design: ErrorSpend

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 Uniform
Accrual Rate 15
Accrual Time 18
Follow-up Time 7.133226
Total Time 25.13323
Max Number of Events 89.07847
Max Sample Size 270
Expected Sample Size (Null Ref) 269.9206
Expected Sample Size (Alt Ref) 263.1141
Accrual Time (Ceiling N) 18
Follow-up Time (Ceiling N) 7.133226
Max Sample Size (Ceiling N) 270

The Number of Events (D) and Sample Sizes (N) table in Output 87.14.7 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 87.14.7: Number of Events and Sample Sizes

Numbers of Events (D) and Sample Sizes (N)
Two-Sample Log-Rank Test
_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 22.27 7.73 14.54 11.2631 168.95 84.47 84.47 5.5674 22.53 7.83 14.71 11.3333 170 85 85 5.6335
2 44.54 15.73 28.81 16.2875 244.31 122.16 122.16 11.1348 45.11 15.93 29.18 16.4000 246 123 123 11.2781
3 66.81 23.93 42.88 20.4926 270.00 135.00 135.00 16.7022 66.81 23.93 42.88 20.4926 270 135 135 16.7022
4 89.08 32.51 56.57 25.1332 270.00 135.00 135.00 22.2696 89.08 32.51 56.57 25.1332 270 135 135 22.2696