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## Power analysis for survival studies

Stata has a suite of tools that provide sample-size and power calculations for survival studies that use Cox proportional-hazards regressions, log-rank tests for two groups, or parametric tests of disparity in two exponential survivor functions.

 power cox power logrank power exponential

power cox estimates required sample size (given power and effect size) or power (given sample size and effect size) or the minimal detectable effect size (given power and sample size) for models with multiple covariates.

power logrank estimates required sample size (given power and effect size) or power (given sample size and effect size) or the minimal detectable effect size (given power and sample size) for studies comparing survivor functions of two groups by using the log-rank test. Both the Freedman (1982) and the Schoenfeld (1981) methods are provided.

power exponential estimates sample size (given power and effect size) or power (given sample size and effect size) for parametric tests of the difference between hazards or log hazards of two groups under the assumption of exponential survivor functions. Both the Lachin and Foulkes (1986) and Rubinstein, Gail, and Santner (1981) methods are provided.

power allows automated production of customizable tables and have options to assist with creating graphs of power curves.

Below are several examples demonstrating some of power’s features:

#### Tabulating results

Study design
Consider a survival study comparing two treatments, a standard treatment and a new, experimental treatment. The survival probability in the control group at the end of the study is expected to be approximately 0.7. We need to estimate the sample size required to detect an increase in survival of the experimental group from 0.7 to 0.8 at the end of the study with power of 80%, 85%, and 90%, using a two-sided log-rank test at the 5% significance level. We use power logrank to obtain the required sample sizes:

. power logrank 0.7 0.8, power(0.8 0.85 0.9)

Estimated sample sizes for two-sample comparison of survivor functions
Log-rank test, Freedman method
Ho: HR = 1  versus  Ha: HR != 1

alpha   power       N      N1      N2       E   delta  hratio      s1

.05      .8     592     296     296     148   .6256   .6256      .7

.05     .85     678     339     339     170   .6256   .6256      .7

.05      .9     794     397     397     199   .6256   .6256      .7

s2    Pr_E

.8     .25

.8     .25

.8     .25



Results
The table reports estimates of the required number of events and sample sizes in the study for three powers given other study parameters. The last row of the table indicates that we need 200 events to be observed in the study (and a sample size of 794 to observe the 200 events in the study) for our log-rank test to have a power of 90%. The increase in survival from 0.7 to 0.8 is equivalent to a hazard ratio of .626 of the experimental to the control group, as shown in the second-to-last column in the table.

#### Producing power curves

If our sample size is predetermined, we may want to find out the smallest effect size or increase in survival expressed as a hazard ratio that can be detected with a given level of power. We can use power to produce power curves as a function of the hazard ratio for several sample sizes.

Suppose that we want to produce power curves as a function of the effect size for sample sizes of 100, 250, and 500 for the study we considered in the first example.

. power logrank 0.7, n(100 250 500) hratio(0.1(0.01)0.9) saving(mypower)

Estimated power for two-sample comparison of survivor functions
Log-rank test, Freedman method
Ho: HR = 1  versus  Ha: HR != 1

alpha   power       N      N1      N2       E   delta  hratio      s1

.05   .9176     100      50      50      17      .1      .1      .7

.05   .9096     100      50      50      17     .11     .11      .7

.05   .9012     100      50      50      18     .12     .12      .7

.05   .8924     100      50      50      18     .13     .13      .7

.05    .883     100      50      50      18     .14     .14      .7

...

.05   .1432     500     250     250     142     .86     .86      .7

.05   .1287     500     250     250     142     .87     .87      .7

.05   .1154     500     250     250     143     .88     .88      .7

.05   .1031     500     250     250     143     .89     .89      .7

.05   .0919     500     250     250     144      .9      .9      .7

s2     Pr_E

.965    .1675

.9615    .1692

.9581    .1709

.9547    .1727

.9513    .1744

...

.7358    .2821

.7332    .2834

.7306    .2847

.728     .286

.7254    .2873

file mypower.dta saved


Powers are computed for each combination of sample-size and hazard-ratio values.

#### Producing customized tables

power also allows you to build your own customized tables. You can choose what to display in a table from a list of results available.

For example, if you prefer to see the probability of a type II error rather than power, and the proportion of subjects in the control group rather than group-sample sizes, reported by default, you can type

power logrank 0.7 0.8, power(0.8 0.85 0.9) table(alpha beta N E hratio s1 s2)

Estimated sample sizes for two-sample comparison of survivor functions
Log-rank test, Freedman method
Ho: HR = 1  versus  Ha: HR != 1

alpha    beta       N       E  hratio      s1      s2

.05      .2     592     148   .6256      .7      .8

.05     .15     678     170   .6256      .7      .8

.05      .1     794     199   .6256      .7      .8



to obtain the table with requested columns displayed in the same order you specified.

Of course, all the above can be done using dialog boxes instead of the command line.

### References

Freedman, L. S. 1982.
Tables of the number of patients required in clinical trials using the logrank test. Statistics in Medicine 1: 121–129.
Lachin, J. M., and M. A. Foulkes. 1986.
Evaluation of sample size and power for analysis of survival with allowance for nonuniform patient entry, losses to follow-up, noncompliance, and stratification. Biometrics 42: 507–519.
Rubinstein, L. V., M. H. Gail, and T. J. Santner. 1981.
Planning the duration of a comparative clinical trial with loss to follow-up and a period of continued observation. Journal of Chronic Diseases 34: 469–479.
Schoenfeld, D. 1981.
The asymptotic properties of nonparametric tests for comparing survival distributions. Biometrika 68: 316–319.