A unified approach to power and sample size determination for log-rank tests under proportional and nonproportional hazards

Log-rank tests have been widely used to compare two survival curves in biomedical research. We describe a unified approach to power and sample size calculation for the unweighted and weighted log-rank tests in superiority, noninferiority and equivalence trials. It is suitable for both time-driven and event-driven trials. A numerical algorithm is suggested. It allows flexible specification of the patient accrual distribution, baseline hazards, and proportional or nonproportional hazards patterns, and enables efficient sample size calculation when there are a range of choices for the patient accrual pattern and trial duration. A confidence interval method is proposed for the trial duration of an event-driven trial. We point out potential issues with several popular sample size formulae. Under proportional hazards, the power of a survival trial is commonly believed to be determined by the number of observed events. The belief is roughly valid for noninferiority and equivalence trials with similar survival and censoring distributions between two groups, and for superiority trials with balanced group sizes. In unbalanced superiority trials, the power depends also on other factors such as data maturity. Surprisingly, the log-rank test usually yields slightly higher power than the Wald test from the Cox model under proportional hazards in simulations. We consider various nonproportional hazards patterns induced by delayed effects, cure fractions, and/or treatment switching. Explicit power formulae are derived for the combination test that takes the maximum of two or more weighted log-rank tests to handle uncertain nonproportional hazards patterns. Numerical examples are presented for illustration.

Comparison of survival distributions in clinical trials: A practical guidance

Background In randomized clinical trials with censored time-to-event outcomes, the logrank test is known to have substantial statistical power under the proportional hazards assumption and is widely adopted as a tool to compare two survival distributions. However, the proportional hazards assumption is impossible to validate in practice until the data are unblinded. However, the statistical analysis plan of a randomized clinical trial and in particular its primary analysis method must be pre-specified before any unblinded information may be reviewed. Purpose The purpose of this article is to guide applied biostatisticians in the prespecification of a desired primary analysis method when a treatment effect with nonproportional hazards is anticipated. While articles proposing alternate statistical tests are aplenty, to the best of our knowledge, there is no article available that attempts to simplify the choice and prespecification of a primary statistical test under specific expected patterns on nonproportional hazards. We provide such guidance by reviewing various tests proposed as more powerful alternatives to the standard logrank test under nonproportional hazards and simultaneously comparing their performance under a wide variety of nonproportional hazards scenarios to elucidate their advantages and disadvantages. Method In order to select the most preferable test for detecting specific differences between survival distributions of interest while controlling false positive rates, we review and assess the performance of weighted and adaptively weighted logrank tests, weighted and adaptively weighted Kaplan–Meier tests and versatile tests under various patterns of nonproportional hazards treatment effects through simulation. Conclusion We validate some of the claimed properties of the proposed extensions and identify tests that may be more preferable under specific expected pattern of nonproportional hazards when such knowledge is available. We show that versatile tests, while achieving robustness to departures from proportional hazards, may lose interpretation of directionality (superiority or inferiority) and can only be seen to test departures from equality. Detailed summary and discussion of the performance of each test in terms of type I error rate and power are provided to formulate specific guidance about their applicability and use.