Sage Journals: Discover world-class research

Abstract

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.

Keywords

Cure fraction data maturity delayed effect exponentiated-Weibull distribution local alternatives maximum combination test noninferiority trial treatment switching

Get full access to this article

View all access options for this article.

References

Schoenfeld

The asymptotic properties of nonparametric tests for comparing survival distributions. Biometrika 1981; 68: 316–319.

Freedman

LS.

Tables of the number of patients required in clinical trials using the logrank test.

Stat Med 1982; 1: 121–129.

Lakatos

Sample sizes based on the log-rank statistic in complex clinical trials. Biometrics 1988; 44: 229–241.

ho Jung

Kang

McCall

, et al. Sample size computation for two-sample noninferiority log-rank test. J Biopharmaceut Stat 2005; 15: 969–979.

ho Jung

Chow

SC.

On sample size calculation for comparing survival curves under general hypothesis testing.

J Biopharmaceut Stat 2012; 22: 485–495.

Porcher

Levy

Chevret

Sample size correction for treatment crossovers in randomized clinical trials with a survival endpoint. Control Clin Trials 2002; 23: 650–661.

Barthel

FMS

Babiker

Royston

, et al. Evaluation of sample size and power for multi-arm survival trials allowing for non-uniform accrual, non-proportional hazards, loss to follow-up and cross-over. Stat Med 2006; 25: 2521–2542.

Zhang

Quan

Power and sample size calculation for log-rank test with a time lag in treatment effect.

Stat Med 2009; 28: 864–879.

Wang

Zhang

Sample size calculation for the proportional hazards cure model.

Stat Med 2012; 31: 3959–3971.

10.

Xiong

A novel sample size formula for the weighted log-rank test under the proportional hazards cure model.

Pharmaceut Stat 2017; 16: 87–94.

11.

Zhen

Park

, et al. Designing therapeutic cancer vaccine trials with delayed treatment effect. Stat Med 2017; 36: 592–605.

12.

Park

Zhen

, et al. Designing cancer immunotherapy trials with random treatment time-lag effect. Stat Med 2018; 37: 4589–609.

13.

Xia

George

Wang

A multi-state model for designing clinical trials for testing overall survival allowing for crossover after progression.

Stat Biopharmaceut Res 2016; 8: 12–21.

14.

Liu

Chu

Rong

. Weighted log-rank test for time-to-event data in immunotherapy trials with random delayed treatment effect and cure rate. Pharmaceut Stat 2018; 17: 541–554.

15.

Luo

Mao

Chen

, et al. Design and monitoring of survival trials in complex scenarios. Stat Med 2019; 38: 192–209.

16.

Yung

Liu

Sample size and power for the weighted log-rank test and Kaplan-Meier based tests with allowance for non-proportional hazards. Biometrics 2020; 76: 939–950.

17.

Wei

Cancer immunotherapy trial design with delayed treatment effect. Pharmaceut Stat 2019; 16: 87–94.

18.

Tang

Complex survival trial design by the product integration method. (in preparation).

19.

Wei

Cancer immunotherapy trial design with cure rate and delayed treatment effect.

Stat Med 2020; 39: 698–708.

20.

Tang

Size and power estimation for the Wilcoxon-Mann-Whitney test for ordered categorical data.

Stat Med 2011; 30: 3461–3470.

21.

Tang

Zhu

An improved sample size calculation method for score tests in generalized linear models. Stat Biopharmaceut Res 2020.

22.

Lee

SH.

On the versatility of the combination of the weighted log-rank statistics. Computat Stat Data Analys 2007; 51: 6557–6564.

23.

Karrison

TG.

Versatile tests for comparing survival curves based on weighted log-rank statistics. Stata J 2016; 16: 678–690.

24.

Lin

Roychoudhury

, et al. Alternative analysis methods for time to event endpoints under nonproportional hazards: a comparative analysis. Stat Biopharmaceut Res 2020; 12: 187–198.

25.

Kantoff

Higano

Shore

, et al. Sipuleucel-T immunotherapy for castration-resistant prostate cancer. New Engl J Med 2010; 363: 411–422.

26.

Chen

Harrington

Ibrahim

JG.

Bayesian cure rate models for malignant melanoma: a case study of ECOG trial E1690. Appl Stat 2002; 51: 135–150.

27.

Gandhi

Rodríguez-Abreu

Gadgeel

, et al. Pembrolizumab plus chemotherapy in metastatic non–small-cell lung cancer. New Engl J Med 2018; 378: 2078–2092.

28.

Cox

DR.

Regression models and life-tables. J Royal Stat Soc Ser B 1972; 34: 187–220.

29.

Sasieni

Maximum weighted partial likelihood estimators for the Cox model. J Am Stat Assoc 1993; 88: 144–152.

30.

Lin

FLeon

Estimation of treatment effects in weighted log-rank tests.

Contemporary Clin Trial 2017; 8: 147–155.

31.

Fleming

Harrington

DP.

A class of hypothesis tests for one and two samples of censored survival data. Commun Stat – Theory Meth 1981; 13: 2469–2486.

32.

Fleming

Harrington

DP.

Counting processes and survival analysis. New York, NY: Wiley, 1991.

33.

Zucker

Lakatos

Weighted log rank type statistics for comparing survival curves when there is a time lag in the effectiveness of treatment. Biometrika 1990; 77: 853–864.

34.

A robust approach to sample size calculation in cancer immunotherapy trials with delayed treatment effect.

Biometrics 2018; 74: 1292–300.

35.

US-FDA. Guidance for industry non-inferiority clinical trials. 2010.

36.

Com-Nougue

Rodary

Patte

How to establish equivalence when data are censored: a randomized trial of treatments for B non-Hodgkin lymphoma.

Stat Med 1993; 12: 1353–64.

37.

US-FDA. Guidance for industry: Scientific considerations in demonstrating biosimilarity to a reference product. 2015.

38.

Abel

Jensen

Karapanagiotou-Schenkel

, et al. Some issues of sample size calculation for time-to-event endpoints using the Freedman and Schoenfeld formulas. J Biopharmaceut Stat 2015; 25: 1285–311.

39.

Hasegawa

Sample size determination for the weighted log-rank test with the Fleming-Harrington class of weights in cancer vaccine studies.

Pharmaceut Stat 2014; 13: 128–135.

40.

Lakatos

Designing complex group sequential survival trials.

Stat Med 2002; 21: 1969–1989.

41.

Hasegawa

Group sequential monitoring based on the weighted log-rank test statistic with the Fleming-Harrington class of weights in cancer vaccine studies.

Pharmaceut Stat 2016; 15: 412–419.

42.

Othus

Barlogie

Leblanc

, et al. Cure models as a useful statistical tool for analyzing survival. Clin Cancer Res 2012; 18: 3731–3736.

43.

Yin

Ibrahim

JG.

Cure rate models: a unified approach. Can J Stat 2005; 33: 559–570.

44.

Putter

Fiocco

Geskus

RB.

Tutorial in biostatistics: competing risks and multi-state models.

Stat Med 2007; 26: 2389–2430.

45.

Meller

Beyersmann

Rufibach

Joint modeling of progression-free and overall survival and computation of correlation measures.

Stat Med 2019; 38: 4270–4289.

46.

Lakatos

Sample size determination in clinical trials with time-dependent rates of losses and noncompliance. Control Clin Trials 1986; 7: 189–199.

47.

Pal

Ali

Woo

. Exponentiated Weibull distribution. Statistica 2006; 2: 139–147.

48.

Hsieh

FY.

Comparing sample size formulae for trials with unbalanced allocation using the logrank test.

Stat Med 1992; 11: 1091–1098.

49.

Wang

Meyerson

Tang

, et al. Statistical methods for the analysis of relapse data in MS clinical trials. J Neurol Sci 2009; 285: 206–211.

50.

Friede

Pohlmann

Schmidli

Blinded sample size reestimation in event-driven clinical trials: Methods and an application in multiple sclerosis.

Pharmaceut Stat 2019; 18: 351–365.

51.

Rubinstein

Gail

Santner

TJ.

Planning the duration of a comparative clinical trial with loss to follow-up and a period of continued observation. J Chronic Dis 1981; 34: 469–479.

52.

Lachin

Foulkes

MA.

Evaluation of sample size and power for analyses of survival with allowance for nonuniform patient entry, losses to follow-up, noncompliance, and stratification. Biometrics 1986; 42: 507–519.

53.

Power and sample size for randomized phase III survival trials under the Weibull model.

J Biopharmaceut Stat 2015; 25: 16–28.

54.

Lakatos

Gordon Lan

KK.

A comparison of sample size methods for the logrank statistic. Stat Med 1992; 44: 179–191.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.03 MB

0.00 MB

0.42 MB

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

Abstract

Keywords

Get full access to this article

References

Supplementary Material