Review Articles

Log-rank and stratified log-rank tests

Ting Ye ,

Department of Biostatistics, University of Washington, Seattle, WA, USA

Jun Shao ,

School of Statistics, East China Normal University, Shanghai, People's Republic of China; Department of Statistics, University of Wisconsin, Madison, WI, USA

Yanyao Yi

Global Statistical Sciences, Eli Lilly and Company, Indianapolis, IN, USA

Pages | Received 03 Feb. 2023, Accepted 20 Sep. 2023, Published online: 05 Oct. 2023,
  • Abstract
  • Full Article
  • References
  • Citations

In randomized clinical trials with right-censored time-to-event outcomes, the popular log-rank test without adjusting for baseline covariates is asymptotically valid for treatment effect under simple randomization of treatments but is too conservative under covariate-adaptive randomization. The stratified log-rank test, which adjusts baseline covariates in the test procedure by stratification, is asymptotically valid regardless of what treatment randomization is applied. In the literature, however, under simple randomization there is no affirmative conclusion about whether the stratified log-rank test is asymptotically more powerful than the unstratified log-rank test. In this article we show when the stratified and unstratified log-rank tests aim for the same null hypothesis and that, under simple randomization, the stratified log-rank test is asymptotically more powerful than the unstratified log-rank test in the region of alternative hypothesis that is specified by a Cox proportional hazards model. We also provide some discussion about why we do not have an affirmative conclusion in general.


  • Antognini, A. B., & Zagoraiou, M. (2015). On the almost sure convergence of adaptive allocation procedures. Bernoulli Journal21(2), 881–908.  
  • Cheng, S., Wei, L., & Ying, Z. (1995). Analysis of transformation models with censored data. Biometrika82(4), 835–845. 
  • Ciolino, J. D., Palac, H. L., Yang, A., Vaca, M., & Belli, H. M. (2019). Ideal vs. real: A systematic review on handling covariates in randomized controlled trials. BMC Medical Research Methodology19(1), 136.
  • DiRienzo, A. G., & Lagakos, S. W. (2002). Effects of model misspecification on tests of no randomized treatment effect arising from cox's proportional hazards model. Journal of the Royal Statistical Society: Series B (Statistical Methodology)63(4), 745–757. 
  • EMA (2015). Guideline on adjustment for baseline covariates in clinical trials. Committee for Medicinal Products for Human Use, European Medicines Agency (EMA).  
  • FDA (2021). Adjusting for covariates in randomized clinical trials for drugs and biological products. Draft Guidance for Industry. Center for Drug Evaluation and Research and Center for Biologics Evaluation and Research, Food and Drug Administration (FDA), U.S. Department of Health and Human Services. May 2021. 
  • ICH E9 (1998). Statistical principles for clinical trials E9. International Council for Harmonisation (ICH). 
  • Kalbfleisch, J. D., & Prentice, R. L. (2011). The statistical analysis of failure time data. Wiley. 
  • Kong, F. H., & Slud, E. (1997). Robust covariate-adjusted logrank tests. Biometrika84(4), 847–862. 
  • Lin, D. Y., & Wei, L. J. (1989). The robust inference for the cox proportional hazards model. Journal of the American Statistical Association84(408), 1074–1078.  
  • Mantel, N. (1966). Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer Chemotherapy Reports50(3), 163–170. 
  • Peto, R., Pike, M. C., Armitage, P., Breslow, N. E., Cox, D. R., Howard, S. V., Mantel, N., McPherson, K., Peto, J., & Smith, P. G. (1976). Design and analysis of randomized clinical trials requiring prolonged observation of each patient. i. introduction and design. British Journal of Cancer34(6), 585–612.  
  • Pocock, S. J., & Simon, R. (1975). Sequential treatment assignment with balancing for prognostic factors in the controlled clinical trial. Biometrics31(1), 103–115.  
  • Schulz, K. F., & Grimes, D. A. (2002). Generation of allocation sequences in randomised trials: Chance, not choice. The Lancet359(9305), 515–519.  
  • Shao, J. (2021). Inference for covariate-adaptive randomization: Aspects of methodology and theory (with discussions). Statistical Theory and Related Fields5(3), 172–186.  
  • Taves, D. R. (1974). Minimization: A new method of assigning patients to treatment and control groups. Clinical Pharmacology and Therapeutics15(5), 443–453. 
  • Taves, D. R. (2010). The use of minimization in clinical trials. Contemporary Clinical Trials31(2), 180–184. 
  • Ye, T., & Shao, J. (2020). Robust tests for treatment effect in survival analysis under covariate-adaptive randomization. Journal of the Royal Statistical Society: Series B (Statistical Methodology)82(5), 1301–1323. 
  • Ye, T., Shao, J., Yi, Y., & Zhao, Q. (2022). Toward better practice of covariate adjustment in analyzing randomized clinical trials. Journal of the American Statistical Association.  
  • Zelen, M. (1974). The randomization and stratification of patients to clinical trials. Journal of Chronic Diseases27(7-8), 365–375.  

To cite this article: Ting Ye, Jun Shao & Yanyao Yi (05 Oct 2023): Log-rank and stratified log-rank tests, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2023.2263720 To link to this article: