On Comparing Survival Curves with Right-Censored Data According to the Events Occur at the Beginning, in the Middle and at the End of Study Period


  • Pinar Gunel Karadeniz Department of Biostatistics Gaziantep, SANKO University, Faculty of Medicine, 27090, Turkey
  • Ilker Ercan Department of Biostatistics Gorukle Campus Bursa, Uludag University, Faculty of Medicine, Turkey




Survival analysis, survival curves, comparison of survival curves, right censored observations.


In clinical practice the event of interest does not always occur equally across the study time period. Depending on the disease being investigated, the event that is of interest can occur intensively in different periods of the follow-up time. In such cases, choosing the correct survival comparison test has importance. This study aims to examine and discuss the results of survival comparison tests under some certain circumstances. A simulation study was conducted. We discussed the result of different tests such as Logrank, Gehan-Wilcoxon, Tarone-Ware, Peto-Peto, Modified Peto-Peto tests and tests belonging to Fleming-Harrington test family with (p, q) values; (1, 0), (0.5, 0.5), (1, 1), (0, 1) ve (0.5, 2) by means of Type I error rate that obtained from simulation study, when the event of interest occurred intensively at the beginning of the study, in the middle of the study and at the end of the study time period. As a result of simulation study, Type I error rate of tests is generally lower or higher than the nominal value. In the light of the results, it is proposed to re-examine the tests for cases where events are observed intensively at the beginning, middle and late periods, to carry out new simulation studies and to develop new tests if necessary.


Fisher LD, Belle GV. Biostatistics, a methodology for the health sciences, John Wiley & Sons Inc, New York, page 1993; 786-807.

Kaplan EL, Meier P. Nonparametric estimation from incomplete observations. Journal of the American Statistical Association 1958; 53(282): 457-481. https://doi.org/10.1080/01621459.1958.10501452 DOI: https://doi.org/10.1080/01621459.1958.10501452

Mantel N, Haenszel W. Statistical aspects of the analysis of data from retrospective studies of disease. Journal of National Cancer Institute 1959; 22(4): 719-748.

Mantel N. Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer Chemotherapy Reports, 1966; 50(3): 163-170.

Gehan EA. A generalized Wilcoxon test for comparing arbi-trarily single-censored samples. Biometrika 1965; 52: 203-223. https://doi.org/10.1093/biomet/52.1-2.203 DOI: https://doi.org/10.2307/2333825

Peto R, Peto J. Asymptotically efficient rank invariant test procedures. Journal of the Royal Statistical Society 1972; 135(2): 185-207. https://doi.org/10.2307/2344317 DOI: https://doi.org/10.2307/2344317

Tarone RE, Ware J. On distribution-free tests for equality of survival distributions. Biometrika 1977; 64(1): 156-160. https://doi.org/10.1093/biomet/64.1.156 DOI: https://doi.org/10.1093/biomet/64.1.156

Fleming TR, Harrington DP. A class of hypothesis tests for one and two samples censored survival data. Communi-cations in Statistics-Theory and Methods 1981; 10(8): 763-794. https://doi.org/10.1080/03610928108828073 DOI: https://doi.org/10.1080/03610928108828073

Harrington DP, Fleming TR. A class of rank test procedures for censored survival data. Biometrika 1982; 69(3): 553-566. https://doi.org/10.1093/biomet/69.3.553 DOI: https://doi.org/10.1093/biomet/69.3.553

Lee JW. Some versatile tests based on the simultaneous use of weighted log-rank statistics. Biometrics 1996; 52(2): 721-725. https://doi.org/10.2307/2532911 DOI: https://doi.org/10.2307/2532911

Martinez RLMC, Naranjo JD. A pretest for choosing between logrank and Wilcoxon tests in the two-sample problem. Metron: International Journal of Statistics 2010; 68(2): 111-125. https://doi.org/10.1007/BF03263529 DOI: https://doi.org/10.1007/BF03263529

Graves TS, Pazdan JL. A permutation test analogue to Tarone's test for trend in survival analysis. Journal of Statistical Computation and Simulation 1995; 53(1-2): 79-89. https://doi.org/10.1080/00949659508811697 DOI: https://doi.org/10.1080/00949659508811697

Leton E, Zuluaga P. Relationships among tests for censored data. Biometrical Journal 2005; 47(3): 377-387. https://doi.org/10.1002/bimj.200410115 DOI: https://doi.org/10.1002/bimj.200410115

Karadeniz PG, Ercan I. Examining Tests for Comparing of Survival Curves with Right Censored Data. Statistics in Transition 2017; 18(2): 311-328. https://doi.org/10.21307/stattrans-2016-072 DOI: https://doi.org/10.21307/stattrans-2016-072

Latta RB. A monte carlo study of some two-sample rank tests with censored data. Journal of American Statistical Association 1981; 76(375): 713-719. https://doi.org/10.1080/01621459.1981.10477710 DOI: https://doi.org/10.1080/01621459.1981.10477710

Hintze JL. NCSS user guide V tabulation, item analysis, proportions, diagnostic tests, and survival / reliability, Published by NCSS, Kaysville, Utah, 2007.

Oller R, Gomez G. A generalized Fleming and Harrington's class of tests for interval-censored data. The Canadian Journal of Statistics 2012; 40(3): 501-516. https://doi.org/10.1002/cjs.11139 DOI: https://doi.org/10.1002/cjs.11139

R Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2013. URL http://www.R-project.org/.

Hintze, J. (2007). NCSS 2007. NCSS, LLC. Kaysville, Utah, USA, 2007. www.ncss.com

WinAutomation Software Solutions, Softotomotive Ltd, Athens, 2014.

Lee ET, Wang JW. Statistical Methods for Survival Data Analysis. New Jersey: John Wiley & Sons Inc, 2003. DOI: https://doi.org/10.1002/0471458546

Fleming TR, Harrington DP, O'Sullivan M. Supremum versions of the log-rank and generalized Wilcoxon statistics. Journal of the American Statistical Association 1987; 82(397): 312-320. https://doi.org/10.1080/01621459.1987.10478435 DOI: https://doi.org/10.1080/01621459.1987.10478435

Buyske S, Fagerstrom R, Ying Z. A class of weighted log-rank tests for survival data when the event is rare. Journal of the American Statistical Association 2000; 95(449): 249-258. https://doi.org/10.1080/01621459.2000.10473918 DOI: https://doi.org/10.1080/01621459.2000.10473918

Pepe MS, Fleming TR. Weighted Kaplan-Meier statistics: a class of distance tests for censored survival data. Biometrics 1989; 45(2): 497-507. https://doi.org/10.2307/2531492 DOI: https://doi.org/10.2307/2531492

Kleinbaum DG, Klein M. Survival Analysis a Self-Learning Text. New York: Springer, 2005. DOI: https://doi.org/10.1007/0-387-29150-4

Klein JP, Rizzo JD, Zhang MJ, Keiding N. Statistical methods for the analysis and presentation of the results of bone marrow transplants. Part I: Unadjusted analysis. Bone Marrow Transplantation 2001; 28(10): 909-915. https://doi.org/10.1038/sj.bmt.1703260 DOI: https://doi.org/10.1038/sj.bmt.1703260

Gomez G, Calle ML, Oller R, Langohr K. Tutorial on methods for interval-censored data and their implementation in R. Statistical Modelling 2009; 9(4): 259-297. https://doi.org/10.1177/1471082X0900900402 DOI: https://doi.org/10.1177/1471082X0900900402




How to Cite

Karadeniz, P. G., & Ercan, I. (2018). On Comparing Survival Curves with Right-Censored Data According to the Events Occur at the Beginning, in the Middle and at the End of Study Period . International Journal of Statistics in Medical Research, 7(4), 117–128. https://doi.org/10.6000/1929-6029.2018.07.04.2



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