Survival Functions in the Presence of Several Events and Competing Risks: Estimation and Interpretation Beyond Kaplan- Meier


  • Patrizia Boracchi University of Milan
  • Annalisa Orenti University of Milan



Survival analysis, competing risks, crude cumulative incidence, net survival, relative survival, breast cancer


Evaluation of a therapeutic strategy is complex when the course of a disease is characterized by the occurrence of different kinds of events. Competing risks arise when the occurrence of specific events prevents the observation of other events. A particular case is semi-competing risks when only fatal events can prevent the observation of the non fatal ones.

Kaplan-Meier is the most popular method to estimate overall or event free survival. On the other hand when a subset of events is considered and net survival is of concern, different estimators have been proposed. Kaplan-Meier method can be used only under the independence assumptions otherwise estimators based on multivariate distribution of times are needed. If causes of death are unknown, relative survival can approximate net survival only under specific assumptions on the mortality pattern.

Kaplan-Meier method cannot be used to estimate crude cumulative incidence of specific events.

The aim of this work is to present the survival functions used in competing risks framework, their non parametric estimators and semi parametric estimators for net survival based on Archimedean Copulas. This would be a help for the reader who is not experienced in competing risks analysis.

A simulation study is performed to evaluate performances of net survival estimators. To illustrate survival functions in presence of different causes of death and of different kind of events a numerical example is given, a literature dataset on prostate cancer and a case series of breast cancer patients have been analysed.

Author Biographies

Patrizia Boracchi, University of Milan

Department of Clinical Sciences and Community Health, Laboratory of Medical Statistics, Epidemiology and
Biometry G. A. Maccacaro

Annalisa Orenti, University of Milan

Department of Clinical Sciences and Community Health, Laboratory of Medical Statistics, Epidemiology and
Biometry G. A. Maccacaro


[1] Zheng M, Klein JP. Estimates of marginal survival for dependent competing risks based on an assumed copula. Biometrika 1995; 82(1): 127-38.
[2] Kaplan EL, Meier P. Nonparametric estimation from incomplete observations. J Am Statist Assoc 1958; 53: 457-81.
[3] Tai P, Joseph K, El-Gayed A, Yu E. Long-term outcome of breast cancer patients with one to two nodes involved - application of nodal ratio. Breast J 2012; 18(6): 542-8.
[4] Tsiatis A. A nonidentifiability aspect of the problem of competing risks. Proc Nat Acad Sci USA 1975; 72(1): 20-2.
[5] Hougaard P. Modelling multivariate survival. Scand J Stat 1987: 291-304.
[6] Kaishev, VK, Dimitrova DS, Haberman S. Modelling the joint distribution of competing risks survival times using copula functions. Insurance: Mathematics and Economics 2007; 41(3): 339-361.
[7] Nelsen RB. An Introduction to Copulas. New York: Springer, 1999.
[8] Martelli G, Boracchi P, Orenti A, et al. Axillary dissection versus no axillary dissection in older T1N0 breast cancer patients: 15-year results of trial and out-trial patients. Eur J Surg Oncol 2014; 40(7): 805-12.
[9] Rutherford MJ, Dickman PW, Lambert PC. Comparison of methods for calculating relative survival in population-based studies. Cancer Epidemiol. 2012; 36(1): 16-21.
[10] Ederer F, Axtell LM, Cutler SJ. The relative survival rate: a statistical methodology. Natl Cancer Inst Monograph 1961; 6: 101-21.
[11] Pohar Perme M, Stare J, Estève J. On estimation in relative survival. Biometrics 2012; 68(1): 113-120.
[12] Moliterni A, Ménard S, Valagussa P, et al. HER2 overexpression and doxorubicin in adjuvant chemotherapy for resectable breast cancer. J Clin Oncol 2003; 21(3): 458- 62.
[13] Fine JP, Jiang H, Chappell R. On semi-competing risks data. Biometrika 2001. 88(4): 907-919.
[14] Kalbfleish JD, Prentice RL. The Statistical Analysis of Failure Time Data. 2nd ed. Hoboken, New Jersery: John Wiley and Sons; 2002.
[15] Marubini E, Valsecchi MG. Analysing Survival Data from Clinical Trials and Observational Studies. Chichester: John Wiley and Sons; 1995.
[16] Fine JP, Gray RJ. A proportional hazards model for the subdistribution of a competing risk. J Am Stat Assoc 1999; 94: 496-509.
[17] Ederer F, Heise H. Instructions to IBM 650 programmers in processing survival computations. Methodological note No. 10, End Results Evaluation Section, National Cancer Institute, Bethesda MD, 1959.
[18] Hakulinen T. Cancer survival corrected for heterogeneity in patient withdrawal. Biometrics 1982; 38: 933-42.
[19] Brenner H, Hakulinen T. On crude and age-adjusted relative survival rates. Journal of clinical epidemiology 2003; 56(12): 1185-1191.
[20] Slud EV, Rubinstein LV. Dependent competing risks and summary survival curves. Biometrika 1983; 70(3): 643-649.
[21] Peterson AV, Bounds for a joint distribution function with fixed sub-distribution functions: Application to competing risks. Proc Nat Acad Sci USA 1976; 73(1): 11-13.
[22] Peterson AV. Dependent competing risks: bounds for net survival functions with fixed crude survival functions. Environment International 1978; 1(6): 351-5.
[23] Klein JP, Moeschberger ML. Bounds on net survival probabilities for dependent competing risks. Biometrics 1988: 529-38.
[24] Dignam JJ, Weissfeld LA, Anderson SJ. Methods for bounding the marginal survival distribution. Stat Med 1995; 14(18): 1985-98.
[25] Rivest LP, Wells MT. A martingale approach to the copulagraphic estimator for the survival function under dependent censoring. J Multiv Anal 2001; 79: 138-55.
[26] de Uña-Álvarez J, Veraverbeke N. Generalized copulagraphic estimator. Test 2013; 22(2): 343-360.
[27] Brown BW, Hollander M, Korwar RM. Nonparametric tests of independence for censored data, with applications to heart transplant studies. Reliability and Biometry 1974: 327-54.
[28] Rotolo F, Legrand C, Van Keilegom I. A simulation procedure based on copulas to generate clustered multi-state survival data. Comput Meth Prog Bio 2013; 109: 305-12.
[29] Byar DP, Green SB. The choice of treatment for cancer patients based on covariate information. Bulletin du cancer 1979; 67(4): 477-490.
[30] Kay R. Treatment effects in competing-risks analysis of prostate cancer data. Biometrics 1986: 203-211.
[31] Veronesi U, Cascinelli N, Mariani L, et al. Twenty-year followup of a randomized study comparing breast-conserving surgery with radical mastectomy for early breast cancer. N Engl J Med 2002; 347(16): 1227-32.
[32] Mariani L, Salvadori B, Marubini E, et al. Ten Year Results of a Randomised Trial Comparing Two Conservative Treatment Strategies for Small Size Breast Cancer. Eur J Cancer 1998; 34(8): 1156-62.
[33] Veronesi U, Marubini E, Mariani L, et al. Radiotherapy after breast-conserving surgery in small breast carcinoma: longterm results of a randomized trial. Ann Oncol 2001; 12: 997- 1003.
[34] Resche-Rigon M, Azoulay E, Chevret S. Evaluating mortality in intensive care units: contribution of competing risks analyses. Crit Care 2006; 10(1): R5.
[35] Kim HT. Cumulative incidence in competing risks data and competing risks regression analysis. Clin Cancer Res 2007; 13(2): 559-65.
[36] Allemani C, Weir HK, Carreira H, et al. Global surveillance of cancer survival 1995-2009: analysis of individual data for 25 676 887 patients from 279 population-based registries in 67 countries (CONCORD-2). Lancet 2014 Nov 26; Available from PIIS0140-6736%2814%2962038-9/abstract
[37] Yen Yen F, Ahmad N, Kassim S. A multistate approach to estimating the net survival function in the presence of competing risks. Malays J Math Sci 2011; 5(1): 125-41.
[38] Peng L, Fine JP. Regression modeling of semicompeting risks data. Biometrics 2007; 63(1): 96-108.
[39] Hsieh JJ, Huang YT. Regression analysis based on conditional likelihood approach under semi-competing risks data. Lifetime Data Anal 2012; 18: 302-20.
[40] Lo SMS, Wilke RA. A regression model for the copulagraphic estimator. Journal of Econometric Methods 2014; 3(1): 21-46.
[41] Elandt-Johnson RC, Johnson NL. Survival models and data analysis. New York: John Wiley & Sons, 1980.
[42] Crowder MJ. Classical competing risks. Boca Raton, FL: Chapman & Hall/CRC, 2001.




How to Cite

Boracchi, P., & Orenti, A. (2015). Survival Functions in the Presence of Several Events and Competing Risks: Estimation and Interpretation Beyond Kaplan- Meier. International Journal of Statistics in Medical Research, 4(1), 121–139.



General Articles