Lifetime Data Analysis

, Volume 22, Issue 3, pp 456–471 | Cite as

Semiparametric model for semi-competing risks data with application to breast cancer study

  • Renke Zhou
  • Hong Zhu
  • Melissa Bondy
  • Jing Ning


For many forms of cancer, patients will receive the initial regimen of treatments, then experience cancer progression and eventually die of the disease. Understanding the disease process in patients with cancer is essential in clinical, epidemiological and translational research. One challenge in analyzing such data is that death dependently censors cancer progression (e.g., recurrence), whereas progression does not censor death. We deal with the informative censoring by first selecting a suitable copula model through an exploratory diagnostic approach and then developing an inference procedure to simultaneously estimate the marginal survival function of cancer relapse and an association parameter in the copula model. We show that the proposed estimators possess consistency and weak convergence. We use simulation studies to evaluate the finite sample performance of the proposed method, and illustrate it through an application to data from a study of early stage breast cancer.


Copula model Informative censoring Model diagnostic Semi-competing risks Simultaneous inference 



The authors thank the editor, the associate editor and two reviewers for their constructive comments that have greatly improved the initial version of this paper. This work was supported in part by Cancer Center Support Grants from the National Institutes of Health (CA142543 to Hong Zhu at UT Southwestern Medical Center and CA016672 to Jing Ning at UT MD Anderson Cancer Center) and by a predoctoral fellowship grant from the Cancer Prevention Research Institute of Texas (RP140103 to Renke Zhou).


  1. Brewster AM, Do KA, Thompson PA, Hahn KM, Sahin AA, Cao Y, Stewart MM, Murray JL, Hortobagyi GN, Bondy ML (2007) Relationship between epidemiologic risk factors and breast cancer recurrence. J Clin Oncol 25:4438–4444CrossRefGoogle Scholar
  2. Chen MC, Bandeen-Roche K (2005) A diagnostic for association in bivariate survival models. Lifetime Data Anal 11:245–264MathSciNetCrossRefMATHGoogle Scholar
  3. Chen YH (2012) Maximum likelihood analysis of semicompeting risks data with semiparametric regression models. Lifetime Data Anal 18:36–57MathSciNetCrossRefMATHGoogle Scholar
  4. Clayton DG (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65:141–151MathSciNetCrossRefMATHGoogle Scholar
  5. Ding AA, Shi G, Wang W, Hsieh JJ (2009) Marginal regression analysis for semi-competing risks data under dependent censoring. Scand J Stat 36:481–500MathSciNetCrossRefMATHGoogle Scholar
  6. Fine JP, Jiang H, Chappell RJ (2001) On semi-competing risks data. Biometrika 88:907–919MathSciNetCrossRefMATHGoogle Scholar
  7. Fleming TR, Harrington DP (2005) Counting processes and survival analysis. Wiley, ChichesterCrossRefMATHGoogle Scholar
  8. Frank M (1979) On the simultaneous associativity of f(x, y) and x + y - f(x, y). Aequ Math 19:194–226MathSciNetCrossRefMATHGoogle Scholar
  9. Gill RD (1980) Censoring and stochastic integrals. Mathematisch Centrum, AmsterdamMATHGoogle Scholar
  10. Gumbel EJ (1960) Bivariate exponential distributions. J Am Stat Assoc 55:698–707MathSciNetCrossRefMATHGoogle Scholar
  11. Hsieh JJ, Wang W, Ding AA (2008) Regression analysis based on semicompeting risks data. J R Stat Soc Series B Stat Methodol 70:3–20MathSciNetMATHGoogle Scholar
  12. Hsieh JJ, Huang YT (2012) Regression analysis based on conditional likelihood approach under semi-competing risks data. Lifetime Data Anal 103:302–320MathSciNetCrossRefMATHGoogle Scholar
  13. Jiang H, Fine JP, Kosorok MR, Chappell RJ (2005) Pseudo self-consistent estimation of a copula model with informative censoring. Scand J Stat 32:1–20MathSciNetCrossRefMATHGoogle Scholar
  14. Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53:457–481MathSciNetCrossRefMATHGoogle Scholar
  15. Oakes D (1982) A model for association in bivariate survival data. J R Stat Soc B 44:414–422MathSciNetMATHGoogle Scholar
  16. Oakes D (1989) Bivariate survival models induced by frailties. J Am Stat Assoc 84:487–493MathSciNetCrossRefMATHGoogle Scholar
  17. Peng L, Fine JP (2007) Regression modeling of semicompeting risks data. Biometrics 63:96–108MathSciNetCrossRefMATHGoogle Scholar
  18. Peng L, Jiang H, Chappell RJ, Fine JP (2007) Statistical advances in the biomedical sciences: clinical trials, epidemiology, survival analysis, and bioinformatics. Wiley, HobokenGoogle Scholar
  19. Shih JH, Louis TA (1995) Inferences on the association parameters in copula models for bivariate survival data. Biometrics 51:1384–1399MathSciNetCrossRefMATHGoogle Scholar
  20. Siegel R, Ma J, Zou Z, Jemal A (2014) Cancer statistics, 2014. CA Cancer J Clin 64:9–29CrossRefGoogle Scholar
  21. Wang W (2003) Estimating the association parameter for copula models under dependent censoring. J R Stat Soc Series B Stat Methodol 65:257–273MathSciNetCrossRefMATHGoogle Scholar
  22. Xu J, Kalbfeisch JD, Tai B (2010) Statistical analysis of illness-death processes and semicompeting risks data. Biometrics 66:716–725MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Renke Zhou
    • 1
    • 2
  • Hong Zhu
    • 3
  • Melissa Bondy
    • 1
  • Jing Ning
    • 4
  1. 1.Duncan Cancer CenterBaylor College of MedicineHoustonUSA
  2. 2.Division of BiostatisticsThe University of Texas School of Public HealthHoustonUSA
  3. 3.Division of Biostatistics, Department of Clinical SciencesThe University of Texas Southwestern Medical CenterDallasUSA
  4. 4.Department of BiostatisticsThe University of Texas MD Anderson Cancer CenterHoustonUSA

Personalised recommendations