Lifetime Data Analysis

, Volume 20, Issue 2, pp 234–251 | Cite as

Calibrated predictions for multivariate competing risks models

  • Malka Gorfine
  • Li Hsu
  • David M. Zucker
  • Giovanni Parmigiani


Prediction models for time-to-event data play a prominent role in assessing the individual risk of a disease, such as cancer. Accurate disease prediction models provide an efficient tool for identifying individuals at high risk, and provide the groundwork for estimating the population burden and cost of disease and for developing patient care guidelines. We focus on risk prediction of a disease in which family history is an important risk factor that reflects inherited genetic susceptibility, shared environment, and common behavior patterns. In this work family history is accommodated using frailty models, with the main novel feature being allowing for competing risks, such as other diseases or mortality. We show through a simulation study that naively treating competing risks as independent right censoring events results in non-calibrated predictions, with the expected number of events overestimated. Discrimination performance is not affected by ignoring competing risks. Our proposed prediction methodologies correctly account for competing events, are very well calibrated, and easy to implement.


Risk prediction Competing risks Frailty model  Multivariate survival model Calibration ROC analysis 



Malka Gorfine’s work was supported by Israel Science Foundation (ISF) Grant 2012898. Li Hsu’s work was supported by NIH Grants P01 CA53996 and R01AG14358. Giovanni Parmigiani’s work was supported by NIH/NCI 5P30 CA006516-46 and Komen KG081303.


  1. Bandeen-Roche K, Liang KY (2002) Modelling multivariate failure times associations in the presence of competing risk. Biometrika 89:299–313CrossRefzbMATHMathSciNetGoogle Scholar
  2. Bandeen-Roche K, Ning J (2008) Nonparametric estimation of bivariate failure time associations in the presence of a competing risk. Biometrika 95:221–232CrossRefzbMATHMathSciNetGoogle Scholar
  3. Chatterjee N, Hartge P, Wacholder S (2003) Adjustment for competing risk in kin-cohort estimation. Genet Epidemiol 25:303–313CrossRefGoogle Scholar
  4. Chen BE, Kramer JL, Greene MH, Rosenberg PS (2008) Competing risks analysis of correlated failure time data. Biometrics 64:172–179CrossRefzbMATHMathSciNetGoogle Scholar
  5. Collaborative Group on Hormonal Factors in Breast Cancer (2001) Familial breast cancer: collaborative reanalysis of individual data from 52 epidemiological studies including 58,209 women with breast cancer and 101,986 women without the disease. Lancet 358:1389–1399CrossRefGoogle Scholar
  6. Gorfine M, Hsu L (2011) Frailty-based competing risks model for multivariate survival data. Biometrics 67:415–426CrossRefzbMATHMathSciNetGoogle Scholar
  7. Gorfine M, Hsu L, Parmigiani G (2013) Frailty models for familial risk with application to breast cancer. J Am Stat Assoc (to appear)Google Scholar
  8. Harrell FE (2001) Regression modeling strategies: with applications to linear models, logistic regression, and survival analysis. Springer, New YorkCrossRefGoogle Scholar
  9. Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  10. Katki HA, Blackford A, Chen S, Parmigiani G (2008) Multiple diseases in carrier probability estimation: accounting for surviving all cancers other than breast and ovary in BRCAPRO. Stat Med 27:4532–4548CrossRefMathSciNetGoogle Scholar
  11. Parmigiani G, Berry D, Iversen J, M\(\ddot{u}\)ller P, Schildkraut J, Winer E (1998) Modeling risk of breast cancer and decisions about genetic testing. In: Gatsonis C et al. (eds) Case studies in Bayesian statistics, vol IV, pp. 173–268.
  12. Pencina MJ, D’Agostino RB, Larson MG, Massaro JM (2009) Predicting the 30-year risk of cardiovascular disease: the Framingham heart study. Circulation 119:3078–3084CrossRefGoogle Scholar
  13. Pharoah PDP, Day NE, Duffy S, Easton DF, Ponder BAJ (1997) Family history and the risk of breast cancer: a systematic review and meta-analysis. Int J Cancer 71:800–809CrossRefGoogle Scholar
  14. Prentice RL, Kalbfleisch JD, Peterson AV, Jr Flournoy N, Farewell VT, Breslow NE (1978) The analysis of failure times in the presence of competing risks. Biometrics 34:541–554Google Scholar
  15. Risch HA, McLaughlin JR, Cole DEC, Rosen B, Bradley L, Fan I, Tang J, Li S, Zhang S, Shaw PA, Narod SA (2006) Population BRCA1 and BRCA2 mutation frequencies and cancer penetrances: a kin-cohort study in Ontario, Canada. J Natl Cancer Inst 98:1694–1706CrossRefGoogle Scholar
  16. Steyerberg EW, Vickers AJ, Cook NR, Gerds T, Gonen M, Obuchowski N, Pencina MJ, Kattan W (2010) Assessing the performance of prediction models: a framework for traditional and novel measures. Epidemiology 21:128–138CrossRefGoogle Scholar
  17. Wolbers M, Koller MT, Witteman JCM, Steyerberg EW (2009) Prognostic models with competing risks methods and application to coronary risk prediction. Epidemiology 20:555–561CrossRefGoogle Scholar
  18. Zhou B, Fine J, Latouche A, Labopin M (2012) Competing risks regression for clustered data. Biostatistics 13:371–383CrossRefzbMATHGoogle Scholar
  19. Zeng D, Lin DY (2007) Maximum likelihood estimation in semiparametric regression models with censored data. J R Stat Soc B 69:507–564CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Malka Gorfine
    • 1
  • Li Hsu
    • 2
  • David M. Zucker
    • 3
  • Giovanni Parmigiani
    • 4
    • 5
  1. 1.Faculty of Industrial Engineering and ManagementTechnion—Israel Institute of TechnologyHaifaIsrael
  2. 2.Division of Public Health SciencesFred Hutchinson Cancer Research CenterSeattleUSA
  3. 3.Department of StatisticsHebrew University of JerusalemJerusalemIsrael
  4. 4.Department of Biostatistics and Computational BiologyDana Farber Cancer Institute BostonUSA
  5. 5.Department of BiostatisticsHarvard School of Public HealthBostonUSA

Personalised recommendations