Estimation of a Concordance Probability for Doubly Censored Time-to-Event Data

Article
  • 22 Downloads

Abstract

Evaluating the relationship between a response variable and explanatory variables is important to establish better statistical models. Concordance probability is one measure of this relationship and is often used in biomedical research. Concordance probability can be seen as an extension of the area under the receiver operating characteristic curve. In this study, we propose estimators of concordance probability for time-to-event data subject to double censoring. A doubly censored time-to-event response is observed when either left or right censoring may occur. In the presence of double censoring, existing estimators of concordance probability lack desirable properties such as consistency and asymptotic normality. The proposed estimators consist of estimators of the left-censoring and the right-censoring distributions as a weight for each pair of cases, and reduce to the existing estimators in special cases. We show the statistical properties of the proposed estimators and evaluate their performance via numerical experiments.

Keywords

Concordance probability Doubly censored data Time-to-event response 

Notes

Acknowledgements

The authors are grateful to Drs. Yasuhiko Sakata, Daisaku Nakatani, and Yasushi Sakata for allowing us to utilize the dataset used in [6]. The authors would like to acknowledge the associate editor and anonymous reviewers for very useful comments and suggestions that improve the presentation of the paper. K. Hayashi is supported by JSPS KAKENHI (Grant-in-Aid for Scientific Research) Grant Number 15K15950. S. Shimizu is supported by JSPS KAKENHI (Grant-in-Aid for Scientific Research) Grant Number 70423085.

References

  1. 1.
    Chang MN (1990) Weak convergence of a self-consistent estimator of the survival function with doubly censored data. Ann Stat 18:391–404MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chang MN, Yang GL (1987) Strong consistency of a nonparametric estimator of the survival function with doubly censored data. Ann Stat 15:1536–1547MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cook NR (2007) Use and misuse of the receiver operating characteristic curve in risk prediction. Circulation 115:928–935CrossRefGoogle Scholar
  4. 4.
    DeLong ER, DeLong DM, Clarke-Pearson DL (1988) Comparing the areas under two or more correlated receiver operating characteristic curves: a nonparametric approach. Biometrics 44:837–845CrossRefMATHGoogle Scholar
  5. 5.
    Gönen M, Heller G (2005) Concordance probability and discriminatory power in proportional hazards regression. Biometrika 92:965–970MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hara M, Sakata Y, Nakatani D, Suna S, Nishino M, Sato H, Kitamura T, Nanto S, Hamasaki T, Hori M, Komuro I (2016) Subclinical elevation of high-sensitive troponin T levels at the convalescent stage is associated with increased 5-year mortality after ST-elevation myocardial infarction. J Cardiol 67:314–320CrossRefGoogle Scholar
  7. 7.
    Harrell FE, Lee KL, Mark DB (1996) Tutorial in biostatistics: multivariate prognostic models: issues in developing models evaluating assumptions and adequacy and measuring and reducing errors. Stat Med 15:361–387CrossRefGoogle Scholar
  8. 8.
    Hilden J, Gerds TT (2014) A note on the evaluation of novel biomarkers: do not rely on integrated discrimination improvement and net reclassification index. Stat Med 33:3405–3414MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ji S, Peng L, Cheng Y, HuiChuan L (2012) Quantile regression for doubly censored data. Biometrics 68:101–112MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Julià O, Gómez G (2011) Simultaneous marginal survival estimators when doubly censored data is present. Lifetime Data Anal 17:347–372MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kyle RT, Rajkumar TV, Offord J, Larson D, Plevak M, Melton LJ III (2002) A long-terms study of prognosis in monoclonal gammopathy of undetermined significance. New Engl J Med 346:564–569CrossRefGoogle Scholar
  12. 12.
    Kim Y, Kim B, Jang W (2010) Asymptotic properties of the maximum likelihood estimator for the proportional hazards model with doubly censored data. J Multivar Anal 101:1339–1351MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Klein JP, Moeschberger ML (2003) Survival analysis: techniques for censored and truncated data. Springer, New YorkMATHGoogle Scholar
  14. 14.
    Mantel N (1967) Ranking procedures for arbitrarily restricted observation. Biometrics 23:65–78MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nolan D, Pollard D (1987) \(U\)-processes: rates of convergence. Ann Stat 15:780–799MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Nolan D, Pollard D (1988) Functional limit theorems for \(U\)-processes. Ann Stat 16:1291–1298MathSciNetMATHGoogle Scholar
  17. 17.
    Pencina MJ, D’Agostino RB (2004) Overall C as a measure of discrimination in survival analysis: model specific population value and confidence interval estimation. Stat Med 23:2109–2123CrossRefGoogle Scholar
  18. 18.
    Pepe MS (2003) The statistical evaluation of medical tests for classification and prediction. Oxford University Press, New YorkMATHGoogle Scholar
  19. 19.
    Pepe MS, Janes H, Li CI (2014) Net risk reclassification p values: valid or misleading? J Natl Cancer Inst 106:dju041CrossRefGoogle Scholar
  20. 20.
    Pepe MS, Thompson LT (2000) Combining diagnostic test results to increasing accuracy. Biostatistics 1:123–140CrossRefMATHGoogle Scholar
  21. 21.
    Peto R (1973) Experimental survival curves for interval-censored data. J R Stat Soc C 22:86–91Google Scholar
  22. 22.
    R Core Team (2014) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Ann Stat 4:1317–1334. http://www.R-project.org/
  23. 23.
    Tsai WY, Crowley J (1985) A large sample study of generalized maximum likelihood estimators from incomplete data via self-consistency. Ann Stat 4:1317–1334MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Turnbull BW (1974) Nonparametric estimation of a survivorship function with doubly censored data. J Am Stat Assoc 69:169–173MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Turnbull BW (1976) The empirical distribution function with arbitrarily grouped censored and truncated data. J R Stat Soc B 38:290–295MathSciNetMATHGoogle Scholar
  26. 26.
    Uno H, Cai H, Pencina MJ, D’Agostino RB, Wei LJ (2011) On the C-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data. Stat Med 30:1105–1117MathSciNetGoogle Scholar
  27. 27.
    Zhang C-H, Li X (1996) Linear regression with doubly censored data. Ann Stat 24:2720–2743MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© International Chinese Statistical Association 2018

Authors and Affiliations

  1. 1.Department of MathematicsKeio UniversityYokohamaJapan
  2. 2.Department of Applied MathematicsWaseda UniversityTokyoJapan

Personalised recommendations