Advertisement

Bootstrapping the Kaplan–Meier estimator on the whole line

  • Dennis DoblerEmail author
Article
  • 142 Downloads

Abstract

This article is concerned with proving the consistency of Efron’s bootstrap for the Kaplan–Meier estimator on the whole support of a survival function. While previous works address the asymptotic Gaussianity of the Kaplan–Meier estimator without restricting time, we enable the construction of bootstrap-based time-simultaneous confidence bands for the whole survival function. Other practical applications include bootstrap-based confidence bands for the mean residual lifetime function or the Lorenz curve as well as confidence intervals for the Gini index. Theoretical results are complemented with a simulation study and a real data example which result in statistical recommendations.

Keywords

Counting process Right censoring Resampling Efron’s bootstrap Mean residual lifetime Lorenz curve Gini index 

Notes

Acknowledgements

The author would like to thank Markus Pauly (Ulm University) for helpful discussions. Furthermore, the discussion with a referee and his / her suggestions to present numerical results and to illustrate the methods with the help of real data have helped to improve this manuscript significantly.

References

  1. Akritas, M. G. (1986). Bootstrapping the Kaplan–Meier Estimator. Journal of the American Statistical Association, 81(396), 1032–1038.MathSciNetzbMATHGoogle Scholar
  2. Akritas, M. G., Brunner, E. (1997). Nonparametric methods for factorial designs with censored data. Journal of the American Statistical Association, 92(438), 568–576.Google Scholar
  3. Allignol, A., Beyersmann, J., Gerds, T., Latouche, A. (2014). A competing risks approach for nonparametric estimation of transition probabilities in a non-Markov illness-death model. Lifetime Data Analysis, 20(4), 495–513.Google Scholar
  4. Andersen, P. K., Borgan, Ø., Gill, R. D., Keiding, N. (1993). Statistical models based on counting processes. New York: Springer.Google Scholar
  5. Billingsley, P. (1999). Convergence of probability measures (2nd ed.). New York: Wiley.CrossRefzbMATHGoogle Scholar
  6. Dobler, D. (2016). Nonparametric inference procedures for multi-state Markovian models with applications to incomplete life science data. PhD thesis Universität Ulm, Deutschland.Google Scholar
  7. Dobler, D., Pauly, M. (2017). Bootstrap- and permutation-based inference for the Mann–Whitney effect for right-censored and tied data. TEST early view:1–20.  https://doi.org/10.1007/s11749-017-0565-z.
  8. Efron, B. (1981). Censored data and the bootstrap. Journal of the American Statistical Association, 76(374), 312–319.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gastwirth, J. L. (1971). A general definition of the Lorenz curve. Econometrica, 39(6), 1037–1039.CrossRefzbMATHGoogle Scholar
  10. Gill, R. D. (1980). Censoring and stochastic integrals. Mathematical Centre Tracts 124, Amsterdam: Mathematisch Centrum.Google Scholar
  11. Gill, R. D. (1983). Large sample behaviour of the product-limit estimator on the whole line. The Annals of Statistics, 11(1), 49–58.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Gill, R. D. (1989). Non- and semi-parametric maximum likelihood estimators and the von mises method (part 1) [with discussion and reply]. Scandinavian Journal of Statistics, 16(2), 97–128.MathSciNetzbMATHGoogle Scholar
  13. Grand, M. K., Putter, H. (2016). Regression models for expected length of stay. Statistics in Medicine, 35(7), 1178–1192.Google Scholar
  14. Horvath, L., Yandell, B. (1987). Convergence rates for the bootstrapped product-limit process. The Annals of Statistics, 15(3), 1155–1173.Google Scholar
  15. Janssen, A., Pauls, T. (2003). How do bootstrap and permutation tests work? The Annals of Statistics, 31(3), 768–806.Google Scholar
  16. Klein, J. P., Moeschberger, M. L. (2003). Survival analysis: Techniques for censored and truncated data (2nd ed.). New York: Springer.Google Scholar
  17. Lo, S.-H., Singh, K. (1986). The product-limit estimator and the bootstrap: Some asymptotic representations. Probability Theory and Related Fields, 71(3), 455–465.Google Scholar
  18. Loprinzi, C. L., Laurie, J. A., Wieand, H. S., Krook, J. E., Novotny, P. J., Kugler, J. W., et al. (1994). Prospective evaluation of prognostic variables from patient-completed questionnaires. Journal of Clinical Oncology, 12(3), 601–607.CrossRefGoogle Scholar
  19. Meilijson, I. (1972). Limiting properties of the mean residual lifetime function. The Annals of Mathematical Statistics, 43(1), 354–357.MathSciNetCrossRefzbMATHGoogle Scholar
  20. Pocock, S. J., Ariti, C. A., Collier, T. J., Wang, D. (2012). The win ratio: A new approach to the analysis of composite endpoints in clinical trials based on clinical priorities. European Heart Journal, 33(2), 176–182.Google Scholar
  21. Pollard, D. (1984). Convergence of stochastic processes. New York: Springer.CrossRefzbMATHGoogle Scholar
  22. R Development Core Team. (2016). R: A language and environment for statistical computing. R Foundation for Statistical Computing Vienna, Austria. http://www.R-project.org.
  23. Revuz, D., Yor, M. (1999). Continuous martingales and Brownian motion (3rd ed.). Berlin: Springer.Google Scholar
  24. Stute, W., Wang, J.-L. (1993). The strong law under random censorship. The Annals of Statistics, 21(3), 1591–1607.Google Scholar
  25. Tattar, P., Vaman, H. (2012). Extension of the Harrington–Fleming tests to multistate models. Sankhya B, 74(1), 1–14.Google Scholar
  26. Therneau, T. M., Lumley, T. (2017). A package for survival analysis in S. http://CRAN.R-project.org/package=survival. version 2.41-3.
  27. Tse, S.-M. (2006). Lorenz curve for truncated and censored data. Annals of the Institute of Statistical Mathematics, 58(4), 675–686.MathSciNetCrossRefzbMATHGoogle Scholar
  28. van der Vaart, A. W., Wellner, J. (1996). Weak convergence and empirical processes. New York: Springer.Google Scholar
  29. Wang, J.-G. (1987). A note on the uniform consistency of the Kaplan–Meier estimator. The Annals of Statistics, 15(3), 1313–1316.MathSciNetCrossRefzbMATHGoogle Scholar
  30. Wellner, J. A. (1978). Limit theorems for the ratio of the empirical distribution function to the true distribution function. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 45(1), 73–88.MathSciNetCrossRefzbMATHGoogle Scholar
  31. Ying, Z. (1989). A note on the asymptotic properties of the product-limit estimator on the whole line. Statistics & Probability Letters, 7(4), 311–314.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2018

Authors and Affiliations

  1. 1.Institute of StatisticsUlm UniversityUlmGermany

Personalised recommendations