Electronic Journal of Statistics

Bootstrapping a change-point Cox model for survival data

Gongjun Xu, Bodhisattva Sen, and Zhiliang Ying

Full-text: Open access

Abstract

This paper investigates the (in)-consistency of various bootstrap methods for making inference on a change-point in time in the Cox model with right censored survival data. A criterion is established for the consistency of any bootstrap method. It is shown that the usual nonparametric bootstrap is inconsistent for the maximum partial likelihood estimation of the change-point. A new model-based bootstrap approach is proposed and its consistency established. Simulation studies are carried out to assess the performance of various bootstrap schemes.

Article information

Source
Electron. J. Statist. Volume 8, Number 1 (2014), 1345-1379.

Dates
First available in Project Euclid: 20 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1408540290

Digital Object Identifier
doi:10.1214/14-EJS927

Mathematical Reviews number (MathSciNet)
MR3263125

Zentralblatt MATH identifier
1298.62172

Subjects
Primary: 62N02: Estimation 62G09: Resampling methods

Keywords
Change-point in time (in)-consistency of bootstrap $m$-out-of-$n$ bootstrap non-standard asymptotics smoothed bootstrap

Citation

Xu, Gongjun; Sen, Bodhisattva; Ying, Zhiliang. Bootstrapping a change-point Cox model for survival data. Electron. J. Statist. 8 (2014), no. 1, 1345--1379. doi:10.1214/14-EJS927. https://projecteuclid.org/euclid.ejs/1408540290


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References

  • Abrevaya, J. and Huang, J. (2005). On the bootstrap of the maximum score estimator. Econometrica, 73, 1175–1204.
  • Andersen, P. and Gill, R. (1982). Cox’s regression model for counting processes: A large sample study. Ann. Statist., 10, 1100–1120.
  • Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
  • Beran, R. (1981). Nonparametric regression with randomly censored survival data. Unpublished technical report, University of California, Berkeley.
  • Bickel, P. J., Götze, F. and van Zwet, W. R. (1997). Resampling fewer than $n$ observations: Gains, losses, and remedies for losses. Statist. Sinica, 7, 1–31.
  • Bilias, Y., Gu, M. and Ying, Z. (1997). A general asymptotic theory for Cox model with staggered entry. Ann. Statist., 25, 662–682.
  • Billingsley, P. (1968). Convergence of Probability Measures. John Wiley & Sons Inc., New York.
  • Bose, A. and Chatterjee, S. (2001). Generalised bootstrap in non-regular $M$-estimation problems. Statist. Probab. Lett., 55, 319–328.
  • Burr, D. (1994). A comparison of certain bootstrap confidence intervals in the Cox model. J. Amer. Statist. Assoc., 89, 1290–1302.
  • Cheng, G. and Huang, J. Z. (2010). Bootstrap consistency for general semiparametric M-estimation. Ann. Statist., 38, 2884–2915.
  • Cox, D. R. (1972). Regression models and life-tables. J. R. Statist. Soc. B, 34, 187–220.
  • Cox, D. R. (1975). Partial likelihood. Biometrika, 62, 269–276.
  • Cox, D. R. and Oakes, D. (1984). Analysis of Survival Data. Chapman & Hall, London.
  • Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge University Press, Cambridge.
  • Delsol, L. and Van Keilegom, I. (2011). Semiparametric M-estimation with non-smooth criterion functions. Technical Report.
  • Dudley, R. M. (2002). Real Analysis and Probability, vol. 74. Cambridge University Press, Cambridge.
  • Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York.
  • Fleming, T. R. and Harrington, D. (1991). Counting Processes and Survival Analysis. John Wiley & Sons Inc., New York.
  • Fryzlewicz, P. (2014). Wild binary segmentation for multiple change-point detection. Ann. Statist. (to appear).
  • Hjort, N. L. and Pollard, D. (2011). Asymptotics for minimisers of convex processes. arXiv:1107.3806.
  • Jacod, J. and Shiryaev, A. (2002). Limit Theorems for Stochastic Processes. Springer, New York.
  • Kalbfleisch, J. D. and Prentice, R. L. (2002). The Statistical Analysis of Failure Time Data. Wiley, New York.
  • Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer Series in Statistics. Springer, New York. URL http://dx.doi.org/10.1007/978-0-387-74978-5.
  • Kosorok, M. R. and Song, R. (2007). Inference under right censoring for transformation models with a change-point based on a covariate threshold. Ann. Statist., 35, 957–989.
  • Lan, Y., Banerjee, M. and Michailidis, G. (2009). Change-point estimation under adaptive sampling. Ann. Statist., 37, 1752–1791.
  • Liang, K.-Y., Self, S. and Liu, X. (1990). The Cox proportional hazards model with change point: An epidemiologic application. Biometrics, 46, 783–793.
  • Luo, X. (1996). The asymptotic distribution of MLE of treatment lag threshold. J. Statist. Plann. Inference, 53, 33–61.
  • Luo, X., Turnbull, B. and Clark, L. (1997). Likelihood ratio tests for a changepoint with survival data. Biometrika, 84, 555–565.
  • Meinert, C. (1986). Clinical Trials: Design, Conduct, and Analysis. Oxford University Press.
  • Neuhaus, G. (1971). On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Statist., 42, 1285–1295.
  • Pollard, D. (1990). Empirical Processes: Theory and Applications. Institute of Mathematical Statistics, Hayward, CA.
  • Pons, O. (2002). Estimation in a Cox regression model with a change-point at an unknown time. Statistics, 36, 101–124.
  • Scott, D. W. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, New York.
  • Seijo, E. and Sen, B. (2011a). Change-point in stochastic design regression and the bootstrap. Ann. Statist., 39, 1580–1607.
  • Seijo, E. and Sen, B. (2011b). A continuous mapping theorem for the smallest argmax functional. Electron. J. Stat., 5, 421–439.
  • Sen, B., Banerjee, M. and Woodroofe, M. (2010). Inconsistency of bootstrap: The grenander estimator. Ann. Statist., 38, 1953–1977.
  • van der Vaart, A. and Wellner, J. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
  • Wells, M. T. (1994). Nonparametric kernel estimation in counting processes with explanatory variables. Biometrika, 81, 795–801.
  • Zucker, D. and Lakatos, E. (1990). Weighted log rank type statistics for comparing survival curves when there is a time lag in the effectiveness of treatment. Biometrika, 77, 853–864.