Electronic Journal of Statistics

Bootstrapping a change-point Cox model for survival data

Gongjun Xu, Bodhisattva Sen, and Zhiliang Ying

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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

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

First available in Project Euclid: 20 August 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62N02: Estimation 62G09: Resampling methods

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


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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  • 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 Unpublished technical report, University of California, Berkeley.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.