The Annals of Statistics

Bayesian bootstrap for proportional hazards models

Yongdai Kim and Jaeyong Lee

Full-text: Open access

Abstract

We propose two Bayesian bootstrap extensions, the binomial and Poisson forms, for proportional hazards models. The binomial form Bayesian bootstrap is the limit of the posterior distribution with a beta process prior as the amount of the prior information vanishes, and thus can be considered as a default nonparametric Bayesian analysis. It is also the same as Lo's Bayesian bootstrap for censored data when covariates are absent. The Poisson form Bayesian bootstrap is equivalent to the Bayesian analysis with Cox's profile likelihood. When the baseline distribution is discrete, thus when the data set has many ties, simulation study suggests that the binomial form Bayesian bootstrap performs better than standard frequentist procedures in the frequentist sense. An advantage of the proposed Bayesian bootstrap procedures over the standard Bayesian analysis is conceptual and computational simplicity. Finally, it is shown that both Bayesian bootstrap posteriors are asymptotically equivalent to the sampling distribution of the maximum likelihood estimator.

Article information

Source
Ann. Statist., Volume 31, Number 6 (2003), 1905-1922.

Dates
First available in Project Euclid: 16 January 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1074290331

Digital Object Identifier
doi:10.1214/aos/1074290331

Mathematical Reviews number (MathSciNet)
MR2036394

Zentralblatt MATH identifier
1042.62030

Subjects
Primary: 62F40: Bootstrap, jackknife and other resampling methods
Secondary: 62F15: Bayesian inference

Keywords
Empirical likelihood survival model beta process

Citation

Kim, Yongdai; Lee, Jaeyong. Bayesian bootstrap for proportional hazards models. Ann. Statist. 31 (2003), no. 6, 1905--1922. doi:10.1214/aos/1074290331. https://projecteuclid.org/euclid.aos/1074290331


Export citation

References

  • Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
  • Choudhuri, N. (1998). Bayesian bootstrap credible sets for multidimensional mean functional. Ann. Statist. 26 2104--2127.
  • Cox, D. R. (1972). Regression models and life tables (with discussion). J. Roy. Statist. Soc. Ser. B 34 187--220.
  • Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1--26.
  • Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209--230.
  • Gasparini, M. (1995). Exact multivariate Bayesian bootstrap distributions of moments. Ann. Statist. 23 762--768.
  • Gilks, W. R. and Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Appl. Statist. 41 337--348.
  • Hjort, N. L. (1990). Nonparametric Bayes estimators based on beta processes in models for life history data. Ann. Statist. 18 1259--1294.
  • Jacobsen, M. (1989). Existence and unicity of MLEs in discrete exponential family distributions. Scand. J. Statist. 16 335--349.
  • James, L. F. (1997). A study of a class of weighted bootstraps for censored data. Ann. Statist. 25 1595--1621.
  • Kalbfleisch, J. D. (1978). Nonparametric Bayesian analysis of survival time data. J. Roy. Statist. Soc. Ser. B 40 214--221.
  • Kalbfleisch, J. D. and Prentice, R. L. (1980). The Statistical Analysis of Failure Time Data. Wiley, New York.
  • Kim, Y. and Lee, J. (2001). On posterior consistency of survival models. Ann. Statist. 29 666--686.
  • Laud, P. W., Damien, P. and Smith, A. F. M. (1998). Bayesian nonparametric and covariate analysis of failure time data. In Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statist. 133 213--225. Springer, New York.
  • Lazar, N. A. (2000). Bayesian empirical likelihood. Technical Report 721, Dept. Statistics, Carnegie Mellon Univ.
  • Lo, A. Y. (1987). A large-sample study of the Bayesian bootstrap. Ann. Statist. 15 360--375.
  • Lo, A. Y. (1988). A Bayesian bootstrap for a finite population. Ann. Statist. 16 1684--1695.
  • Lo, A. Y. (1993). A Bayesian bootstrap for censored data. Ann. Statist. 21 100--123.
  • Mason, D. M. and Newton, M. A. (1992). A rank statistics approach to the consistency of a general bootstrap. Ann. Statist. 20 1611--1624.
  • Owen, A. B. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18 90--120.
  • Peto, R. (1972). Contribution to the discussion of ``Regression models and life tables,'' by D. R. Cox. J. Roy. Statist. Soc. Ser. B 34 205--207.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • Præstgaard, J. and Wellner, J. A. (1993). Exchangeably weighted bootstraps of the general empirical process. Ann. Probab. 21 2053--2086.
  • Prentice, R. L. and Gloeckler, L. A. (1978). Regression analysis of grouped survival data with application to breast cancer data. Biometrics 34 57--67.
  • Rubin, D. B. (1981). The Bayesian bootstrap. Ann. Statist. 9 130--134.
  • Sethuraman, J. (1961). Some limit theorems for joint distributions. Sankyhā Ser. A 23 379--386.
  • Tsiatis, A. A. (1981). A large sample study of Cox's regression model. Ann. Statist. 9 93--108.
  • Volinsky, T. C., Madigan, D., Raftery, A. E. and Kronmal, R. A. (1996). Bayesian model averaging in proportional hazard models: Assessing stroke risk. Technical Report 302, Dept. Statistics, Univ. Washington.
  • Wellner, J. A. and Zhan, Y. (1996). Bootstrapping $Z$-estimators. Technical Report 308, Dept. Statistics, Univ. Washington.
  • Weng, C.-S. (1989). On a second-order asymptotic property of the Bayesian bootstrap mean. Ann. Statist. 17 705--710.