Bernoulli

  • Bernoulli
  • Volume 21, Number 1 (2015), 401-419.

Beyond first-order asymptotics for Cox regression

Donald A. Pierce and Ruggero Bellio

Full-text: Open access

Abstract

To go beyond standard first-order asymptotics for Cox regression, we develop parametric bootstrap and second-order methods. In general, computation of $P$-values beyond first order requires more model specification than is required for the likelihood function. It is problematic to specify a censoring mechanism to be taken very seriously in detail, and it appears that conditioning on censoring is not a viable alternative to that. We circumvent this matter by employing a reference censoring model, matching the extent and timing of observed censoring. Our primary proposal is a parametric bootstrap method utilizing this reference censoring model to simulate inferential repetitions of the experiment. It is shown that the most important part of improvement on first-order methods – that pertaining to fitting nuisance parameters – is insensitive to the assumed censoring model. This is supported by numerical comparisons of our proposal to parametric bootstrap methods based on usual random censoring models, which are far more unattractive to implement. As an alternative to our primary proposal, we provide a second-order method requiring less computing effort while providing more insight into the nature of improvement on first-order methods. However, the parametric bootstrap method is more transparent, and hence is our primary proposal. Indications are that first-order partial likelihood methods are usually adequate in practice, so we are not advocating routine use of the proposed methods. It is however useful to see how best to check on first-order approximations, or improve on them, when this is expressly desired.

Article information

Source
Bernoulli, Volume 21, Number 1 (2015), 401-419.

Dates
First available in Project Euclid: 17 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1426597075

Digital Object Identifier
doi:10.3150/13-BEJ572

Mathematical Reviews number (MathSciNet)
MR3322324

Zentralblatt MATH identifier
06436799

Keywords
censoring conditional inference Cox regression higher-order asymptotics parametric bootstrap partial likelihood

Citation

Pierce, Donald A.; Bellio, Ruggero. Beyond first-order asymptotics for Cox regression. Bernoulli 21 (2015), no. 1, 401--419. doi:10.3150/13-BEJ572. https://projecteuclid.org/euclid.bj/1426597075


Export citation

References

  • [1] Alvarez-Andrade, S., Balakrishnan, N. and Bordes, L. (2009). Proportional hazards regression under progressive type-II censoring. Ann. Inst. Statist. Math. 61 887–903.
  • [2] Andersen, P.K., Borgan, Ø., Gill, R.D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. New York: Springer.
  • [3] Barndorff-Nielsen, O. (1983). On a formula for the distribution of the maximum likelihood estimator. Biometrika 70 343–365.
  • [4] Barndorff-Nielsen, O.E. (1986). Inference on full or partial parameters based on the standardized signed log likelihood ratio. Biometrika 73 307–322.
  • [5] Barndorff-Nielsen, O.E. (1991). Modified signed log likelihood ratio. Biometrika 78 557–563.
  • [6] Barndorff-Nielsen, O.E. and Cox, D.R. (1994). Inference and Asymptotics. London: Chapman & Hall.
  • [7] Brazzale, A.R., Davison, A.C. and Reid, N. (2007). Applied Asymptotics: Case Studies in Small-sample Statistics. Cambridge: Cambridge Univ. Press.
  • [8] Canty, A. and Ripley, B. (2013). boot: Bootstrap R (S-Plus) functions. R package version 1.3-9.
  • [9] Cox, D.R. (1972). Regression models and life-tables. J. Roy. Statist. Soc. Ser. B 34 187–220. With discussion and reply by D.R. Cox.
  • [10] Cox, D.R. (1975). Partial likelihood. Biometrika 62 269–276.
  • [11] Cox, D.R. and Hinkley, D.V. (1974). Theoretical Statistics. London: Chapman & Hall.
  • [12] Crowley, J. (1974). A note on some recent likelihoods leading to the log rank test. Biometrika 61 533–538.
  • [13] Davison, A.C. and Hinkley, D.V. (1997). Bootstrap Methods and Their Application. Cambridge: Cambridge Univ. Press.
  • [14] Davison, A.C., Hinkley, D.V. and Young, G.A. (2003). Recent developments in bootstrap methodology. Statist. Sci. 18 141–157.
  • [15] DiCiccio, T.J., Martin, M.A. and Stern, S.E. (2001). Simple and accurate one-sided inference from signed roots of likelihood ratios. Canad. J. Statist. 29 67–76.
  • [16] Fleming, T.R. and Harrington, D.P. (1991). Counting Processes and Survival Analysis. New York: Wiley.
  • [17] Fraser, D.A.S., Reid, N. and Wu, J. (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika 86 249–264.
  • [18] Hu, F. and Kalbfleisch, J.D. (2000). The estimating function bootstrap. Canad. J. Statist. 28 449–499. With discussion and rejoinder by the authors.
  • [19] Jiang, W. and Kalbfleisch, J.D. (2008). Permutation methods in relative risk regression models. J. Statist. Plann. Inference 138 416–431.
  • [20] Johnson, M.E., Tolley, H.D., Bryson, M.C. and Goldman, A.S. (1982). Covariate analysis of survival data: A small-sample study of Cox’s model. Biometrics 38 685–698.
  • [21] Kalbfleisch, J.D. and Prentice, R.L. (1973). Marginal likelihoods based on Cox’s regression and life model. Biometrika 60 267–278.
  • [22] Kalbfleisch, J.D. and Prentice, R.L. (2002). The Statistical Analysis of Failure Time Data, 2nd ed. Hoboken, NJ: Wiley.
  • [23] Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data, 2nd ed. Hoboken, NJ: Wiley.
  • [24] Lee, S.M.S. and Young, G.A. (2005). Parametric bootstrapping with nuisance parameters. Statist. Probab. Lett. 71 143–153.
  • [25] Mykland, P.A. (1999). Bartlett identities and large deviations in likelihood theory. Ann. Statist. 27 1105–1117.
  • [26] Pierce, D.A. and Bellio, R. (2006). Effects of the reference set on frequentist inferences. Biometrika 93 425–438.
  • [27] Pierce, D.A. and Peters, D. (1992). Practical use of higher order asymptotics for multiparameter exponential families. J. Roy. Statist. Soc. Ser. B 54 701–737. With discussion and a reply by the authors.
  • [28] Pierce, D.A. and Peters, D. (1994). Higher-order asymptotics and the likelihood principle: One-parameter models. Biometrika 81 1–10.
  • [29] R Core Team (2013). R: A language and environment for statistical computing. Available at http://www.R-project.org/.
  • [30] Samuelsen, S.O. (2003). Exact inference in the proportional hazard model: Possibilities and limitations. Lifetime Data Anal. 9 239–260.
  • [31] Sartori, N. (2003). Modified profile likelihoods in models with stratum nuisance parameters. Biometrika 90 533–549.
  • [32] Severini, T.A. (1999). An empirical adjustment to the likelihood ratio statistic. Biometrika 86 235–247.
  • [33] Severini, T.A. (2000). Likelihood Methods in Statistics. Oxford: Oxford Univ. Press.
  • [34] Severini, T.A. (2002). Modified estimating functions. Biometrika 89 333–343.
  • [35] Skovgaard, I.M. (1985). A second-order investigation of asymptotic ancillarity. Ann. Statist. 13 534–551.
  • [36] Skovgaard, I.M. (1996). An explicit large-deviation approximation to one-parameter tests. Bernoulli 2 145–165.
  • [37] Skovgaard, I.M. (2001). Likelihood asymptotics. Scand. J. Statist. 28 3–32.