Lifetime Data Analysis

, 3:13

A Flexible Approach to Time-varying Coefficients in the Cox Regression Setting

  • Daniel J. Sargent
Article

Abstract

Research on methods for studying time-to-event data (survival analysis) has been extensive in recent years. The basic model in use today represents the hazard function for an individual through a proportional hazards model (Cox, 1972). Typically, it is assumed that a covariate's effect on the hazard function is constant throughout the course of the study. In this paper we propose a method to allow for possible deviations from the standard Cox model, by allowing the effect of a covariate to vary over time. This method is based on a dynamic linear model. We present our method in terms of a Bayesian hierarchical model. We fit the model to the data using Markov chain Monte Carlo methods. Finally, we illustrate the approach with several examples.

Hierarchical models Markov chain Monte Carlo Dynamic linear model Smoothing Survival analysis 

References

  1. O.O. Aalen, “Nonparametric inference for a family of counting processes,” Annals of Statistics 6, 701–726, 1978.MATHMathSciNetGoogle Scholar
  2. H. Akaike, “Information theory and an extension of the entropy maximization principle”, in Proceedings of the Second International Symposium on Information Theory, eds. B.N. Petrov and R. Csak. Kiado: Akademica, 1973.Google Scholar
  3. P.K. Anderson and R.D. Gill, “Cox's regression model for counting processes: A large sample study,” Annals of Statistics 10, 1100–1120, 1982.MathSciNetGoogle Scholar
  4. D.R. Cox, “Regression models and life tables (with discussion),” Journal of the Royal Statistical Society, Series B 34, 187–220, 1972.MATHGoogle Scholar
  5. D.R. Cox, “Partial Likelihood,” Biometrika 62, 269–275, 1975.MATHMathSciNetCrossRefGoogle Scholar
  6. B. Efron, “The efficiency of Cox's likelihood function for censored data,” Journal of the American Statistical Association 72, 557–565, 1977.MATHMathSciNetCrossRefGoogle Scholar
  7. D. Gamerman, “Dynamic Bayesian models for survival data,” Applied Statistics 40, 63–79, 1991.MATHCrossRefGoogle Scholar
  8. A.E. Gelfand and A.F.M. Smith, “Sampling based approaches to calculating marginal densities,” Journal of the American Statistical Association 85, 398–409, 1990.MATHMathSciNetCrossRefGoogle Scholar
  9. A. Gelman, G. Roberts, W. Gilks, “Efficient Metropolis jumping rules,” in Bayesian Statistics 5, eds. J.O. Berger, J.M. Bernardo, A.P. Dawid, A.F.M. Smith, Oxford: University Press, 1996.Google Scholar
  10. A. Gelman and D.B. Rubin, “Inference from iterative simulation using multiple sequences (with discussion),” Statistical Science 7, 457–511, 1992.Google Scholar
  11. I.J. Good and R.A Gaskins, “Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data (with discussion),” Journal of the American Statistical Association 75, 42–56, 1980.MATHMathSciNetCrossRefGoogle Scholar
  12. P.M. Grambsch and T.M. Therneau, “Proportional hazards tests and diagnostics based on weighted residuals,” Biometrika 81, 515–526, 1994.MATHMathSciNetCrossRefGoogle Scholar
  13. R.J. Gray, “Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis,” Journal of the American Statistical Association 87, 942–951, 1992.CrossRefGoogle Scholar
  14. R.J. Gray, “Spline-based tests in survival analysis,” Biometrics 50, 640–652, 1994.MATHMathSciNetCrossRefGoogle Scholar
  15. T.J. Hastie and R.J. Tibshirani, Generalized additive models, New York: Chapman and Hall, 1990.MATHGoogle Scholar
  16. T.J. Hastie and R.J. Tibshirani, “Varying-coefficient models,” Journal of the Royal Statistical Society, Series B 55, 86-95, 1993.MathSciNetGoogle Scholar
  17. S. Johansen, “An extension of Cox's regression model,” International Statistical Review, 51, 165-174, 1983.MATHMathSciNetCrossRefGoogle Scholar
  18. J.D. Kalbfleisch, “Nonparametric Bayesian analysis of survival time data,” Journal of the Royal Statistical Society, Series B 40, 214–221, 1978.MATHMathSciNetGoogle Scholar
  19. J.D. Kalbfleisch and R.L. Prentice, The Statistical Analysis of Failure Time Data. New York: Wiley, 1978.Google Scholar
  20. D.Y. Lin, “Goodness-of-fit for the Cox regression model based on a class of parameter estimators,” Journal of the American Statistical Association 86, 725–728, 1991.MATHMathSciNetCrossRefGoogle Scholar
  21. D.V. Lindley and A.F.M Smith, “Bayes estimates for the linear model (with discussion),” Journal of the Royal Statistical Society, Series B, 34, 1–41, 1972.MATHMathSciNetGoogle Scholar
  22. P. Müller, “A generic approach to posterior integration and Gibbs sampling,” Technical Report 91-009, Department of Statistics, Purdue University.Google Scholar
  23. F. O'sullivan, “Nonparametric estimation in the Cox model,” The Annals of Statistics 21, 124–145, 1993.MATHMathSciNetGoogle Scholar
  24. D.J. Sargent, “A general framework for random effects survival analysis in the Cox proportional hazards setting,” Research Report 95–004, Division of Biostatistics, University of Minnesota, 1995.Google Scholar
  25. D. Schoenfeld, “Partial residuals for the proportional hazards regression model,” Biometrika 69, 239–241, 1982.CrossRefGoogle Scholar
  26. A.F.M. Smith and G.O. Roberts, “Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion),” Journal of the Royal Statistical Society, Series B 55, 3–23, 1993.MATHMathSciNetGoogle Scholar
  27. T.M. Therneau, “A package of survival functions for S,” Technical Report No. 53, Section of Biostatistics, Mayo Clinic, 1994.Google Scholar
  28. L. Tierney, “Markov chains for exploring posterior distributions (with discussion),” Annals of Statistics 22, 1701–1762, 1994.MATHMathSciNetGoogle Scholar
  29. P.J.M. Verweij and H.C. van Houwelingen, “Time-dependent effects of fixed covariates in Cox regression,” Biometrics 51 1550–1556, 1995.MATHCrossRefGoogle Scholar
  30. M. West, P.J. Harrison, H.S. Migon, “Dynamic generalized linear models and Bayesian forecasting (with discussion),” Journal of the American Statistical Association 80, 73–97, 1985.MATHMathSciNetCrossRefGoogle Scholar
  31. D.M. Zucker and A.F. Karr, “Nonparametric survival analysis with time-dependent covariate effects: a penalized partial likelihood approach,” The Annals of Statistics 18, 329–353, 1990.MATHMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Daniel J. Sargent
    • 1
  1. 1.Cancer Center Statistics, Plummer 4Mayo ClinicRochester

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