Lifetime Data Analysis

, Volume 11, Issue 2, pp 213–232 | Cite as

Bayesian Model Selection and Averaging in Additive and Proportional Hazards Models

  • David B. DunsonEmail author
  • Amy H. Herring


Although Cox proportional hazards regression is the default analysis for time to event data, there is typically uncertainty about whether the effects of a predictor are more appropriately characterized by a multiplicative or additive model. To accommodate this uncertainty, we place a model selection prior on the coefficients in an additive-multiplicative hazards model. This prior assigns positive probability, not only to the model that has both additive and multiplicative effects for each predictor, but also to sub-models corresponding to no association, to only additive effects, and to only proportional effects. The additive component of the model is constrained to ensure non-negative hazards, a condition often violated by current methods. After augmenting the data with Poisson latent variables, the prior is conditionally conjugate, and posterior computation can proceed via an efficient Gibbs sampling algorithm. Simulation study results are presented, and the methodology is illustrated using data from the Framingham heart study.


additive hazards Cox model gibbs sampler order restricted inference posterior probability proportional hazards survival analysis variable selection 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. O. O. Aalen, ‘‘A model for non-parametric regression analysis of counting processes,’’ in Lecture Notes in Statistics-2: Mathematical Statistics and Probability Theory, W. Klonecki, A. Kozek and J. Rosinski (eds), pp. 1–25, 1980.Google Scholar
  2. E. Beamonte and J. D. Bermúdez, ‘‘A Bayesian semiparametric analysis for additive hazard models with censored observations’’, Test vol. 12.2 pp. 101–117, 2003.Google Scholar
  3. H. Chipman, E. I. George, and R. E. McCulloch, ‘‘The practical implementation of Bayesian model selection’’, IMS Lecture Notes–Monograph Series vol. 38, 2001.Google Scholar
  4. D. Clayton, ‘‘Bayesian analysis of frailty models’’, Technical Report, Medical Research Council Biostatistics Unit, Cambridge, 1994.Google Scholar
  5. Cox, DR 1972‘‘Regression models and life-tables (with discussions)’‘Journal of the Royal Statistical Society B34187202Google Scholar
  6. Cox, DR, Oaks, D.A 1984Oakes, Analysis of Survival DataChapman and HallLondonGoogle Scholar
  7. Dawber, T. R., Meadurs, G. F., Moore, F.E.J 1951‘‘Epidemiological approaches to heart disease: The Framingham study’‘American Journal of Public Health41279286Google Scholar
  8. Dunson, DB, Herring, AH 2003‘‘Bayesian inferences in the Cox model for order restricted hypotheses’‘Biometrics59918925Google Scholar
  9. Faraggi, D, Simon, R 1998‘‘Bayesian variable selection method for censored survival data’‘ Biometrics5414751485Google Scholar
  10. George, EI, McCulloch, RE 1997‘‘Approaches for Bayesian variable selection’‘Statistica Sinica7339373Google Scholar
  11. J. Geweke, ‘‘Variable selection and model comparison in regression’’. in Bayesian Statistics vol. 5, J.O. Berger, J.M. Bernardo, A.P. Dawid, and A.F.M. Smith (eds.), Oxford University Press, pp. 609–620, 1996.Google Scholar
  12. Kass, RE, Raftery, AE 1995‘‘Bayes factors’‘Journal of the American Statistical Association90773795Google Scholar
  13. Ibrahim, J.G., Chen, M.-H., MacEachern, S.N. 1999‘‘Bayesian variable selection for proportional hazards models’‘Canadian Journal of Statistics27701717Google Scholar
  14. Ibrahim, J.G., Chen, M.-H., MacEachern, S.N. 2001Bayesian Survival AnalysisSpringerNew YorkGoogle Scholar
  15. Ibrahim, JG, Chen, MH, Sinha, D 2001‘‘Criterion-based methods for Bayesian model assessment’‘Statistica Sinica 11419443Google Scholar
  16. Lin, DY, Ying, ZL 1994‘‘Semiparametric analysis of the additive risk model’‘Biometrika816171Google Scholar
  17. Lin, DY, Ying, ZL 1995‘‘Semiparametric analysis of general additive-multiplicative hazard models for counting processes’‘Annals of Statistics2317121734Google Scholar
  18. Martinussen, T, Scheike, TH 2002‘‘A flexible additive multiplicative hazard model’‘Biometrika89283298Google Scholar
  19. McKeague, IW, Sasieni, PD 1994‘‘A partly parametric additive risk model’‘Biometrika81501514Google Scholar
  20. Scheike, TH, Zhang, MJ 2002‘‘An additive-multiplicative Cox-Aalen regression model’‘Scandinavian Journal of Statistics297588Google Scholar
  21. Sinha, D, Chen, MH, Ghosh, SK 1999‘‘Bayesian analysis and model selection for interval-censored survival data’‘Biometrics55585590Google Scholar
  22. Volinsky, CT, Madigan, D, Raftery, AE, Kronmal, RA 1997‘‘Bayesian model averaging in proportional hazards: Assessing the risk of a stroke’‘Applied Statistics46433448Google Scholar
  23. Volinsky, CT, Raftery, AE 2000‘‘Bayesian information criterion for censored survival models’‘Biometrics56256262Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Biostatistics BranchNational Institute of Environmental Health SciencesResearch Triangle ParkUSA
  2. 2.Department of BiostatisticsThe University of North CarolinaChapel HillUSA

Personalised recommendations