Journal of Statistical Theory and Practice

, Volume 12, Issue 1, pp 23–41 | Cite as

Assessing covariate effects using Jeffreys-type prior in the Cox model in the presence of a monotone partial likelihood

  • Jing Wu
  • Mário de Castro
  • Elizabeth D. Schifano
  • Ming-Hui ChenEmail author


In medical studies, the monotone partial likelihood is frequently encountered in the analysis of time-to-event data using the Cox model. For example, with a binary covariate, the subjects can be classified into two groups. If the event of interest does not occur (zero event) for all the subjects in one of the groups, the resulting partial likelihood is monotone and consequently, the covariate effects are difficult to estimate. In this article, we develop both Bayesian and frequentist approaches using a data-dependent Jeffreys-type prior to handle the monotone partial likelihood problem. We first carry out an in-depth examination of the conditions of the monotone partial likelihood and then characterize sufficient and necessary conditions for the propriety of the Jeffreys-type prior. We further study several theoretical properties of the Jeffreys-type prior for the Cox model. In addition, we propose two variations of the Jeffreys-type prior: the shifted Jeffreys-type prior and the Jeffreys-type prior based on the first risk set. An efficient Markov-chain Monte Carlo algorithm is developed to carry out posterior computation. We perform extensive simulations to examine the performance of parameter estimates and demonstrate the applicability of the proposed method by analyzing real data from the SEER prostate cancer study.


Bayesian estimates cause-specific hazards model first risk set penalized maximum likelihood shifted Jeffreys-type prior zero events 

AMS Subject Classification



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  1. Beyersmann, J., and T. Scheike. 2013. Classical regression models for competing risks. In ed. J. P. Klein, H. C. van Houwelingen, J. G. Ibrahim, and Τ. Η. Scheike, Handbook of survival analysis, 157–77. Boca Raton, FL: Chapman & Hall/CRC.Google Scholar
  2. Bryson, M. C., and Μ. Ε. Johnson. 1981. The incidence of monotone likelihood in the Cox model. Technometrics 23:381–83.CrossRefGoogle Scholar
  3. Chen, M.-H., Q.-M. Shao, and J. G. Ibrahim. 2000. Monte Carlo methods in Bayesian computation. New York, NY: Springer.CrossRefGoogle Scholar
  4. Chen, M.-H., J. G. Ibrahim, and Q.-M. Shao. 2006. Posterior propriety and computation for the Cox regression model with applications to missing covariates. Biometrika 93:791–807.MathSciNetCrossRefGoogle Scholar
  5. Chen, M.-H., J. G. Ibrahim, and Q.-M. Shao. 2009. Maximum likelihood inference for the Cox regression model with applications to missing covariates. Journal of Multivariate Analysis 100: 2018–30.MathSciNetCrossRefGoogle Scholar
  6. Chen, M.-H., Μ. de Castro, Μ. Ge, and Y. Zhang. 2013. Bayesian regression models for competing risks. In ed. J. P. Klein, H. C. van Houwelingen, J. G. Ibrahim, and Τ. Η. Scheike, Handbook of survival analysis, 179–98. Boca Raton, FL: Chapman & Hall/CRC.Google Scholar
  7. Cox, D. R. 1972. Regression models and life-tables. Journal of the Royal Statistical Society Β 34:187–220.MathSciNetzbMATHGoogle Scholar
  8. Cox, D. R. 1975. Partial likelihood. Biometrika 62:269–76.MathSciNetCrossRefGoogle Scholar
  9. Fine, J., and Β. Η. Lindqvist. 2014. Competing risks. Lifetime Data Analysis 20:159–60.MathSciNetCrossRefGoogle Scholar
  10. Firth, D. 1993. Bias reduction of maximum likelihood estimates. Biometrika 80:27–38.MathSciNetCrossRefGoogle Scholar
  11. Gaynor, J. J., E. J. Feuer, C. C. Tan, D. H. Wu, C. R Little, D. J. Straus, B. D. Clarkson, and M. F. Brennan. 1993. On the use of cause-specific failure and conditional failure probabilities: Examples from clinical oncology data. Journal of the American Statistical Association 88:400–9.CrossRefGoogle Scholar
  12. Ge, M., and M.-H. Chen. 2012. Bayesian inference of the fully specified subdistribution model for survival data with competing risks. Lifetime Data Analysis 18:339–63.MathSciNetCrossRefGoogle Scholar
  13. Greenland, S., and M. A. Mansournia. 2015. Penalization, bias reduction, and default priors in logistic and related categorical and survival regressions. Statistics in Medicine 34:3133–43.MathSciNetCrossRefGoogle Scholar
  14. Heinze, G., and D. Dunkler. 2008. Avoiding infinite estimates of time-dependent effects in small-sample survival studies. Statistics in Medicine 27:6455–69.MathSciNetCrossRefGoogle Scholar
  15. Heinze, G., and M. Ploner. 2002. SAS and SPLUS programs to perform Cox regression without convergence problems. Computer Methods and Programs in Biomedicine 67:217–23.CrossRefGoogle Scholar
  16. Heinze, G., and M. Schemper. 2001. A solution to the problem of monotone likelihood in Cox regression. Biometrics 57:114–19.MathSciNetCrossRefGoogle Scholar
  17. Ibrahim, J. G., and P. W. Laud. 1991. On Bayesian analysis of generalized linear models using Jeffreys’s prior. Journal of the American Statistical Association 86:981–86.MathSciNetCrossRefGoogle Scholar
  18. Kalbfleisch, J. D. 1978. Non-parametric Bayesian analysis of survival time data. Journal of the Royal Statistical Society Β 40:214–21.MathSciNetzbMATHGoogle Scholar
  19. Kalbfleisch, J. D., and R L. Prentice. 2011. The statistical analysis of failure time data, 2nd ed. Hoboken, NJ: Wiley.zbMATHGoogle Scholar
  20. Ploner, M., and G. Heinze. 2010. coxphf: Cox regression with Firth’s penalized likelihood. R package version 1.05
  21. Roberts, G. O., and J. S. Rosenthal. 2009. Examples of adaptive MCMC. Journal of Computational and Graphical Statistics 18:349–67.MathSciNetCrossRefGoogle Scholar
  22. Roy, V., and J. P. Hobert. 2007. Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression. Journal of the Royal Statistical Society B, 69:607–623.MathSciNetCrossRefGoogle Scholar
  23. Sinha, D., J. G. Ibrahim, and M.-H. Chen. 2003. A Bayesian justification of Cox’s partial likelihood. Biometrika 90:629–41.MathSciNetCrossRefGoogle Scholar
  24. Tierney, L. 1994. Markov chains for exploring posterior distributions. Annals of Statistics 22:1701–62.MathSciNetCrossRefGoogle Scholar
  25. Zhang, F. 1999. Matrix theory. Basic results and techniques. New York, NY: Springer.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing, 20 Middlefield Ct, Greensboro, NC 27455 2018

Authors and Affiliations

  • Jing Wu
    • 1
  • Mário de Castro
    • 2
  • Elizabeth D. Schifano
    • 1
  • Ming-Hui Chen
    • 1
    Email author
  1. 1.Department of StatisticsUniversity of ConnecticutStorrsUSA
  2. 2.Universidade de São PauloInstituto de Ciências Matemáticas e de ComputaçãoSão CarlosBrazil

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