Article ID: MTJPAM-D-20-00025

Title: On the Estimation of Parametric Cause Specific Hazard Function with Bayesian Approach under Informative Priors

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00025; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 25-38

Document Type: Research Paper

Author(s): Navin Chandra a , Habbiburr Rehman b

aDepartment of Statistics, Ramanujan School of Mathematical Sciences, Pondicherry University, Puducherry-605 014, India

bDepartment of Statistics, Ramanujan School of Mathematical Sciences, Pondicherry University, Puducherry-605 014, India

Received: 24 August 2020, Accepted: 1 December 2020, Published: 25 April 2021.

Corresponding Author: Navin Chandra (Email address:

Full Text: PDF


In this article we analysed the competing risks data using cause specific hazard approach. We proposed the Cox proportional hazards regression technique of analysing the survival data in the presence of competing risks setting using baseline modified Weibull distribution. For estimating the cumulative cause specific hazard function we used maximum likelihood as well Bayesian methods of estimation. Under Bayesian scenario, we used three types of informative priors such as gamma, Weibull and log-normal for baseline parameters and standard normal prior for regression parameters. The Comparison of Bayes estimates is made based on two different loss functions like, squared error and LINEX loss functions. Simulation study shows the appropriate convergence and identifiability of the proposed model. The bladder cancer data is utilized for the validation of proposed study.

Keywords: Competing risks, Modified Weibull distribution, Informative priors, Squared error loss function, LINEX loss function, Makov Chain Monte Carlo simulation

  1. D. F. Andrews and A. M. Herzberg, Data: A Collection of Problems from many Fields for the Student and Research Worker, Springer Science & Business Media, 2012.
  2. S. Anjana and P. Sankaran, Parametric analysis of lifetime data with multiple causes of failure using cause specific reversed hazard rates, Calcutta Stat. Assoc. Bull. 67 (3-4), 129-142, 2015.
  3. J. Benichou and M. H. Gail, Estimates of absolute cause-specific risk in cohort studies, Biometrics 46 (03), 813-826, 1990.
  4. J. Beyersmann, A. Allignol and M. Schumacher, Competing risks and multistate models with R, Springer Science & Business Media, 2012.
  5. J. Bryant and J. J. Dignam, Semiparametric models for cumulative incidence functions, Biometrics 60 (1), 182-190, 2004.
  6. D. R. Cox, The analysis of exponentially distributed life-times with two types of failure, J. R. Stat. Soc., Ser. B 21 (2), 411-421, 1959.
  7. D. R. Cox, Regression models and life-tables, J. R. Stat. Soc., Ser. B 34 (2), 187-202, 1972.
  8. J. J. Gaynor, E. J. Feuer, C. C. Tan, et al., On the use of cause-specific failure and conditional failure probabilities: examples from clinical oncology data, J. Am. Stat. Assoc. 88 (422), 400-409, 1993.
  9. M. Ge and M. H. Chen, Bayesian inference of the fully specified subdistribution model for survival data with competing risks, Lifetime Data Anal. 18 (3) 339-363, 2012.
  10. S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Transactions on pattern analysis and machine intelligence PAMI-6 (6) 721-741, 1984.
  11. C. B. Guure and N. A. Ibrahim, Bayesian analysis of the survival function and failure rate of Weibull distribution with censored data, Math. Probl. Eng. 2012, 2012.
  12. B. Haller, G. Schmidt and K. Ulm, Applying competing risks regression models: an overview, Lifetime Data Anal. 19 (1), 33-58, 2013.
  13. W. K. Hastings, Monte carlo sampling methods using Markov chains and their applications, Biometrica 57 (1), 97-109 ,1970.
  14. X. Huang, G. Li, R. M. Elashoff, et al., A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects, Lifetime Data Anal. 17 (1), 80-100, 2011.
  15. J. H. Jeong and J. Fine, Direct parametric inference for the cumulative incidence function, J. R. Stat. Soc., Ser. C, Appl. Stat. 55 (2), 187-200, 2006.
  16. J. H. Jeong and J. P. Fine, Parametric regression on cumulative incidence function, Biostatistics 8 (2), 184-196, 2007.
  17. H. Jiang, M. Xie and L. Tang, Markov chain Monte Carlo methods for parameter estimation of the modified Weibull distribution, J. Appl. Stat. 35 (6), 647-658, 2008.
  18. J. D. Kalbfleisch and R. L. Prentice, The Statistical Analysis of Failure Time Data, John Wiley & Sons, New York, 2002.
  19. C. Lai, M. Xie and D. Murthy, A modified Weibull distribution, IEEE Transactions on reliability 52 (1), 33-37, 2003.
  20. J. F. Lawless, Parametric models in survival analysis, American Cancer Society, 2014,
    Available from:
  21. M. Lee, Parametric inference for quantile event times with adjustment for covariates on competing risks data, J. Appl. Stat. 46 (12), 2128-2144, 2019.
  22. D. Lunn, C. Jackson, N. Best, et al., The BUGS book: A Practical Introduction to Bayesian Analysis, Chapman and Hall/CRC, 2012.
  23. H. K. T. Ng, Parameter estimation for a modified Weibull distribution, for progressively type-II censored samples, IEEE Transactions on Reliability 54 (3), 374-380, 2005.
  24. R. L. Prentice, J. D. Kalbfleisch, Jr. Peterson, et al., The analysis of failure times in the presence of competing risks, Biometrics 34 (4), 541-554, 1978.
  25. A. Sen, M. Banerjee, Y. Li, et al., A Bayesian approach to competing risks analysis with masked cause of death, Statistics in medicine 29 (16), 1681-1695, 2010.
  26. S. Sinha, Bayesian Estimation, New Age International (P) Limited Publisher, 1998.
  27. E. Sreedevi and P. Sankaran, A semiparametric Bayesian approach for the analysis of competing risks data, Commun. Stat., Theory Methods 41 (15) 2803-2818, 2012.
  28. A. Tsiatis, A nonidentifiability aspect of the problem of competing risks, Proc. Nat. Acad. Sci. USA 72 (1), 20-22, 1975.
  29. S. Upadhyay and A. Gupta, A Bayes analysis of modified weibull distribution via Markov chain Monte Carlo simulation, J. Stat. Comput. Simulation 80 (3), 241-254, 2010.
  30. L. J. Wei, D. Y. Lin and L. Weissfeld, Regression analysis of multivariate incomplete failure time data by modeling marginal distributions, J. Am. Stat. Assoc. 84 (408), 1065-1073, 1989.