Skip to main content
SpringerLink
Log in

RETRACTED ARTICLE: Positive Stable Shared Frailty Models Based on Additive Hazards

  • Published:
Statistics in Biosciences Aims and scope Submit manuscript

This article was retracted on 15 June 2023

This article has been updated

Abstract

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiment, twin or family data), the shared frailty models were suggested. These models are based on the assumption that frailty acts multiplicatively to hazard rate. In this paper, we assume that frailty acts additively to hazard rate. We introduce the positive stable shared frailty models with three different baseline distributions namely, the generalized log-logistic and the generalized Weibull distributions. We introduce the Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We apply these models to a real-life bivariate survival data set of McGilchrist and Aisbett (Biometrics 47:461–466, 1991) related to the kidney infection data and a better model is suggested for the data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

References

  1. Cox DR (1972) Regression models and life tables (with discussion). J R Stat Soc Ser B 34:187–220

    MATH  Google Scholar 

  2. Vaupel JW, Manton KG, Stallaed E (1979) The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16:439–454

    Article  Google Scholar 

  3. Clayton’s DG (1978) A model for association in bivariate life tables and its applications to epidemiological studies of familial tendency in chronic disease incidence. Biometrica 65:141–151

    Article  MathSciNet  Google Scholar 

  4. Oakes D (1982) A model for association in bivariate survival data. J R Stat Soc B 44:414–422

    MathSciNet  MATH  Google Scholar 

  5. Clayton DG, Cuzick J (1985) Multivariate generalizations of the proportional hazards model (with discussion). J R Stat Soc Ser A 148:82–117

    Article  MATH  Google Scholar 

  6. Crowder M (1985) A distributional model for repeated failure time measurements. J R Stat Soc Ser B 47:447–452

    MathSciNet  Google Scholar 

  7. Hougaard P (1986) A class of multivariate failure time distributions. Biometrika 73:671–678

    MathSciNet  MATH  Google Scholar 

  8. Hougard P (1987) Modelling multivariate survival. Scand J Stat 14(4):291–304

    MathSciNet  Google Scholar 

  9. Oakes D (1989) Bivariate survival models induced by frailties. J Am Stat Assoc 84(406):487–493

    Article  MathSciNet  MATH  Google Scholar 

  10. Genest C, Mackay J (1986) Joy of copulas: bivariate distributions with uniform marginals. Am Stat 40(4):280–283

    MathSciNet  Google Scholar 

  11. Vaupel JW (1991a) Relatives’ risks: frailty models of life history data. Theor Popul Biol 37(1):220–234

    Article  MathSciNet  MATH  Google Scholar 

  12. Vaupel JW (1991b) Kindred lifetimes: frailty models in population genetics. In: Adams J et al (eds) Convergent questions in genetics and demography. Oxford University Press, Oxford, pp 155–170

    Google Scholar 

  13. Vaupel JW, Yashin AI, Hauge M, Harvald B, Holm N, Liang X (1991) Survival analysis in genetics: Danish twins data applied to gerontological question. In: Klein JP, Goel PK (eds) Survival analysis: state of the art. Kluwer Academic Publisher, Dordrecht, pp 121–138

    Google Scholar 

  14. Vaupel JW, Yashin AI, Hauge M, Harvald B, Holm N, Liang X (1992) Strategies of modelling genetics in survival analysis. What can we learn from twins data? Paper presented on PAA meeting in Denver, CO, April 30-May 2, 1992

  15. Nielsen GG, Gill RD, Andersen PK, Sorensen TIA (1992) A counting process approach to maximum likelihood estimation in frailty models. Scand J Stat 19:25–43

    MathSciNet  MATH  Google Scholar 

  16. Hanagal DD (2005) A positive stable frailty regression model in bivariate survival data. J Indian Soc Probab Stat 9:35–44

    Google Scholar 

  17. Marshall AW, Olkin I (1967) A multivariate distribution. J Am Stat Assoc 62:30–44

    Article  MATH  Google Scholar 

  18. Hanagal DD (2006) A gamma frailty regression model in bivariate survival data. IAPQR Trans 31:73–83

    Google Scholar 

  19. Hanagal DD (2007) Gamma frailty regression models in mixture distributions. Econ Qual Control 22(2):295–302

    Article  MathSciNet  MATH  Google Scholar 

  20. Hanagal DD (2010) Modeling heterogeneity for bivariate survival data by the compound Poisson distribution with random scale. Stat Probab Lett 80:1781–1790

    Article  MathSciNet  MATH  Google Scholar 

  21. Hanagal DD, Pandey A (2014) Inverse Gaussian shared frailty for modeling kidney infection data. Adv Reliab 1:1–14

    Google Scholar 

  22. Hanagal DD, Pandey A (2015a) Gamma frailty models for bivarivate survival data. J Stat Comput Simul 85(15):3172–3189

    Article  MathSciNet  MATH  Google Scholar 

  23. Hanagal DD, Sharma R (2013) Analysis of diabetic retinopathy data using shared inverse Gaussian frailty model. Model Assist Stat Appl 8:103–119

    Google Scholar 

  24. Hanagal DD, Sharma R (2015) Analysis of bivariate survival data using shared inverse Gaussian frailty model. Commun Stat Theory Methods 44(7):1351–1380

    Article  MathSciNet  MATH  Google Scholar 

  25. Hanagal DD, Bhambure SM (2014a) Shared inverse Gaussian frailty model based on reversed hazard rate for modeling Australian twin data. J Indian Soc Probab Stat 15:9–37

    Google Scholar 

  26. Hanagal DD, Bhambure SM (2016) Modeling bivariate survival data using shared inverse Gaussian frailty model. Commun Stat Theory Methods 45(17):4969–4987

    Article  MathSciNet  MATH  Google Scholar 

  27. Hanagal DD, Pandey A (2015b) Inverse Gaussian shared frailty models with generalized exponential and generalized inverted exponential as baseline distributions. J Data Sci 13(2):569–602

    Google Scholar 

  28. Hanagal DD, Pandey A (2016a) Inverse Gaussian shared frailty models based on reversed hazard rate. Model Assist Stat Appl 11:137–151

    MATH  Google Scholar 

  29. Hanagal DD, Pandey A (2017a) Correlated gamma frailty models for bivariate survival data based on reversed hazard rate. Int J Data Sci 2(4):301–324

    Article  Google Scholar 

  30. Hanagal DD, Pandey A, Ganguly A (2017) Correlated gamma frailty models for bivariate survival data. Commun Stat Simul Comput 46(5):3627–3644

    MathSciNet  MATH  Google Scholar 

  31. Hanagal DD, Pandey A (2016b) Shared gamma frailty models based on additive hazards. J Indian Soc Probab Stat 17(2):161–184

    Article  Google Scholar 

  32. Hanagal DD, Pandey A (2017b) Shared inverse Gaussian frailty models based on additive hazards. Commun Stat Theory Methods 46(22):11143–11162

    Article  MathSciNet  MATH  Google Scholar 

  33. Hanagal DD (2011) Modeling survival data using frailty models. Chapman & Hall/CRC, Boca Rotan

    Book  MATH  Google Scholar 

  34. Hanagal DD (2017) Frailty models in public health. In: Rao ASRS, Pyne S, Rao CR (eds) Handbook of statistics, vol 37(B). Elsevier Publishers, Amsterdam, pp 209–247

    Google Scholar 

  35. Hanagal DD (2019) Modeling survival data using frailty models, 2nd edn. Springer, Singapore

    Book  MATH  Google Scholar 

  36. Lin DY, Ying Z (1994) Semiparametric analysis of the additive risk model. Biometrika 81(1):61–71

    Article  MathSciNet  MATH  Google Scholar 

  37. Bin H (2010) Additive hazards model with time-varying regression coefficients. Acta Math Sci 30B(4):1318–1326

    Article  MathSciNet  MATH  Google Scholar 

  38. Hosmer DW, Royston P (2002) Using Aalen’s linear hazards model to investigate time-varying effects in the proportional hazards regression model. Strata J 2(4):331–350

    Google Scholar 

  39. O’Neill TJ (1986) Inconsistency of misspecified proportional hazards model. Stat Probab Lett 4:219–222

    Article  MathSciNet  MATH  Google Scholar 

  40. Yin G, Cai J (2004) Additive hazards model with multivariate failure time data. Biometrika 91(4):801–818

    Article  MathSciNet  MATH  Google Scholar 

  41. Xie X, Strickler HD, Xue X (2013) Additive hazard regression models: an application to the natural history of human papillomavirus. Comput Math Methods Med 2013:1–7

    Article  MathSciNet  Google Scholar 

  42. Vahidpour M (2016). Cure rate models. Ph.D. dissertation, Ecole Polytechnique de Montreal

  43. Nelson W (1982) Applied life data analysis. Wiley, New York

    Book  MATH  Google Scholar 

  44. Boag JW (1949) Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J R Stat Soc B 11:15–53

    MATH  Google Scholar 

  45. Aalen OO (1980) A model for non-parametric regression analysis of counting processes. Lect Not Stat 2:1–25

    Article  MathSciNet  MATH  Google Scholar 

  46. Aalen OO (1989) A linear regression model for the analysis of lifetimes. Stat Med 8(8):907–925

    Article  Google Scholar 

  47. Shih JH (1998) A goodness-of-fit test for association in a bivariate survival model. Biometrika 85:189–200

    Article  MathSciNet  MATH  Google Scholar 

  48. Glidden DV (1999) Checking the adequacy of the gamma frailty model for multivariate failure times. Biometrika 86:381–94

    Article  MathSciNet  MATH  Google Scholar 

  49. Fan JJ, Hsu L, Prentice RL (2000) Dependence estimation over a finite bivariate failure region. Lifetime Data Anal 6:343–55

    Article  MathSciNet  MATH  Google Scholar 

  50. Hougaard P (2000) Analysis of multivariate survival data. Springer, New York

    Book  MATH  Google Scholar 

  51. Fine JP, Glidden DV, Lee KE (2003) A simple estimator for a shared frailty regression model. J R Stat Soc B 65:317–329

    Article  MathSciNet  MATH  Google Scholar 

  52. Hanagl DD, Bhambure SM (2014b) Analysis of kidney infection data using shared positive stable frailty models. Adv Reliab 1:21–39

    Google Scholar 

  53. Hanagal DD, Kamble AT (2014) Bayesian estimation in shared positive stable frailty models. J Data Sci 13:615–640

    Google Scholar 

  54. Hanagal DD, Bhambure SM (2017) Modeling Australian twin data using shared positive stable frailty models based on reversed hazard rate. Commun Stat Theory Methods 46(8):3754–3771

    Article  MathSciNet  MATH  Google Scholar 

  55. Bacon RW (1993) A note on the use of the log-logistic functional form for modeling saturation effects. Oxf Bull Econ Stat 55:355–361

    Article  Google Scholar 

  56. Deshpande JV, Purohit SG (2006) Life time data: statistical models and methods. World Scientific, New Jersey

    Book  MATH  Google Scholar 

  57. Govind MS, Srivastava DK (1993) Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Trans Reliab 42(2):299–302

    Article  MATH  Google Scholar 

  58. Govind MS, Kumar SD, Georgia KD (1995) The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics 37(4):436–445

    Article  MATH  Google Scholar 

  59. Govind MS, Kumar SD, Kollia GD (1996) A generalization of the Weibull distribution with application to the analysis of survival data. J Am Stat Assoc 91(436):1575–1583

    Article  MathSciNet  MATH  Google Scholar 

  60. Kheiri S, Kimber A, Meshkani MR (2007) Bayesian analysis of an inverse Gaussian correlated frailty model. Comput Stat Data Anal 51:5317–5326

    Article  MathSciNet  MATH  Google Scholar 

  61. Ibrahim JG, Chen M-H, Sinha D (2001) Bayesian survival analysis. Springer, New York

    Book  MATH  Google Scholar 

  62. Santos CA, Achcar JA (2010) A Bayesian analysis for multivariate survival data in the presence of covariates. J Stat Theory Appl 9:233–253

    MathSciNet  Google Scholar 

  63. Geweke J (1992) Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In: Bernardo JM, Berger J, Dawid AP, Smith AFM (eds) Bayesian statistics, vol 4. Oxford University Press, Oxford, pp 169–193

    Google Scholar 

  64. Gelman A, Rubin DB (1992) A single series from the Gibbs sampler provides a false sense of security. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics, vol 4. Oxford University Press, Oxford, pp 625–632

    Google Scholar 

  65. McGilchrist CA, Aisbett CW (1991) Regression with frailty in survival analysis. Biometrics 47:461–466

    Article  Google Scholar 

Download references

Acknowledgements

I thank both the referees and the associate editor for the valuable and constructive suggestions and comments which improved the earlier version of the manuscript.

Author information

Authors and Affiliations

  1. Department of Statistics, Savitribai Phule Pune University, Pune, 411007, India

    David D. Hanagal

Authors
  1. David D. Hanagal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David D. Hanagal.

Additional information

This article has been retracted. Please see the retraction notice for more detail:https://doi.org/10.1007/s12561-023-09380-y

Appendix: Summary of Tables

Appendix: Summary of Tables

See Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10

Table 1 Baseline distribution generalized logistic distribution Model-I with positive stable frailty (simulation for Model I)
Full size table
Table 2 Baseline distribution generalized Weibull distribution Model III with positive stable frailty (simulation for Model III)
Full size table
Table 3 p Values of K–S statistics for goodness of fit test for kidney infection data set
Full size table
Table 4 Posterior summary for kidney infection data set Model I
Full size table
Table 5 Posterior summary for kidney infection data set Model II
Full size table
Table 6 Posterior summary for kidney infection data set Model III
Full size table
Table 7 Posterior summary for kidney infection data set Model IV
Full size table
Table 8 Comparison of AIC, BIC, and DIC values in different frailty models based on additive hazards
Full size table
Table 9 Bayes factors for four models
Full size table
Table 10 Comparison of AIC, BIC, and DIC values in the positive stable frailty models
Full size table

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hanagal, D.D. RETRACTED ARTICLE: Positive Stable Shared Frailty Models Based on Additive Hazards. Stat Biosci 13, 431–453 (2021). https://doi.org/10.1007/s12561-020-09299-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12561-020-09299-8

Keywords

Mathematics Subject Classification

Search

Navigation