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Inference in mixed proportional hazard models with K random effects

Abstract

A general formulation of mixed proportional hazard models with K random effects is provided. It enables to account for a population stratified at K different levels. I then show how to approximate the partial maximum likelihood estimator using an EM algorithm. In a Monte Carlo study, the behavior of the estimator is assessed and I provide an application to the ratification of ILO conventions. Compared to other procedures, the results indicate an important decrease in computing time, as well as improved convergence and stability.

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References

  1. 1.

    Abbring J, Van den Berg G (2006) The unobserved heterogeneity distribution in duration analysis. Discussion Paper 059/3, Tinbergen Institute

  2. 2.

    Bolstad W and Manda S (2001). Investigating child mortality in Malawi using family and community random effects: a Bayesian analysis. J Am Stat Assoc 96: 12–19

    Article  MathSciNet  Google Scholar 

  3. 3.

    Boockmann B (2001). The ratification of ILO conventions: a hazard rate analysis. Econ Polit 13: 281–309

    Article  Google Scholar 

  4. 4.

    Clayton D and Cuzick J (1985). Multivariate generalizations of the proportional hazards model. J R Stat Soc Ser A 148: 82–108

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

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

    MATH  Google Scholar 

  6. 6.

    de Montricher G, Tapia R and Thompson J (1975). Nonparametric maximum likelihood estimation of probability densities by penalty function methods. Ann Stat 3: 1329–1348

    MATH  Article  MathSciNet  Google Scholar 

  7. 7.

    Feller W (1971). An introduction to probability theory and its applications, vol 2, 2nd edn. Wiley, New York

    Google Scholar 

  8. 8.

    Gill R (1985). Discussion of the paper by D. Clayton and J. Cuzick. J R Stat Soc Ser A 148: 108–109

    Google Scholar 

  9. 9.

    Gouriéroux C and Peaucelle I (1990). Hétérogénéité I. Étude des biais d’estimation dans le cas linéaire. Annales d’Économie et de Statistique 17: 163–184

    Google Scholar 

  10. 10.

    Horny G, Boockmann B, Djurdjevic D, Laisney F (2005) Bayesian estimation of Cox models with non-nested random effects: An application to the ratification of ILO conventions by developing countries. Discussion Paper 05–23, ZEW

  11. 11.

    Lancaster T (1990). The Econometric Analysis of Transition Data. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  12. 12.

    Louis T (1982). Finding the observed information matrix when using the EM algorithm. J R Stat Soc Ser B 44: 226–233

    MATH  MathSciNet  Google Scholar 

  13. 13.

    Manda S and Meyer R (2005). Age at first marriage in Malawi: a Bayesian multilevel analysis using a discrete time model. J R Stat Soc Ser A 168: 439–455

    MATH  Article  MathSciNet  Google Scholar 

  14. 14.

    Meilijson I (1989). A fast improvement to the EM algorithm on its own terms. J R Stat Soc Ser B 51: 127–138

    MATH  MathSciNet  Google Scholar 

  15. 15.

    Moulton B (1986). Random group effects and the precision of regression estimates. J Econ 32: 385–397

    MATH  Google Scholar 

  16. 16.

    Nelson W (1969). Hazard plotting for incomplete failure time data. J Qual Technol 1: 27–52

    Google Scholar 

  17. 17.

    Ng S, Krishnan T and McLachlan G (2004). Handbook of Computational Statistics Chap. The EM Algorithm. Springer, Heidelberg

    Google Scholar 

  18. 18.

    Pakes A (1983). On group effects and errors in variables in aggregation. Rev Econ Stat 65: 168–173

    Article  Google Scholar 

  19. 19.

    Parner E (1997) Inference in semiparametric frailty models. Ph.D. thesis, University of Aarhus

  20. 20.

    Rondeau V, Commenges D and Joly P (2003). Maximum penalized likelihood estimation in a gamma-frailty model. Lifetime Data Anal 9: 139–153

    MATH  Article  MathSciNet  Google Scholar 

  21. 21.

    Sastry N (1997). A nested frailty model for survival data, with an application to thestudy of child survival in northeast Brazil. J Am Stat Assoc 92: 426–435

    MATH  Article  Google Scholar 

  22. 22.

    Team RDC (2005) R: A language and environment for statistical computing, R Foundation for Statistical Computing. Vienna, Austria, ISBN 3-900051-07-0

  23. 23.

    Therneau T, Grambsch P and Pankratz V (2003). Penalized survival models and frailty. J Comput Graph Stat 12: 156–175

    Article  MathSciNet  Google Scholar 

  24. 24.

    Van den Berg G (2001). Duration models: specification, identification and multiple durations. In: Heckman, JJ and Leamer, E (eds) Handbook of Econometrics, vol 5, Chap. 55, pp 3381–3463. Elsevier, Amsterdam

    Google Scholar 

  25. 25.

    Vu H (2003). Parametric and semiparametric conditional shared gamma frailty models with events before study entry. Commun Stat Simul Comput 32(4): 1223–1248

    MATH  Article  MathSciNet  Google Scholar 

  26. 26.

    Vu H (2004). Estimation in semiparametric conditional shared frailty models with events before study entry. Comput Stat Data Anal 45: 621–637

    MATH  Article  MathSciNet  Google Scholar 

  27. 27.

    Vu H and Knuiman M (2002). Estimation in semiparametric marginal shared gamma frailty models. Aust N Z J Stat 44(4): 489–501

    MATH  Article  MathSciNet  Google Scholar 

  28. 28.

    Vu H and Knuiman M (2002). A hybrid ML-EM algorithm for calculation of maximum likelihood estimates in semiparametric shared frailty model. Comput Stat Data Anal 40: 173–187

    MATH  Article  MathSciNet  Google Scholar 

  29. 29.

    Vu H, Segal M, Knuiman M and James I (2001). Asymptotic and small sample statistical properties of random frailty variance estimates for shared gamma frailty models. Commun Stat Simul Comput 30(3): 581–595

    MATH  Article  MathSciNet  Google Scholar 

  30. 30.

    Wei G and Tanner M (1990). Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithm. J Am Stat Assoc 85: 699–704

    Article  Google Scholar 

  31. 31.

    Xue X and Brookmeyer R (1996). Bivariate frailty model for the analysis of multivariate survival time. Lifetime Data Anal 2: 277–289

    MATH  Article  Google Scholar 

  32. 32.

    Yau K (2001). Multilevel models for survival analysis with random effects. Biometrics 57: 96–102

    Article  MathSciNet  Google Scholar 

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Correspondence to Guillaume Horny.

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Horny, G. Inference in mixed proportional hazard models with K random effects. Stat Papers 50, 481–499 (2009). https://doi.org/10.1007/s00362-007-0087-y

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Keywords

  • EM algorithm
  • Penalized likelihood
  • Partial likelihood
  • Frailties
  • Duration analysis