Statistical Papers

, Volume 50, Issue 3, pp 481–499 | Cite as

Inference in mixed proportional hazard models with K random effects

Regular Article


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.


EM algorithm Penalized likelihood Partial likelihood Frailties Duration analysis 


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  1. 1.
    Abbring J, Van den Berg G (2006) The unobserved heterogeneity distribution in duration analysis. Discussion Paper 059/3, Tinbergen InstituteGoogle Scholar
  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 CrossRefMathSciNetGoogle Scholar
  3. 3.
    Boockmann B (2001). The ratification of ILO conventions: a hazard rate analysis. Econ Polit 13: 281–309 CrossRefGoogle 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 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cox D (1972). Regression models and life tables. J R Stat Soc Ser B 34: 187–220 MATHGoogle 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 MATHCrossRefMathSciNetGoogle 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, ZEWGoogle Scholar
  11. 11.
    Lancaster T (1990). The Econometric Analysis of Transition Data. Cambridge University Press, Cambridge MATHGoogle 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 MATHMathSciNetGoogle 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 MATHCrossRefMathSciNetGoogle 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 MATHMathSciNetGoogle Scholar
  15. 15.
    Moulton B (1986). Random group effects and the precision of regression estimates. J Econ 32: 385–397 MATHGoogle 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 CrossRefGoogle Scholar
  19. 19.
    Parner E (1997) Inference in semiparametric frailty models. Ph.D. thesis, University of AarhusGoogle Scholar
  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 MATHCrossRefMathSciNetGoogle 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 MATHCrossRefGoogle Scholar
  22. 22.
    Team RDC (2005) R: A language and environment for statistical computing, R Foundation for Statistical Computing. Vienna, Austria, ISBN 3-900051-07-0Google Scholar
  23. 23.
    Therneau T, Grambsch P and Pankratz V (2003). Penalized survival models and frailty. J Comput Graph Stat 12: 156–175 CrossRefMathSciNetGoogle 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 MATHCrossRefMathSciNetGoogle 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 MATHCrossRefMathSciNetGoogle 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 MATHCrossRefMathSciNetGoogle 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 MATHCrossRefMathSciNetGoogle 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 MATHCrossRefMathSciNetGoogle 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 CrossRefGoogle 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 MATHCrossRefGoogle Scholar
  32. 32.
    Yau K (2001). Multilevel models for survival analysis with random effects. Biometrics 57: 96–102 CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.BETAUniversity Louis Pasteur (Strasbourg I)Strasbourg CedexFrance

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