Additive Hazards Regression Models for Survival Data

  • D. Y. Lin
  • Zhiliang Ying
Part of the Lecture Notes in Statistics book series (LNS, volume 123)


The additive hazards regression model relates the conditional hazard function of the failure time linearly to the covariates. This formulation complements the familiar proportional hazards model in that it describes the association between the covariates and failure time in terms of the risk difference rather than the risk ratio. In this paper, we provide a closed-form semiparametric estimator for the (vector-valued) regression parameter of the additive hazards model with right-censored data, which is consistent and asymptotically normal with a simple variance estimator. We also demonstrate how the additive hazards framework can be used effectively to incorporate frailty and to handle interval-censored data, the resulting semiparametric inference procedures being much simpler than their counterparts under the proportional hazards framework.


Hazard Function Failure Time Frailty Model Failure Time Data Current Status Data 
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  1. Aalen, O.O. (1980). A model for nonparametric regression analysis of counting processes. In Lecture Notes in Statistics, Vol. 2, pp. 1–25. New York: Springer.Google Scholar
  2. Aalen, O. O. (1989). A linear regression model for the analysis of life times. Statist. Med., 8, 907–925.CrossRefGoogle Scholar
  3. Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. New York: Springer-Verlag.CrossRefMATHGoogle Scholar
  4. Andersen, P. K. and Gill, R. D. (1982). Cox’s regression model for counting processes: a large sample study. Ann. Statist., 10, 1100–1120.MathSciNetCrossRefMATHGoogle Scholar
  5. Aranda-Ordaz, F. J. (1983). An extension of the proportional-hazards model for grouped data. Biometrics, 39, 109–117.MathSciNetCrossRefMATHGoogle Scholar
  6. Breslow, N. E. (1985). Cohort analysis in epidemiology. In A Celebration of Statistics, Eds. A. C. Atkinson and S. E. Fienberg. pp. 109–143. New York: Springer.CrossRefGoogle Scholar
  7. Breslow, N. E. and Day, N. E. (1980). Statistical Methods in Cancer Research, Vol. I, The Design and Analysis of Case-Control Studies. Lyon: IARC.Google Scholar
  8. Breslow, N. E. and Day, N. E. (1987). Statistical Methods in Cancer Research, Vol. II, The Design and Analysis of Cohort Studies. Lyon: IARC.Google Scholar
  9. Clayton, D. and Cuzick J. (1985). Multivariate generalizations of the proportional hazards model (with discussion). J. Roy. Statist. Soc. A, 148, 82–117.MathSciNetCrossRefMATHGoogle Scholar
  10. Cox, D. R. (1972). Regression models and life-tables (with discussion). J. R. Statist. Soc. B, 34, 187–220.MATHGoogle Scholar
  11. Cox, D. R. and Oakes, D. (1984). Analysis of Survival Data. London: Chapman and Hall.Google Scholar
  12. Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. New York: Wiley.MATHGoogle Scholar
  13. Groeneboom, P. and Wellner, J.A. (1992). Information Bounds and Nonparametric Maximum Likelihood estimation. Basel: Birkhäuser Verlag.CrossRefMATHGoogle Scholar
  14. Heckman, J.J. and Singer, B. (1982). Population heterogeneity in demographic models. In Multidimensional Mathematical Demography, Ch. 12, pp. 567–599, Eds. K. C. Land and A. Rogers. New York: Academic Press.Google Scholar
  15. Hougaard, P. (1987). Modelling multivariate survival. Scand. J. Statist., 14, 291–304.MathSciNetMATHGoogle Scholar
  16. Huang, J. (1996). Efficient estimation for the Cox model with interval censoring. Ann. Statist., 24, 540–568.MathSciNetCrossRefMATHGoogle Scholar
  17. Huffer, F. W. and McKeague, I. W. (1991). Weighted least squares estimation for Aalen’s additive risk model. J. Amer. Statist. Assoc, 86, 114–129.CrossRefGoogle Scholar
  18. Klein, R.W. and Spady, R.H. (1993). An efficient semiparametric estimator for binary response models. Econometrica, 61, 387–421.MathSciNetCrossRefMATHGoogle Scholar
  19. Lee, E.W., Wei, L.J. and Amato, D. (1992). Cox-type regression analysis for large number of small groups of correlated failure time observations. In Survival Analysis, State of the Art, Eds. J. P. Klein and P. K. Goel, pp. 237–247. Netherlands: Kluwer Academic Publishers.Google Scholar
  20. Lin, D. Y. and Ying, Z. (1994). Semiparametric analysis of the additive risk model. Biometrika, 81, 61–71.MathSciNetCrossRefMATHGoogle Scholar
  21. Lin, D.Y. and Ying, Z. (1995). Semiparametric analysis of general additive-multiplicative intensity models for counting processes. Ann. Statist., 23, 1712–1734.MathSciNetCrossRefMATHGoogle Scholar
  22. Murphy, S. (1995). Asymptotic theory for the frailty model. Ann. Statist., 23, 182–198.MathSciNetCrossRefMATHGoogle Scholar
  23. Nielsen, G.G., Gill, R.D., Andersen, P.K. and Sorensen, T.I.A. (1992). A counting process approach to maximum likelihood estimation in frailty models. Scand. J. Statist., 19, 25–43.MathSciNetMATHGoogle Scholar
  24. Oakes, D. (1989). Bivariate survival models induced by frailties. J. Amer. Statist. Assoc, 84, 487–493.MathSciNetCrossRefMATHGoogle Scholar
  25. Prentice, R. L. and Self, S. G. (1983). Asymptotic distribution theory for Cox-type regression models with general risk form. Ann. Statist., 11, 804–813.MathSciNetCrossRefMATHGoogle Scholar
  26. Rabinowitz, D., Tsiatis, A. and Aragon, J. (1995). Regression with interval censored data. Biometrika, 82, 501–513.MathSciNetCrossRefMATHGoogle Scholar
  27. Thomas, D. C. (1986). Use of auxiliary information in fitting nonproportional hazards models. In Modern Statistical Methods in Chronic Disease Epidemiology, Ed. S. H. Moolgavkar and R. L. Prentice, pp. 197–210. New York: Wiley.Google Scholar
  28. Turnbull, B.W. (1976). The empirical distribution function with arbitrarily grouped, censored and truncated data. J. Roy. Statist. Soc. Ser. B, 38, 290–295.MathSciNetMATHGoogle Scholar
  29. Wei, L. J., Lin, D. Y. and Weissfeld, L. (1989). Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. J. Amer. Statist. Assoc, 84, 1065–1073.MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag New York, Inc. 1997

Authors and Affiliations

  • D. Y. Lin
  • Zhiliang Ying

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