Hazard Regression Analysis

  • M. Luz Gámiz
  • K. B. Kulasekera
  • Nikolaos Limnios
  • Bo Henry Lindqvist
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)


This chapter is devoted to the analysis of certain models that describe in some way the relationship between the time-to-failure of a system and a set of explanatory variables (hereafter called covariates) that represent endogenous or exogenous information relevant (or maybe not) to the deterioration process of the system. The relationship between lifetime and covariates is expressed in terms of the hazard function and we consider as starting point the semi-parametric Cox proportional hazard model that is extended in several ways leading to more flexible models that may cover a wider range of practical situations. In each case, we present different techniques of estimation of the model. Finally, we focus the problem from a nonparametric point of view. Although the nonparametric estimation of the hazard function may be tackled in several ways, we consider an estimator obtained as the ratio of nonparametric estimators of the conditional density an the survival function


Hazard Function Quantile Regression Baseline Hazard Partial Likelihood Bandwidth Selection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Aalen OO (1980) A model for nonparametric regression analysis of counting processes. Lecture Notes in Statistics. vol 2, Springer, New York, pp 1–25Google Scholar
  2. 2.
    Aalen OO (1989) A linear regression model for the analysis of life times. Stat Med 8:907–925CrossRefGoogle Scholar
  3. 3.
    Aalen OO (1993) Further results on the nonparametric linear regression model in survival analysis. Stat Med 12:1569–1588CrossRefGoogle Scholar
  4. 4.
    Akaike H (1970) Statistical predictor identification. Ann Inst Stat Math 22:203–217MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New YorkMATHGoogle Scholar
  6. 6.
    Boos DD, Stefanski LA, Wu Y (2009) Fast FSR variable selection with applications to clinical trials. Biometrics 65:692–700MATHCrossRefGoogle Scholar
  7. 7.
    Breslow NE (1975) Analysis of survival data under the proportional hazards model. Int Stat Rev 43:45–58CrossRefGoogle Scholar
  8. 8.
    Carrión A, Solano H, Gámiz ML, Debón (2010) A evaluation of the reliability of a water supply network from right-censored and left-truncated break data. Water Resourc Manage, doi: 10.1007/s11269-010-9587-y
  9. 9.
    Chen YQ, Jewell NP (2001) On a general class of semiparametric hazards regression models. Biometrika 88(3):687–702MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chen YQ, Wang MC (2000) Analysis of accelerated hazards models. J Am Stat Assoc 95:608–618MATHCrossRefGoogle Scholar
  11. 11.
    Cho HJ, Hong S-M (2008) Median regression tree for analysis of censored survival data. IEEE Trans Syst Man Cybern A Syst Hum 38(3):715–726CrossRefMathSciNetGoogle Scholar
  12. 12.
    Clayton D, Cuzik J (1985) The EM-algorithm for Cox’s regression model using GLIM. Appl Stat 34:148–156CrossRefGoogle Scholar
  13. 13.
    Cox DR (1972) Regression models and life-tables (with discussion). J R Stat Soc B 34:187–220MATHGoogle Scholar
  14. 14.
    Crowder MJ, Kimber AC, Smith RL, Sweeting TJ (1991) Statistical analysis of reliability data. Chapman and Hall, LondonGoogle Scholar
  15. 15.
    Dabrowska DM (1997) Smoothed Cox regression. Ann Stat 25(4):1510–1540MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ebrahimi N (2007) Accelerated life tests: nonparametric approach. In: Ruggeri F, Kenett R, Faltin FW (eds) Encyclopedia of statistics in quality and reliability. Wiley, New YorkGoogle Scholar
  17. 17.
    Efron B (1967) Two sample problem with censored data. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, vol IV, pp 831–853Google Scholar
  18. 18.
    Fan J, Gibels I, King M (1997) Local likelihood and local partial likelihood in hazard regression. Ann Stat 25(4):1661–1690MATHCrossRefGoogle Scholar
  19. 19.
    Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96:1348–1360MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Fan J, Li R (2002) Variable selection for Cox’s proportional hazard model and frailty model. Ann Stat 30:74–99MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Fan J, Lin H, Zhou Y (2006) Local partial-likelihood estimation for lifetime data. Ann Stat 34(1):290–325MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Fan J, Yao Q, Tong H (1996) Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83(1):189–206MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Fox J (2008) Cox proportional-hazards regression for survival data. Appendix to an R and S-PLUS companion to applied regression. Sage Publications, LondonGoogle Scholar
  24. 24.
    Gentleman R, Crowley J (1991) Local full likelihood estimation for the proportional hazards model. Biometrics 47:1283–1296CrossRefMathSciNetGoogle Scholar
  25. 25.
    González-Manteiga W, Cadarso-Suárez C (1994) Asymptotic properties of a generalized KaplanMeier estimator with some applications. J Nonparametr Stat 4:65–78MATHCrossRefGoogle Scholar
  26. 26.
    Hall P, Racine J, Li Q (2004) Cross-validation and the estimation of conditional probability densities. J Am Stat Assoc 99:1015–1026MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Hayfield T, Racine JS (2008) Nonparametric econometrics: the np package. J Stat Softw 27(5)
  28. 28.
    Honoré B, Khan S, Powell JL (2002) Quantile regression under random censoring. J Econometr 109:67–105MATHCrossRefGoogle Scholar
  29. 29.
    Jones MC, Davies SJ, Park BU (1994) Versions of kernel-type regression estimator. J Am Stat Assoc 89:825–832MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Karr AF (1986) Point processes and their statistical applications. Marcel Dekker, New YorkGoogle Scholar
  31. 31.
    Kalbfleisch JD, Prentice RL (2002) The Statistical analysis of failure time data, 2nd edn. John Wiley and sons, New JerseyGoogle Scholar
  32. 32.
    Klein M, Moeschberger W (1997) Survival analysis. Techniques for censored and truncated data. Springer, New YorkMATHGoogle Scholar
  33. 33.
    Kleinbaum DG, Klein M (2005) Survival analysis: a self-learning text. Springer, New YorkMATHGoogle Scholar
  34. 34.
    Koenker R (2005) Quantile regression. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  35. 35.
    Koenker R (2009) Quantreg, quantile regression
  36. 36.
    Koenker R, Bassett GS (1978) Regression quantiles. Econometrica 46:33–50MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Krivtsov VV, Tananko DE, Davis TP (2002) Regression approach to tire reliability analysis 78(3):267–273Google Scholar
  38. 38.
    Kvaløy JT (1999) Nonparametric estimation in Cox-models: time transformation methods versus partial likelihood methodsGoogle Scholar
  39. 39.
    Kvaløy JT, Lindqvist BH (2003) Estimation and inference in nonparametric Cox-models: time transformation methods. Comput Stat 18:205–221Google Scholar
  40. 40.
    Kvaløy JT, Lindqvist BH (2004) The covariate order method for nonparametric exponential regression and some applications in other lifetime models. In: Nikulin MS, Balakrishnan N, Mesbah M, Limnios N (eds) Parametric and semiparametric models with applications to reliability, survival analysis and quality of life. Birkhäuser, pp 221–237Google Scholar
  41. 41.
    Lawless JF (2002) Statistical models and methods for lifetime data, 2nd edn. Wiley, LondonGoogle Scholar
  42. 42.
    Mallows CL (1973) Some comments on c p. Technometrics 15:661–675MATHCrossRefGoogle Scholar
  43. 43.
    Martinussen T, Scheike TH (2006) Dynamic regression models for survival data. Springer, BerlinMATHGoogle Scholar
  44. 44.
    Meeker WQ, Escobar LA (1998) Statistical methods for reliability data. Wiley, New YorkMATHGoogle Scholar
  45. 45.
    Nelson W (1990) Accelerated testing: statistical models, test plans, and data analyses. Wiley, New YorkGoogle Scholar
  46. 46.
    Nelson W (2004) Applied life data analysis. Wiley, New JerseyMATHGoogle Scholar
  47. 47.
    Nelson W (2005) A bibliography of accelerated test plans. IEEE Trans Reliabil 54(2):194–197CrossRefGoogle Scholar
  48. 48.
    Nelson W (2005) A bibliography of accelerated test plans. Part II-References. IEEE Trans Reliabil 54(3):370–373CrossRefGoogle Scholar
  49. 49.
    Nielsen JP (1998) Marker dependent kernel hazard estimation from local linear estimation. Scand Actuar J 2:113–124Google Scholar
  50. 50.
    Nielsen JP, Linton OB (1995) Kernel estimation in a nonparametric marker dependent hazard model. Ann Stat 23:1735–1748MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Portnoy S (2003) Censored regression quintiles. J Am Stat Assoc 98:1001–1012MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Royston B, Parmar MKB (2002) Flexible parametric proportional-hazards and proportional-odds models for censored survival data, with appliction to prognostic modelling and estimation of treatment effects. Stat Med 21:2175–2197CrossRefGoogle Scholar
  53. 53.
    Scheike T, Martinussen T, Silver J (2010) Timereg: timereg package for Flexible regression models for survival data
  54. 54.
    Schwarz G (1978) Estimating the dimensions of a model. Ann Stat 6:461–464MATHCrossRefGoogle Scholar
  55. 55.
    Shi P, Tsai C (2002) Regression model selection—a residual likelihood approach. J R Stat Soc B 64:237–252MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Spierdijk L (2008) Nonparametric conditional hazard rate estimation: a local linear approach. Comput Stat Data Anal 52:2419–2434MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Therneau TM, Grambsch PM (2000) Modeling survival data. Extending the Cox model. Springer, BerlinMATHGoogle Scholar
  58. 58.
    Therneau T, Lumley T (2009) survival: Survival analysis, including penalised likelihood
  59. 59.
    Tibshirani R (1996) Regression shrinkage and selection via the LASSO. J R Stat Soc B 58:267–288MATHMathSciNetGoogle Scholar
  60. 60.
    Toomet O, Henningsen A, Graves S (2009) maxLik: maximum likelihood estimation
  61. 61.
    Wang HJ, Wang L (2009) Locally weighted censored quanitile regression. J Am Stat Assoc 104:1117–1128CrossRefGoogle Scholar
  62. 62.
    Wang JL (2003) Smoothing hazard rates. Encyclopedia of biostatisticsGoogle Scholar
  63. 63.
    Wells MT (1994) Nonparametric kernel estimation in counting processes with explanatory variables. Biometrika 81:759–801CrossRefGoogle Scholar
  64. 64.
    Wu Y, Boos DD, Stefanski LA (2007) Controlling variable selection by the addition of pseudovariables. J Am Stat Assoc 102:235–243MATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    Yang S (1999) Censored median regression using weighted empirical survival and hazard functions. J Am Stat Assoc 94(445):137–145MATHCrossRefGoogle Scholar
  66. 66.
    Ying Z, Jung SH, Wei LJ (1995) Survival analysis with median regression models. J Am Stat Assoc 90(429):178–184MATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    Zhang H, Lu W (2007) Adaptive-LASSO for Cox’s proportional hazards model Biometrika 94:691–703MATHMathSciNetGoogle Scholar
  68. 68.
    Zhao Y, Chen F (2008) Empirical likelihood inference for censored median regression model via nonparametric kernel estimation. J Multivar Anal 99:215–231MATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    Zou H (2006) The adaptive LASSO and its Oracle properties. J Am Stat Assoc 101:1418–1429MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • M. Luz Gámiz
    • 1
  • K. B. Kulasekera
    • 2
  • Nikolaos Limnios
    • 3
  • Bo Henry Lindqvist
    • 4
  1. 1.Facultad Ciencias, Depto. Estadistica e Investigacion OperativaUniversidad GranadaGranadaSpain
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA
  3. 3.Centre de Recherches de Royallieu, Laboratoire de Mathématiques AppliquéesUniversité de Technologie de CompiègneCompiègneFrance
  4. 4.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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