Lifetime Data Analysis

, Volume 19, Issue 4, pp 442–462

An introduction to survival models: in honor of Ross Prentice



I review some key ideas and models in survival analysis with emphasis on modeling the effects of covariates on survival times. I focus on the proportional hazards model of Cox (J R Stat Soc B 34:187–220, 1972), its extensions and alternatives, including the accelerated life model. I briefly describe some models for competing risks data, multiple and repeated event-time data and multivariate survival data.


Cox model Failure time data Lifetime data analysis Ross Prentice 


  1. Aalen OO (1978) Nonparametric inference for a family of counting processes. Ann Stat 6:701–726MathSciNetCrossRefMATHGoogle Scholar
  2. Aalen OO, Borgan O, Gjessing HK (2008) Survival and event history analysis: a process point of view. Springer, New YorkCrossRefGoogle Scholar
  3. Andersen PK, Gill RD (1982) Cox’s regression models for counting processes: a large sample study. Ann Stat 10:1100–1120MathSciNetCrossRefMATHGoogle Scholar
  4. Arnold BC, Brockett PL (1983) When does the \(\beta \)’th percentile life function determine the distribution? Oper Res 31:391–396MathSciNetCrossRefMATHGoogle Scholar
  5. Bickel PJ, Klassen CA, Ritov Y, Wellner JA (1993) Efficient and adaptive estimation for semiparametric models. Johns Hopkins UP, BaltimoreMATHGoogle Scholar
  6. Cai J, Prentice RL (1995) Estimating equations for hazard ratio parameters based on correlated failure time data. Biometrika 82:151–164MathSciNetCrossRefMATHGoogle Scholar
  7. Chambers JM, Mallows CL, Stuck BW (1976) A method for simulating stable random variables. J Am Stat Assoc 71:340–344MathSciNetCrossRefMATHGoogle Scholar
  8. Chen K, Jin Z, Ying Z (2002) Semiparametric analysis of transformation models with censored data. Biometrika 93:627–640MathSciNetGoogle Scholar
  9. Chen SC, Wei LJ, Ying Z (1995) Analysis of transformation models with censored data. Biometrika 82:835–845MathSciNetCrossRefGoogle Scholar
  10. Clayton DG (1978) A model for association in bivariate life tables and its application in epidemiologic studies of familial tendency in chronic disease incidence. Biometrika 65:141–151MathSciNetCrossRefMATHGoogle Scholar
  11. Cox DR (1972) Regression models and life tables (with discussion). J R Stat Soc B 34:187–220MATHGoogle Scholar
  12. Cox DR (1975) Partial likelihood. Biometrika 62:269–276MathSciNetCrossRefMATHGoogle Scholar
  13. Cox DR, Oakes D (1984) Analysis of survival data. Chapman and Hall, LondonGoogle Scholar
  14. Dempster AP, Laird NM, Rubin DR (1977) Maximum likelihood from incomplete data via the EM algorithm (with discussion). J R Stat Soc B 39:1–38MathSciNetMATHGoogle Scholar
  15. Devroye L (2009) On exact simulation algorithms for some distributions related to Jacobi theta functions. Stat Probab Lett 79:2251–2259MathSciNetCrossRefMATHGoogle Scholar
  16. Doksum KA, Nabeya S (1984) Estimation in proportional hazard and loglinear models. J Stat Plan Inference 9:297–303MathSciNetCrossRefMATHGoogle Scholar
  17. Farewell VT, Cox DR (1979) A note on multiple time scales in life-testing. Appl Stat 28:73–75CrossRefGoogle Scholar
  18. Feller W (1971) An introduction to probability theory and its applications, vol 2, 2nd edn. Wiley, New YorkMATHGoogle Scholar
  19. Fine JP, Gray RJ (1999) A proportional hazards model for the subdistribution of a competing risk. J Am Stat Assoc 94:496–509MathSciNetCrossRefMATHGoogle Scholar
  20. Gumbel EI (1924) Eine Darstellung der Sterbetafel. Biometrika 16:283–196CrossRefGoogle Scholar
  21. Hall WJ, Wellner JA (1981) Mean residual life. In: Csorgo DA, Dawson DA, Rao JNK, Saleh AK, Md E (eds) Statistics and related topics. Elsevier/North Holland, New York, pp 169–184Google Scholar
  22. Haller B, Schmidt G, Ulm (2013) Applying competing risks models: an overview. Lifetime Data Anal 19:33–58MathSciNetCrossRefGoogle Scholar
  23. Holt JD, Prentice RL (1974) Survival analysis in twin studies and matched pair experiments. Biometrika 61:17–30MathSciNetCrossRefMATHGoogle Scholar
  24. Horvitz DG, Thompson DJ (1952) A generalization of sampling without replacement from a finite universe. J Am Stat Assoc 47:663–685MathSciNetCrossRefMATHGoogle Scholar
  25. Hougaard P (1986) A class of multivariate failure time distributions. Biometrika 75:395 (Biometrika 73:671–678 Correction)Google Scholar
  26. Jung S-H, Jeong J-H, Bandos H (2009) Regression and quantile residual life. Biometrics 65:1203–1212MathSciNetCrossRefMATHGoogle Scholar
  27. Kalbfleisch JD, Prentice RL (1980) The statistical analysis of failure time data. Wiley, New YorkMATHGoogle Scholar
  28. Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. Wiley, New YorkCrossRefMATHGoogle Scholar
  29. Kanter M (1975) Stable distributions under change of scale and total variation inequalities. Ann Probab 3:697–707MathSciNetCrossRefMATHGoogle Scholar
  30. Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53:457–481MathSciNetCrossRefMATHGoogle Scholar
  31. Lawless JF (1982) Statistical models and methods for lifetime data. Wiley, New YorkMATHGoogle Scholar
  32. Lee M-LT, Whitmore GA (2006) Threshold regression for survival analysis: modeling event times by a stochastic process reaching a boundary. Stat Sci 21:501–513MathSciNetCrossRefMATHGoogle Scholar
  33. Lin DY, Ying Z (1994) Semiparametric analysis of the additive risk model. Biometrika 81:61–71MathSciNetCrossRefMATHGoogle Scholar
  34. Maguluri G, Zhang C (1994) Estimation in the mean residual life regression model. J R Stat Soc B 56:477–489MathSciNetMATHGoogle Scholar
  35. Martinussen T, Scheike TH (2002) A flexible additive multiplicative hazard model. Biometrika 89:283–298MathSciNetCrossRefMATHGoogle Scholar
  36. McKeague IW, Sasieni PD (1994) A partly parametric additive risk model. Biometrika 81:501–514MathSciNetCrossRefMATHGoogle Scholar
  37. Murphy SA, Sen PK (1991) Time-dependent coefficients in a Cox-type regression model. Stoch Process Appl 39:153–180MathSciNetCrossRefMATHGoogle Scholar
  38. Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New YorkMATHGoogle Scholar
  39. Oakes D (1982) A model for association in bivariate survival data. J R Stat Soc B 44:414–422MathSciNetMATHGoogle Scholar
  40. Oakes D (1989) Bivariate survival models induced by frailties. J Am Stat Assoc 84:487–493MathSciNetCrossRefMATHGoogle Scholar
  41. Oakes D (1992) Frailty models for multiple event times. In: Klein JP, Goel PK (eds) Survival analysis: state of the art. Kluwer, Dordrecht, pp 371–379CrossRefGoogle Scholar
  42. Oakes D (1995) Multiple time scales in survival analysis. Lifetime Data Anal 1:7–18CrossRefMATHGoogle Scholar
  43. Oakes D (2006) On potential risk reversal due to heterogeneity. Student 5:205–210Google Scholar
  44. Oakes D, Dasu T (1990) A note on residual life. Biometrika 77:409–410MathSciNetCrossRefMATHGoogle Scholar
  45. Prentice RL (1975) Discrimination among some parametric models. Biometrika 62:607–614MathSciNetCrossRefMATHGoogle Scholar
  46. Prentice RL (1986) A case–cohort design for epidemiologic studies and disease prevention studies. Biometrika 73:1–11MathSciNetCrossRefMATHGoogle Scholar
  47. Prentice RL, Cai J (1992) Covariance and survivor function estimation using censored multivariate failure time data. Biometrika 79:495–512MathSciNetCrossRefMATHGoogle Scholar
  48. Prentice RL, Williams BJ, Peterson AV (1981) On the regression analysis of multivariate time data. Biometrika 68:373–379MathSciNetCrossRefMATHGoogle Scholar
  49. Pyke R (1965) Spacings (with discussion). J R Stat Soc B 27:395–449Google Scholar
  50. Reid N (1994) A conversation with Sir David Cox. Stat Sci 9:439–455CrossRefMATHGoogle Scholar
  51. Robins JM, Rotnitzky A (1992). Recovery of information and adjustment for dependent censoring using surrogate markers. In: Jewell N, Dietz K, Farewell V (eds) AIDS epidemiology methodological issues. Birkhauser, Boston, pp 297–331Google Scholar
  52. Robins JM, Tsiatis AA (1992) Semiparametric estimation of an accelerated failure time model with time-dependent covariates. Biometrika 79:311–319MathSciNetMATHGoogle Scholar
  53. Sasieni PD (1996) Proportional excess hazards. Biometrika 83:127–141MathSciNetCrossRefMATHGoogle Scholar
  54. Shaw M, Mitchell R, Dorling D (2000) Time for a smoke? One cigarette reduces your life by 11 minutes. BMJ 320:53CrossRefGoogle Scholar
  55. Therneau TM, Grambsch PM (2000) Modeling survival data: extending the Cox model. Springer, New YorkCrossRefGoogle Scholar
  56. van der Laan MJ, Robins JM (2003) Unified methods for censored longitudinal data and causality. Springer, New YorkCrossRefMATHGoogle Scholar
  57. Venzon DJ, Moolgavkar SH (1988) Origin-invariant relative risk functions for case–control and survival studies. Biometrika 75:325–333CrossRefMATHGoogle Scholar
  58. Wei LJ, Lin DY, Weissfeld L (1989) Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. J Am Stat Assoc 84:1065–1073MathSciNetCrossRefGoogle Scholar
  59. Wienke A (2011) Frailty models in survival analysis. Chapman and Hall/CRC, Boca RatonGoogle Scholar
  60. Yang Y, Ying Z (2001) Marginal proportional hazards models for multiple event-time data. Biometrika 88:581–586MathSciNetCrossRefMATHGoogle Scholar
  61. Yu QQ, Chappell R, Wong GYC, Hsu Y, Mazur M (2008) Relationship between the Cox, Lehmann, Weibull and accelerated life models. Commun Stat Theo Meth 37:1456–1470MathSciNetGoogle Scholar
  62. Zeng D, Lin DY (2006) Efficient estimation of semiparametric transformation models for counting processes. Biometrika 93:627–640MathSciNetCrossRefMATHGoogle Scholar
  63. Zucker DM, Karr AF (1990) Nonparametric survival analysis with time-dependent covariate effects, a penalized partial likelihood approach. Ann Stat 18:329–353MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Biostatistics and Computational BiologyUniversity of Rochester Medical CenterRochesterUSA

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