Survival Analysis



Survival analysis concerns the time from a well-defined origin to some end event, such as the time from surgery to death of a cancer patient, the time from wedding to divorce, time from layoff to finding a new job, and time between the first and second suicide attempts. Although originated in and driven much by research on lifetime, or survival of an object such as light bulbs and other electric devices in the early days, modern applications of survival analysis include many non-survival events such as the aforementioned examples. Thus, survival analysis may be more appropriately called the time-to-event analysis. However, in this chapter we continue to use the classic term survival analysis in our discussion of this statistical model and its applications.


  1. .
    Aalen OO (1978) Nonparametric estimation of partial transition probabilities in multiple decrement models. Ann Stat 6:534–545CrossRefGoogle Scholar
  2. .
    Aalen OO, Andersen PK, Borgan Ø, Gill RD, Keiding N (2009) History of applications of martingales in survival analysis. Electron J Hist Probab Stat 5: 1–28.Google Scholar
  3. .
    Aalen OO, Borgan Ø, Gjessing HK (2008) Survival and event history analysis. Springer, New YorkCrossRefGoogle Scholar
  4. .
    Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New YorkCrossRefGoogle Scholar
  5. .
    Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Stat 10:1100–1120CrossRefGoogle Scholar
  6. .
    Breslow NE (1972) Discussion of Professor Cox’s paper. J Roy Stat Soc B 34:216–217Google Scholar
  7. .
    Cai J, Prentice RL (1995) Estimating equations for hazard ratio parameters based on correlated failure time data. Biometrika 82:151–164CrossRefGoogle Scholar
  8. .
    Cox DR (1972) Regression models and life tables (with discussion). J Roy Stat Soc B 34:187–220Google Scholar
  9. .
    Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J Roy Stat Soc B 39:1–38Google Scholar
  10. .
    Dudley RM (1999) Uniform central limit theorems. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  11. .
    Efron B (1977) The efficiency of Cox’s likelihood function for censored data. J Am Stat Assoc 72:557–565CrossRefGoogle Scholar
  12. .
    Elbers C, Ridder G (1982) True and spurious duration dependence: the identifiability of the proportional hazard model. Rev Econ Stud 49:403–409CrossRefGoogle Scholar
  13. .
    Feng C, Wang H, Tu XM (2012) A unified definition of conditional expectation and its applications in survival analysis. Adv Appl Stat 27:81–95Google Scholar
  14. .
    Fine JP, Gray RJ (1999) A proportional hazards model for the subdistribution of a competing risk. J Am Stat Assoc 94:496–509CrossRefGoogle Scholar
  15. .
    Fitzmaurice GM, Laird NM, Ware JH (2004) Applied longitudinal analysis. Wiley, Hoboken, NJGoogle Scholar
  16. .
    Fleming TR, Harrington DP (1991) Counting processes and survival analysis. Wiley, Chichester, UKGoogle Scholar
  17. .
    Gill R (1980) Censoring and stochastic integrals. Tract 124, Amsterdam Mathematical Centre, AmsterdamGoogle Scholar
  18. .
    Gill RD, Johansen S (1990) A survey of product-integration with a view towards applications in survival analysis. Ann Stat 18:1501–1555CrossRefGoogle Scholar
  19. .
    Greenwood M (1926) The natural duration of cancer. Reports of Public Health and Related Subjects, vol 33. HMSO, LondonGoogle Scholar
  20. .
    Hall WJ, Wellner JA (1980) Confidence bands for a survival curve from censored data. Biometrika 67:133–143CrossRefGoogle Scholar
  21. .
    Hogan JW, Roy J, Korkontzelou C (2004) Handling drop-out in longitudinal studies. Stat Med 23:1455–1497PubMedCrossRefGoogle Scholar
  22. .
    Hougaard P (2000) Analysis of multivariate survival data. Springer, New YorkCrossRefGoogle Scholar
  23. .
    Jacod J (1975) Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales. Zeitschriftfür Wahrschein-lichkeitstheorie und verwandte Gebiete 31:235–253Google Scholar
  24. .
    Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn, Wiley, New YorkCrossRefGoogle Scholar
  25. .
    Kaplan EL, Meier P (1958) Nonparametric estimator from incomplete observations. J Am Stat Assoc 53:457–481CrossRefGoogle Scholar
  26. .
    Klein JP (1988) Small sample properties of censored data estimators of the cumulative hazard rate, survivor function, and estimators of their variance. Research Report 88/7. Statistical Uunit, University of Copenhagen.Google Scholar
  27. .
    Klein JP (1991) Small sample moments of some estimators of the variance of the Kaplan-Meier and Nelson-Aalen estimators. Scandinavian J Stat 18:333–340Google Scholar
  28. .
    Kosorok MR (2008) Introduction to empirical processes and semiparametric inference. Springer, New YorkCrossRefGoogle Scholar
  29. .
    Kowalski J, Tu XM (2007) Modern Applied U Statistics. Wiley, New YorkCrossRefGoogle Scholar
  30. .
    Liang KY, Zeger S (1986) Longitudinal data analysis using generalized linear models. Biometrika 73:13–22CrossRefGoogle Scholar
  31. .
    Little RJA, Rubin DB (2002) Statistical analysis with missing data, 2nd edn. Wiley, New YorkGoogle Scholar
  32. .
    Lu X, Tsiatis AA (2008) Improving the efficiency of the logrank test using auxiliary covariates. Biometrika 95:679–694CrossRefGoogle Scholar
  33. .
    Mantel N (1966) Evaluation of survival data and two new rank order statistics arising from its consideration. Canc Chemother Rep 50:163–170Google Scholar
  34. .
    Mantel N, Haenszel W (1959) Statistical aspects of the analysis of data from the retrospective analysis of disease. J Natl Canc Inst 22(4):719–748Google Scholar
  35. .
    Mantel N, Bohidar NR, Ciminera JL (1977) Mantel-Haenszel analyses of litter-matched time-to-response data, with modifications for recovery of interlitter information. Canc Res 37:3863–3868Google Scholar
  36. .
    Murphy SA (1994) Consistency in a proportional hazards model incorporating a random effect. Ann Stat 22:712–731CrossRefGoogle Scholar
  37. .
    Murphy SA (1995) Asymptotic theory for the frailty model. Ann Stat 23:182–198CrossRefGoogle Scholar
  38. .
    Nelson W (1969) Hazard plotting for incomplete failure data J Qual Tech 1:27–52Google Scholar
  39. .
    Nelson W (1972) Theory and applications of hazard plotting for censored failure data. Technometrics 14:945–965CrossRefGoogle Scholar
  40. .
    Nielsen GG, Gill RD, Andersen PK, Sørensen TIA (1992) A counting process approach to maximum likelihood estimation in frailty models. Scandinavian J Stat 19:25–44Google Scholar
  41. .
    Oakes D (1989) Bivariate survival models induced by frailties. J Amer Stat Assoc 84:487–493CrossRefGoogle Scholar
  42. .
    Oakes D (2001) Biometrika centenary: survival analysis. Biometrika 88:99–142CrossRefGoogle Scholar
  43. .
    Parner E (1998) Asymptotic theory for the correlated gamma-frailty model. Ann Stat 26:183–214CrossRefGoogle Scholar
  44. .
    Peto R (1959) Rank tests of maximal power against Lehmann-type alternatives. Biometrika 59:472–475CrossRefGoogle Scholar
  45. .
    Petersen JH, Andersen PK, Gill RD (1996) Variance components models for survival data. Statistica Neerlandica 50:193–211CrossRefGoogle Scholar
  46. .
    Peto R, Peto J (1972) Asymptotically efficient rank invariant test procedures (with discussion). J Roy Stat Soc A 135:185–206CrossRefGoogle Scholar
  47. .
    Pollard D (1984) Convergence of stochastic processes. Springer, New YorkCrossRefGoogle Scholar
  48. .
    Pollard D (1990) Empirical processes: theory and applications. NSF-CBMS regional conference series in probability and statistics, vol. 2. Institute of Mathematical StatisticsGoogle Scholar
  49. .
    Tang W, He H, Tu XM (2012) Applied categorical data analysis. Chapman & Hall/CRC, New YorkGoogle Scholar
  50. .
    van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, CambridgeGoogle Scholar
  51. .
    van der Vaart AW, Wellner J (1996) Weak convergence and empirical processes. Springer, New YorkGoogle Scholar
  52. .
    Vaupel JW, Manton KG, Stallard E (1979) The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16:439–454PubMedCrossRefGoogle Scholar
  53. .
    Wong WH (1986) Theory of partial likelihood. Ann Stat 14:88–123CrossRefGoogle Scholar
  54. .
    Zhang H, Xia Y, Chen R, Lu N, Tang W, Tu XM (2011) On modeling longitudinal binomial responses—implications from two dueling paradigms. Appl Stat 38:2373–2390CrossRefGoogle Scholar
  55. .
    Zhang H, Yu Q, Feng C, Gunzler D, Wu P, Tu XM (2012) A new look at the difference between GEE and GLMM when modeling longitudinal count responses. J Appl Stat 39:2067–2079CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Biostatistics and Computational BiologyUniversity of RochesterRochesterUSA

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