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Introduction to Survival Analysis

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Regression Modeling Strategies

Part of the book series: Springer Series in Statistics ((SSS))

Abstract

Suppose that one wished to study the occurrence of some event in a population of subjects. If the time until the occurrence of the event were unimportant, the event could be analyzed as a binary outcome using the logistic regression model. For example, in analyzing mortality associated with open heart surgery, it may not matter whether a patient dies during the procedure or he dies after being in a coma for two months. For other outcomes, especially those concerned with chronic conditions, the time until the event is important. In a study of emphysema, death at eight years after onset of symptoms is different from death at six months. An analysis that simply counted the number of deaths would be discarding valuable information and sacrificing statistical power.

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Notes

  1. 1.

    An exception to this is the case in which once an event occurs for the first time, that event is likely to recur multiple times for any patient. Then the latter occurrences are redundant.

References

  1. O. O. Aalen. Nonparametric inference in connection with multiple decrement models. Scan J Stat, 3:15–27, 1976.

    MathSciNet  Google Scholar 

  2. O. O. Aalen, E. Bjertness, and T. Sønju. Analysis of dependent survival data applied to lifetimes of amalgam fillings. Stat Med, 14:1819–1829, 1995.

    Article  Google Scholar 

  3. B. Altschuler. Theory for the measurement of competing risks in animal experiments. Math Biosci, 6:1–11, 1970.

    Article  MathSciNet  Google Scholar 

  4. F. Ambrogi, E. Biganzoli, and P. Boracchi. Estimates of clinically useful measures in competing risks survival analysis. Stat Med, 27:6407–6425, 2008.

    Article  MathSciNet  Google Scholar 

  5. P. K. Andersen and R. D. Gill. Cox’s regression model for counting processes: A large sample study. Ann Stat, 10:1100–1120, 1982.

    Article  MathSciNet  Google Scholar 

  6. E. Arjas. A graphical method for assessing goodness of fit in Cox’s proportional hazards model. J Am Stat Assoc, 83:204–212, 1988.

    Article  Google Scholar 

  7. D. M. Berridge and J. Whitehead. Analysis of failure time data with ordinal categories of response. Stat Med, 10:1703–1710, 1991.

    Article  Google Scholar 

  8. C. Berzuini and D. Clayton. Bayesian analysis of survival on multiple time scales. Stat Med, 13:823–838, 1994.

    Article  Google Scholar 

  9. W. B. Bilker and M. Wang. A semiparametric extension of the Mann-Whitney test for randomly truncated data. Biometrics, 52:10–20, 1996.

    Article  Google Scholar 

  10. C. Binquet, M. Abrahamowicz, A. Mahboubi, V. Jooste, J. Faivre, C. Bonithon-Kopp, and C. Quantin. Empirical study of the dependence of the results of multivariable flexible survival analyses on model selection strategy. Stat Med, 27:6470–6488, 2008.

    Article  MathSciNet  Google Scholar 

  11. E. H. Blackstone. Analysis of death (survival analysis) and other time-related events. In F. J. Macartney, editor, Current Status of Clinical Cardiology, pages 55–101. MTP Press Limited, Lancaster, UK, 1986.

    Google Scholar 

  12. J. Bryant and J. J. Dignam. Semiparametric models for cumulative incidence functions. Biometrics, 69:182–190, 2004.

    Article  MathSciNet  Google Scholar 

  13. K. Bull and D. Spiegelhalter. Survival analysis in observational studies. Stat Med, 16:1041–1074, 1997.

    Article  Google Scholar 

  14. S. C. Cheng, J. P. Fine, and L. J. Wei. Prediction of cumulative incidence function under the proportional hazards model. Biometrics, 54:219–228, 1998.

    Article  MathSciNet  Google Scholar 

  15. A. Cnaan and L. Ryan. Survival analysis in natural history studies of disease. Stat Med, 8:1255–1268, 1989.

    Article  Google Scholar 

  16. D. Collett. Modelling Survival Data in Medical Research. Chapman and Hall, London, 1994.

    Book  Google Scholar 

  17. D. R. Cox. Regression models and life-tables (with discussion). J Roy Stat Soc B, 34:187–220, 1972.

    Google Scholar 

  18. D. R. Cox and D. Oakes. Analysis of Survival Data. Chapman and Hall, London, 1984.

    Google Scholar 

  19. E. R. DeLong, C. L. Nelson, J. B. Wong, D. B. Pryor, E. D. Peterson, K. L. Lee, D. B. Mark, R. M. Califf, and S. G. Pauker. Using observational data to estimate prognosis: an example using a coronary artery disease registry. Stat Med, 20:2505–2532, 2001.

    Article  Google Scholar 

  20. J. A. Dubin, H. Müller, and J. Wang. Event history graphs for censored data. Stat Med, 20:2951–2964, 2001.

    Article  Google Scholar 

  21. J. P. Fine and R. J. Gray. A proportional hazards model for the subdistribution of a competing risk. J Am Stat Assoc, 94:496–509, 1999.

    Article  MathSciNet  Google Scholar 

  22. D. M. Finkelstein and D. A. Schoenfeld. Combining mortality and longitudinal measures in clinical trials. Stat Med, 18:1341–1354, 1999.

    Article  Google Scholar 

  23. M. Fiocco, H. Putter, and H. C. van Houwelingen. Reduced-rank proportional hazards regression and simulation-based predictino for multi-state models. Stat Med, 27:4340–4358, 2008.

    Article  MathSciNet  Google Scholar 

  24. T. R. Fleming and D. P. Harrington. Nonparametric estimation of the survival distribution in censored data. Comm Stat Th Meth, 13(20):2469–2486, 1984.

    Article  MathSciNet  Google Scholar 

  25. T. R. Fleming and D. P. Harrington. Counting Processes & Survival Analysis. Wiley, New York, 1991.

    Google Scholar 

  26. B. Francis and M. Fuller. Visualization of event histories. J Roy Stat Soc A, 159:301–308, 1996.

    Article  Google Scholar 

  27. M. H. Gail. Does cardiac transplantation prolong life? A reassessment. Ann Int Med, 76:815–817, 1972.

    Article  Google Scholar 

  28. J. J. Gaynor, E. J. Feuer, C. C. Tan, D. H. Wu, C. R. Little, D. J. Straus, D. D. Clarkson, and M. F. Brennan. On the use of cause-specific failure and conditional failure probabilities: Examples from clinical oncology data. J Am Stat Assoc, 88:400–409, 1993.

    Article  Google Scholar 

  29. R. B. Geskus. Cause-specific cumulative incidence estimation and the Fine and Gray model under both left truncation and right censoring. Biometrics, 67(1):39–49, 2011.

    Article  MathSciNet  Google Scholar 

  30. A. I. Goldman. EVENTCHARTS: Visualizing survival and other timed-events data. Am Statistician, 46:13–18, 1992.

    Google Scholar 

  31. T. A. Gooley, W. Leisenring, J. Crowley, and B. E. Storer. Estimation of failure probabilities in the presence of competing risks: New representations of old estimators. Stat Med, 18:695–706, 1999.

    Article  Google Scholar 

  32. U. S. Govindarajulu, H. Lin, K. L. Lunetta, and R. B. D’Agostino. Frailty models: Applications to biomedical and genetic studies. Stat Med, 30(22):2754–2764, 2011.

    Article  MathSciNet  Google Scholar 

  33. A. J. Gross and V. A. Clark. Survival Distributions: Reliability Applications in the Biomedical Sciences. Wiley, New York, 1975.

    Google Scholar 

  34. S. T. Gross and T. L. Lai. Nonparametric estimation and regression analysis with left-truncated and right-censored data. J Am Stat Assoc, 91:1166–1180, 1996.

    Article  MathSciNet  Google Scholar 

  35. R. Henderson. Problems and prediction in survival-data analysis. Stat Med, 14:161–184, 1995.

    Article  Google Scholar 

  36. J. E. Herndon and F. E. Harrell. The restricted cubic spline hazard model. Comm Stat Th Meth, 19:639–663, 1990.

    Article  MathSciNet  Google Scholar 

  37. J. E. Herndon and F. E. Harrell. The restricted cubic spline as baseline hazard in the proportional hazards model with step function time-dependent covariables. Stat Med, 14:2119–2129, 1995.

    Article  Google Scholar 

  38. J. W. Hogan and N. M. Laird. Mixture models for the joint distribution of repeated measures and event times. Stat Med, 16:239–257, 1997.

    Article  Google Scholar 

  39. J. W. Hogan and N. M. Laird. Model-based approaches to analysing incomplete longitudinal and failure time data. Stat Med, 16:259–272, 1997.

    Article  Google Scholar 

  40. M. Hollander, I. W. McKeague, and J. Yang. Likelihood ratio-based confidence bands for survival functions. J Am Stat Assoc, 92:215–226, 1997.

    Article  MathSciNet  Google Scholar 

  41. P. Hougaard. Fundamentals of survival data. Biometrics, 55:13–22, 1999.

    Article  Google Scholar 

  42. Y. Huang and M. Wang. Frequency of recurrent events at failure times: Modeling and inference. J Am Stat Assoc, 98:663–670, 2003.

    Article  Google Scholar 

  43. H. Jiang, R. Chapell, and J. P. Fine. Estimating the distribution of nonterminal event time in the presence of mortality or informative dropout. Controlled Clin Trials, 24:135–146, 2003.

    Article  Google Scholar 

  44. N. L. Johnson, S. Kotz, and N. Balakrishnan. Distributions in Statistics: Continuous Univariate Distributions, volume 1. Wiley-Interscience, New York, second edition, 1994.

    Google Scholar 

  45. J. D. Kalbfleisch and R. L. Prentice. The Statistical Analysis of Failure Time Data. Wiley, New York, 1980.

    Google Scholar 

  46. E. L. Kaplan and P. Meier. Nonparametric estimation from incomplete observations. J Am Stat Assoc, 53:457–481, 1958.

    Article  MathSciNet  Google Scholar 

  47. T. Karrison. Restricted mean life with adjustment for covariates. J Am Stat Assoc, 82:1169–1176, 1987.

    Article  MathSciNet  Google Scholar 

  48. T. G. Karrison. Use of Irwin’s restricted mean as an index for comparing survival in different treatment groups—Interpretation and power considerations. Controlled Clin Trials, 18:151–167, 1997.

    Article  Google Scholar 

  49. R. Kay. Treatment effects in competing-risks analysis of prostate cancer data. Biometrics, 42:203–211, 1986.

    Article  Google Scholar 

  50. P. J. Kelly and L. Lim. Survival analysis for recurrent event data: An application to childhood infectious diseases. Stat Med, 19:13–33, 2000.

    Article  Google Scholar 

  51. J. P. Klein, N. Keiding, and E. A. Copelan. Plotting summary predictions in multistate survival models: Probabilities of relapse and death in remission for bone marrow transplantation patients. Stat Med, 12:2314–2332, 1993.

    Article  Google Scholar 

  52. J. P. Klein and M. L. Moeschberger. Survival Analysis: Techniques for Censored and Truncated Data. Springer, New York, 1997.

    Book  Google Scholar 

  53. M. T. Koller, H. Raatz, E. W. Steyerberg, and M. Wolbers. Competing risks and the clinical community: irrelevance or ignorance? Stat Med, 31(11–12):1089–1097, 2012.

    Article  MathSciNet  Google Scholar 

  54. C. Kooperberg and D. B. Clarkson. Hazard regression with interval-censored data. Biometrics, 53:1485–1494, 1997.

    Article  Google Scholar 

  55. C. Kooperberg, C. J. Stone, and Y. K. Truong. Hazard regression. J Am Stat Assoc, 90:78–94, 1995.

    Article  MathSciNet  Google Scholar 

  56. E. L. Korn and F. J. Dorey. Applications of crude incidence curves. Stat Med, 11:813–829, 1992.

    Article  Google Scholar 

  57. R. Lancar, A. Kramar, and C. Haie-Meder. Non-parametric methods for analysing recurrent complications of varying severity. Stat Med, 14:2701–2712, 1995.

    Article  Google Scholar 

  58. M. G. Larson and G. E. Dinse. A mixture model for the regression analysis of competing risks data. Appl Stat, 34:201–211, 1985.

    Article  MathSciNet  Google Scholar 

  59. J. F. Lawless. Statistical Models and Methods for Lifetime Data. Wiley, New York, 1982.

    Google Scholar 

  60. J. F. Lawless. The analysis of recurrent events for multiple subjects. Appl Stat, 44:487–498, 1995.

    Article  Google Scholar 

  61. J. F. Lawless and C. Nadeau. Some simple robust methods for the analysis of recurrent events. Technometrics, 37:158–168, 1995.

    Article  MathSciNet  Google Scholar 

  62. E. T. Lee. Statistical Methods for Survival Data Analysis. Lifetime Learning Publications, Belmont, CA, second edition, 1980.

    Google Scholar 

  63. J. J. Lee, K. R. Hess, and J. A. Dubin. Extensions and applications of event charts. Am Statistician, 54:63–70, 2000.

    Google Scholar 

  64. D. Y. Lin. Cox regression analysis of multivariate failure time data: The marginal approach. Stat Med, 13:2233–2247, 1994.

    Article  Google Scholar 

  65. D. Y. Lin. Non-parametric inference for cumulative incidence functions in competing risks studies. Stat Med, 16:901–910, 1997.

    Article  Google Scholar 

  66. J. C. Lindsey and L. M. Ryan. Tutorial in biostatistics: Methods for interval-censored data. Stat Med, 17:219–238, 1998.

    Article  Google Scholar 

  67. M. Lunn and D. McNeil. Applying Cox regression to competing risks. Biometrics, 51:524–532, 1995.

    Article  Google Scholar 

  68. M. Mandel. Censoring and truncation—Highlighting the differences. Am Statistician, 61(4):321–324, 2007.

    Article  MathSciNet  Google Scholar 

  69. N. Mantel and D. P. Byar. Evaluation of response-time data involving transient states: An illustration using heart-transplant data. J Am Stat Assoc, 69:81–86, 1974.

    Article  Google Scholar 

  70. E. Marubini and M. G. Valsecchi. Analyzing Survival Data from Clinical Trials and Observational Studies. Wiley, Chichester, 1995.

    Google Scholar 

  71. R. G. Miller. What price Kaplan–Meier? Biometrics, 39:1077–1081, 1983.

    Article  MathSciNet  Google Scholar 

  72. G. S. Mudholkar, D. K. Srivastava, and G. D. Kollia. A generalization of the Weibull distribution with application to the analysis of survival data. J Am Stat Assoc, 91:1575–1583, 1996.

    Article  MathSciNet  Google Scholar 

  73. W. B. Nelson. Theory and applications of hazard plotting for censored failure data. Technometrics, 14:945–965, 1972.

    Article  Google Scholar 

  74. M. Nishikawa, T. Tango, and M. Ogawa. Non-parametric inference of adverse events under informative censoring. Stat Med, 25:3981–4003, 2006.

    Article  MathSciNet  Google Scholar 

  75. M. K. B. Parmar and D. Machin. Survival Analysis: A Practical Approach. Wiley, Chichester, 1995.

    Google Scholar 

  76. M. S. Pepe. Inference for events with dependent risks in multiple endpoint studies. J Am Stat Assoc, 86:770–778, 1991.

    Article  MathSciNet  Google Scholar 

  77. M. S. Pepe and J. Cai. Some graphical displays and marginal regression analyses for recurrent failure times and time dependent covariates. J Am Stat Assoc, 88:811–820, 1993.

    Article  Google Scholar 

  78. M. S. Pepe, G. Longton, and M. Thornquist. A qualifier Q for the survival function to describe the prevalence of a transient condition. Stat Med, 10: 413–421, 1991.

    Article  Google Scholar 

  79. M. S. Pepe and M. Mori. Kaplan–Meier, marginal or conditional probability curves in summarizing competing risks failure time data? Stat Med, 12: 737–751, 1993.

    Article  Google Scholar 

  80. R. L. Prentice, J. D. Kalbfleisch, A. V. Peterson, N. Flournoy, V. T. Farewell, and N. E. Breslow. The analysis of failure times in the presence of competing risks. Biometrics, 34:541–554, 1978.

    Article  Google Scholar 

  81. H. Putter, M. Fiocco, and R. B. Geskus. Tutorial in biostatistics: Competing risks and multi-state models. Stat Med, 26:2389–2430, 2007.

    Article  MathSciNet  Google Scholar 

  82. B. D. Ripley and P. J. Solomon. Statistical models for prevalent cohort data. Biometrics, 51:373–374, 1995.

    Google Scholar 

  83. Y. Shen and P. F. Thall. Parametric likelihoods for multiple non-fatal competing risks and death. Stat Med, 17:999–1015, 1998.

    Article  Google Scholar 

  84. R. Simon and R. W. Makuch. A non-parametric graphical representation of the relationship between survival and the occurrence of an event: Application to responder versus non-responder bias. Stat Med, 3:35–44, 1984.

    Article  Google Scholar 

  85. J. D. Singer and J. B. Willett. Modeling the days of our lives: Using survival analysis when designing and analyzing longitudinal studies of duration and the timing of events. Psych Bull, 110:268–290, 1991.

    Article  Google Scholar 

  86. C. J. Stone, M. H. Hansen, C. Kooperberg, and Y. K. Truong. Polynomial splines and their tensor products in extended linear modeling (with discussion). Ann Stat, 25:1371–1470, 1997.

    Article  MathSciNet  Google Scholar 

  87. D. Strauss and R. Shavelle. An extended Kaplan–Meier estimator and its applications. Stat Med, 17:971–982, 1998.

    Article  Google Scholar 

  88. B. Tai, D. Machin, I. White, and V. Gebski. Competing risks analysis of patients with osteosarcoma: a comparison of four different approaches. Stat Med, 20:661–684, 2001.

    Article  Google Scholar 

  89. T. Therneau and P. Grambsch. Modeling Survival Data: Extending the Cox Model. Springer-Verlag, New York, 2000.

    Book  Google Scholar 

  90. T. M. Therneau, P. M. Grambsch, and T. R. Fleming. Martingale-based residuals for survival models. Biometrika, 77:216–218, 1990.

    Article  MathSciNet  Google Scholar 

  91. T. M. Therneau and S. A. Hamilton. rhDNase as an example of recurrent event analysis. Stat Med, 16:2029–2047, 1997.

    Google Scholar 

  92. W. Y. Tsai, N. P. Jewell, and M. C. Wang. A note on the product limit estimator under right censoring and left truncation. Biometrika, 74:883–886, 1987.

    Article  Google Scholar 

  93. B. W. Turnbull. Nonparametric estimation of a survivorship function with doubly censored data. J Am Stat Assoc, 69:169–173, 1974.

    Article  MathSciNet  Google Scholar 

  94. Ü. Uzuno=gullari and J.-L. Wang. A comparison of hazard rate estimators for left truncated and right censored data. Biometrika, 79:297–310, 1992.

    Google Scholar 

  95. M. Wang and S. Chang. Nonparametric estimation of a recurrent survival function. J Am Stat Assoc, 94:146–153, 1999.

    Article  MathSciNet  Google Scholar 

  96. L. J. Wei, D. Y. Lin, and L. Weissfeld. Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. J Am Stat Assoc, 84:1065–1073, 1989.

    Article  MathSciNet  Google Scholar 

  97. A. S. Whittemore and J. B. Keller. Survival estimation using splines. Biometrics, 42:495–506, 1986.

    Article  MathSciNet  Google Scholar 

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  1. Department of Biostatistics, School of Medicine Vanderbilt University, Nashville, TN, USA

    Frank E. Harrell Jr.

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Harrell, F.E. (2015). Introduction to Survival Analysis. In: Regression Modeling Strategies. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-19425-7_17

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