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Survival Analysis

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Fundamentals of Clinical Trials

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Abstract

This chapter reviews some of the fundamental concepts and basic methods in survival analysis. Frequently, event rates such as mortality or occurrence of nonfatal myocardial infarction are selected as primary response variables. The analysis of such event rates in two groups could employ the chi-square statistic or the equivalent normal statistic for the comparison of two proportions. However, when the length of observation is different for each participant, estimating an event rate is more complicated. Furthermore, simple comparison of event rates between two groups is not necessarily the most informative type of analysis. For example, the 5-year survival for two groups may be nearly identical, but the survival rates may be quite different at various times during the 5 years. This is illustrated by the survival curves in Fig. 15.1. This figure shows survival probability on the vertical axis and time on the horizontal axis. For Group A, the survival rate (or one minus the mortality rate) declines steadily over the 5 years of observation. For Group B, however, the decline in the survival rate is rapid during the first year and then levels off. Obviously, the survival experience of the two groups is not the same, although the mortality rate at 5 years is nearly the same. If only the 5-year survival rate is considered, Group A and Group B appear equivalent. Curves such as these might reasonably be expected in a trial of surgical versus medical intervention, where surgery might carry a high initial operative mortality.

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References

  1. Altman DG. Practical Statistics for Medical Research. Chapman and Hall, 2015.

    Google Scholar 

  2. Armitage P, Berry G, Mathews J. Statistical Methods in Medical Research, ed 4th. Malden MA, Blackwell Publishing, 2002.

    Book  Google Scholar 

  3. Breslow NE. Comparison of survival curves; in Buyse B, Staquet M, Sylvester R (eds): The Practice of Clinical Trials in Cancer. Oxford, Oxford University Press, 1982.

    Google Scholar 

  4. Brown BW, Hollander M. Statistics: A Biomedical Introduction. Wiley, 2009.

    Google Scholar 

  5. Fisher L, Van Belle G, Heagerty PL, Lumley TS. Biostatistics—A Methodology for the Health Sciences. New York, John Wiley and Sons, 2004.

    Google Scholar 

  6. Woolson RF, Clarke WR. Statistical Methods for the Analysis of Biomedical Data. Wiley, 2011.

    Google Scholar 

  7. Crowley J, Breslow N. Statistical Analysis of Survival Data. Annu Rev Public Health 1984;5:385–411.

    Article  Google Scholar 

  8. Cox DR, Oakes D. Analysis of Survival Data. Taylor & Francis, 1984.

    Google Scholar 

  9. Fleming TR, Harrington DP. Counting Processes and Survival Analysis. Wiley, 2011.

    Google Scholar 

  10. Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. Wiley, 2011.

    Google Scholar 

  11. Miller RG. Survival Analysis. Wiley, 2011.

    Google Scholar 

  12. Cutler SJ, Ederer F. Maximum utilization of the life table method in analyzing survival. J Chronic Dis 1958;8:699–712.

    Article  Google Scholar 

  13. Kaplan EL, Meier P. Nonparametric Estimation from Incomplete Observations. J Am Stat Assoc 1958;53:457–481.

    Article  MathSciNet  Google Scholar 

  14. Chan MCY, Giannetti N, Kato T, et al. Severe tricuspid regurgitation after heart transplantation. J Heart Lung Transplant 2001;20:709–717.

    Article  Google Scholar 

  15. Kumagai R, Kubokura M, Sano A, et al. Clinical evaluation of percutaneous endoscopic gastrostomy tube feeding in Japanese patients with dementia. Psychiatry Clin Neurosci 2012;66:418–422.

    Article  Google Scholar 

  16. Lara-Gonzalez JH, Gomez-Flores R, Tamez-Guerra P, et al. In Vivo Antitumor Activity of Metal Silver and Silver Nanoparticles in the L5178Y-R Murine Lymphoma Model. Br J Med Med Res 2013;3:1308–1316.

    Article  Google Scholar 

  17. Miyamoto K, Aida A, Nishimura M, et al. Gender effect on prognosis of patients receiving long-term home oxygen therapy. The Respiratory Failure Research Group in Japan. Am J Respir Crit Care Med 1995;152:972–976.

    Article  Google Scholar 

  18. Sarris GE, Robbins RC, Miller DC, et al. Randomized, prospective assessment of bioprosthetic valve durability. Hancock versus Carpentier-Edwards valves. Circulation 1993;88:II55-II64.

    Google Scholar 

  19. Greenwood M. The natural duration of cancer. Reports on Public Health and Medical Subjects 1926;33:1–26.

    Google Scholar 

  20. Everitt BS, Rabe-Hesketh S. Handbook of Statistical Analyses Using Stata, Fourth Edition. Taylor & Francis, 2006.

    Google Scholar 

  21. R Core Team. R: A Language and Environment for Statistical Computing. Vienna, Austria, R Foundation for Statistical Computing, 2013.

    Google Scholar 

  22. SAS Institute: SAS/STAT 12.1 User’s Guide: Survival Analysis. SAS Institute, 2012.

    Google Scholar 

  23. TIBCO Software I: SPLUS. TIBCO Softward Inc., 2008.

    Google Scholar 

  24. Nelson W. Hazard plotting for incomplete failure data. Journal of Quality Technology 1969;1:27–52.

    Article  Google Scholar 

  25. Brookmeyer R, Crowley J. A Confidence Interval for the Median Survival Time. Biometrics 1982;38:29–41.

    Article  Google Scholar 

  26. Gehan EA. A Generalized Wilcoxon Test for Comparing Arbitrarily Singly-Censored Samples. Biometrika 1965;52:203–223.

    Article  MathSciNet  Google Scholar 

  27. Mantel N. Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer chemotherapy reports Part 1 1966;50:163–170.

    Google Scholar 

  28. Cochran WG. Some Methods for Strengthening the Common-ç2 Tests. Biometrics 1954;10:417–451.

    Article  MathSciNet  Google Scholar 

  29. Mantel N, Haenszel W. Statistical Aspects of the Analysis of Data From Retrospective Studies of Disease. J Natl Cancer Inst 1959;22:719–748.

    Google Scholar 

  30. Crowley J, Breslow N. Remarks on the Conservatism of Sigma(0-E)2/E in Survival Data. Biometrics 1975;31:957–961.

    Article  Google Scholar 

  31. Peto R, Pike MC. Conservatism of the Approximation Sigma (O-E)2-E in the Logrank Test for Survival Data or Tumor Incidence Data. Biometrics 1973;29:579–584.

    Article  MathSciNet  Google Scholar 

  32. Mantel N. Ranking Procedures for Arbitrarily Restricted Observation. Biometrics 1967;23:65-78.

    Article  MathSciNet  Google Scholar 

  33. Breslow NE. A Generalized Kruskal-Wallis Test for Comparing K Samples Subject to Unequal Patterns of Censorship. Biometrika 1970;57:579–594.

    Article  Google Scholar 

  34. Peto R, Peto J. Asymptotically Efficient Rank Invariant Test Procedures. J R Stat Soc Ser A 1972;135:185–207.

    Article  Google Scholar 

  35. Harrington DP, Fleming TR. A Class of Rank Test Procedures for Censored Survival Data. Biometrika 982;69:553–566.

    Google Scholar 

  36. Leurgans S. Three Classes of Censored Data Rank Tests: Strengths and Weaknesses under Censoring. Biometrika 1983;70:651–658.

    Article  MathSciNet  Google Scholar 

  37. Oakes D. The Asymptotic Information in Censored Survival Data. Biometrika 1977;64:441–448.

    Article  MathSciNet  Google Scholar 

  38. Prentice RL. Linear Rank Tests with Right Censored Data. Biometrika 1978;65:167–179.

    Article  MathSciNet  Google Scholar 

  39. Schoenfeld DA. The Asymptotic Properties of Nonparametric Tests for Comparing Survival Distributions. Biometrika 1981;68:316–319.

    Article  MathSciNet  Google Scholar 

  40. Tarone RE, Ware J. On Distribution-Free Tests for Equality of Survival Distributions. Biometrika 1977;64:156–160.

    Article  MathSciNet  Google Scholar 

  41. Simon R. Confidence Intervals for Reporting Results of Clinical Trials. Ann Intern Med 1986;105:429–435.

    Article  Google Scholar 

  42. Cox DR. Regression Models and Life-Tables. J R Stat Soc Series B Stat Methodol 1972;34:187–220.

    MathSciNet  Google Scholar 

  43. Feigl P, Zelen M. Estimation of Exponential Survival Probabilities with Concomitant Information. Biometrics 1965;21:826–838.

    Article  Google Scholar 

  44. Prentice RL, Kalbfleisch JD. Hazard Rate Models with Covariates. Biometrics 1979;35:25–39.

    Article  MathSciNet  Google Scholar 

  45. Zelen M. Application of Exponential Models to Problems in Cancer Research. J R Stat Soc Ser A 1966;129:368–398.

    Article  Google Scholar 

  46. Fisher L, Van Belle G. Biostatistics—A Methodology for the Health Sciences. New York, John Wiley and Sons, 1993.

    Google Scholar 

  47. Breslow NE. Covariance Analysis of Censored Survival Data. Biometrics 1974;30:89–99.

    Article  Google Scholar 

  48. Breslow NE. Analysis of Survival Data under the Proportional Hazards Model. International Statistical Review/Revue Internationale de Statistique 1975;43:45–57.

    Google Scholar 

  49. Efron B. The Efficiency of Cox’s Likelihood Function for Censored Data. J Am Stat Assoc 1977;72:557–565.

    Article  MathSciNet  Google Scholar 

  50. Kalbfleisch JD, Prentice RL. Marginal Likelihoods Based on Cox’s Regression and Life Model. Biometrika 1973;60:267–278.

    Article  MathSciNet  Google Scholar 

  51. Kay R. Proportional Hazard Regression Models and the Analysis of Censored Survival Data. J R Stat Soc Ser C Appl Stat 1977;26:227–237.

    Google Scholar 

  52. Pocock SJ. Interim analyses for randomized clinical trials: the group sequential approach. Biometrics 1982;38:153–162.

    Article  Google Scholar 

  53. Prentice RL, Gloeckler LA. Regression Analysis of Grouped Survival Data with Application to Breast Cancer Data. Biometrics 1978;34:57–67.

    Article  Google Scholar 

  54. Schoenfeld DA. Chi-Squared Goodness-of-Fit Tests for the Proportional Hazards Regression Model. Biometrika 1980;67:145–153.

    Article  MathSciNet  Google Scholar 

  55. Tsiatis AA. A Large Sample Study of Cox’s Regression Model. Ann Stat 1981;9:93–108.

    Article  MathSciNet  Google Scholar 

  56. Lagakos SW, Schoenfeld DA. Properties of Proportional-Hazards Score Tests under Misspecified Regression Models. Biometrics 1984;40:1037–1048.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. North Bethesda, MD, USA

    Lawrence M. Friedman

  2. Division of Public Health Sciences, Wake Forest School of Medicine, Winston-Salem, NC, USA

    Curt D. Furberg

  3. Department Biostatistics and Medical Informatics, University of Wisconsin, Madison, WI, USA

    David L. DeMets

  4. Department of Biostatistics, Wake Forest School of Medicine, Winston-Salem, NC, USA

    David M. Reboussin

  5. Department of Medicine, Duke University, Durham, NC, USA

    Christopher B. Granger

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  1. Lawrence M. Friedman
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  2. Curt D. Furberg
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  3. David L. DeMets
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  4. David M. Reboussin
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  5. Christopher B. Granger
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Friedman, L.M., Furberg, C.D., DeMets, D.L., Reboussin, D.M., Granger, C.B. (2015). Survival Analysis. In: Fundamentals of Clinical Trials. Springer, Cham. https://doi.org/10.1007/978-3-319-18539-2_15

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