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Cox Proportional Hazards Regression Model

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Regression Modeling Strategies

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

Abstract

The Cox proportional hazards model 132 is the most popular model for the analysis of survival data. It is a semiparametric model; it makes a parametric assumption concerning the effect of the predictors on the hazard function, but makes no assumption regarding the nature of the hazard function λ(t) itself. The Cox PH model assumes that predictors act multiplicatively on the hazard function but does not assume that the hazard function is constant (i.e., exponential model), Weibull, or any other particular form. The regression portion of the model is fully parametric; that is, the regressors are linearly related to log hazard or log cumulative hazard. In many situations, either the form of the true hazard function is unknown or it is complex, so the Cox model has definite advantages. Also, one is usually more interested in the effects of the predictors than in the shape of λ(t), and the Cox approach allows the analyst to essentially ignore λ(t), which is often not of primary interest.

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References

  1. O. O. Aalen. Further results on the non-parametric linear regression model in survival analysis. Stat Med, 12:1569–1588, 1993.

    Article  Google Scholar 

  2. M. Abrahamowicz, T. MacKenzie, and J. M. Esdaile. Time-dependent hazard ratio: Modeling and hypothesis testing with applications in lupus nephritis. JAMA, 91:1432–1439, 1996.

    MATH  Google Scholar 

  3. D. G. Altman and P. K. Andersen. A note on the uncertainty of a survival probability estimated from Cox’s regression model. Biometrika, 73:722–724, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. G. Altman and P. Royston. What do we mean by validating a prognostic model? Stat Med, 19:453–473, 2000.

    Article  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  MATH  Google Scholar 

  6. J. A. Anderson and A. Senthilselvan. A two-step regression model for hazard functions. Appl Stat, 31:44–51, 1982.

    Article  Google Scholar 

  7. 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 

  8. N. E. Breslow. Covariance analysis of censored survival data. Biometrics, 30:89–99, 1974.

    Article  Google Scholar 

  9. N. E. Breslow, N. E. Day, K. T. Halvorsen, R. L. Prentice, and C. Sabai. Estimation of multiple relative risk functions in matched case-control studies. Am J Epi, 108:299–307, 1978.

    Google Scholar 

  10. N. E. Breslow, L. Edler, and J. Berger. A two-sample censored-data rank test for acceleration. Biometrics, 40:1049–1062, 1984.

    Article  MATH  Google Scholar 

  11. B. W. Brown, M. Hollander, and R. M. Korwar. Nonparametric tests of independence for censored data, with applications to heart transplant studies. In F. Proschan and R. J. Serfling, editors, Reliability and Biometry, pages 327–354. SIAM, Philadelphia, 1974.

    Google Scholar 

  12. R. M. Califf, F. E. Harrell, K. L. Lee, J. S. Rankin, and Others. The evolution of medical and surgical therapy for coronary artery disease. JAMA, 261:2077–2086, 1989.

    Google Scholar 

  13. W. H. Carter, G. L. Wampler, and D. M. Stablein. Regression Analysis of Survival Data in Cancer Chemotherapy. Marcel Dekker, New York, 1983.

    Google Scholar 

  14. J. M. Chambers and T. J. Hastie, editors. Statistical Models in S. Wadsworth and Brooks/Cole, Pacific Grove, CA, 1992.

    MATH  Google Scholar 

  15. R. Chappell. A note on linear rank tests and Gill and Schumacher’s tests of proportionality. Biometrika, 79:199–201, 1992.

    Article  Google Scholar 

  16. S. C. Cheng, L. J. Wei, and Z. Ying. Predicting Survival Probabilities with Semiparametric Transformation Models. JASA, 92(437):227–235, Mar. 1997.

    Article  MathSciNet  MATH  Google Scholar 

  17. B. Choodari-Oskooei, P. Royston, and M. K. B. Parmar. A simulation study of predictive ability measures in a survival model II: explained randomness and predictive accuracy. Stat Med, 31(23):2644–2659, 2012.

    Article  MathSciNet  Google Scholar 

  18. B. Choodari-Oskooei, P. Royston, and M. K. B. Parmar. A simulation study of predictive ability measures in a survival model I: Explained variation measures. Stat Med, 31(23):2627–2643, 2012.

    Article  MathSciNet  Google Scholar 

  19. W. S. Cleveland. Robust locally weighted regression and smoothing scatterplots. J Am Stat Assoc, 74:829–836, 1979.

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  21. J. B. Copas. Cross-validation shrinkage of regression predictors. J Roy Stat Soc B, 49:175–183, 1987.

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  24. L. A. Cupples, D. R. Gagnon, R. Ramaswamy, and R. B. D’Agostino. Age-adjusted survival curves with application in the Framingham Study. Stat Med, 14:1731–1744, 1995.

    Article  Google Scholar 

  25. D. M. Dabrowska, K. A. Doksum, N. J. Feduska, R. Husing, and P. Neville. Methods for comparing cumulative hazard functions in a semi-proportional hazard model. Stat Med, 11:1465–1476, 1992.

    Article  Google Scholar 

  26. B. Efron. The two sample problem with censored data. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, volume 4, pages 831–853. 1967.

    Google Scholar 

  27. B. Efron. The efficiency of Cox’s likelihood function for censored data. J Am Stat Assoc, 72:557–565, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  28. G. E. Eide, E. Omenaas, and A. Gulsvik. The semi-proportional hazards model revisited: Practical reparameterizations. Stat Med, 15:1771–1777, 1996.

    Article  Google Scholar 

  29. J. H. Friedman. A variable span smoother. Technical Report 5, Laboratory for Computational Statistics, Department of Statistics, Stanford University, 1984.

    Google Scholar 

  30. M. Gönen and G. Heller. Concordance probability and discriminatory power in proportional hazards regression. Biometrika, 92(4):965–970, Dec. 2005.

    Article  MathSciNet  MATH  Google Scholar 

  31. S. M. Gore, S. J. Pocock, and G. R. Kerr. Regression models and non-proportional hazards in the analysis of breast cancer survival. Appl Stat, 33:176–195, 1984.

    Article  Google Scholar 

  32. P. Grambsch and T. Therneau. Proportional hazards tests and diagnostics based on weighted residuals. Biometrika, 81:515–526, 1994. Amendment and corrections in 82: 668 (1995).

    Google Scholar 

  33. R. J. Gray. Some diagnostic methods for Cox regression models through hazard smoothing. Biometrics, 46:93–102, 1990.

    Article  MATH  Google Scholar 

  34. R. J. Gray. Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis. J Am Stat Assoc, 87:942–951, 1992.

    Article  Google Scholar 

  35. R. J. Gray. Spline-based tests in survival analysis. Biometrics, 50:640–652, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  36. F. E. Harrell. The PHGLM Procedure. In SUGI Supplemental Library Users Guide, pages 437–466. SAS Institute, Inc., Cary, NC, Version 5 edition, 1986.

    Google Scholar 

  37. F. E. Harrell, R. M. Califf, D. B. Pryor, K. L. Lee, and R. A. Rosati. Evaluating the yield of medical tests. JAMA, 247:2543–2546, 1982.

    Article  Google Scholar 

  38. F. E. Harrell and K. L. Lee. Verifying assumptions of the Cox proportional hazards model. In Proceedings of the Eleventh Annual SAS Users Group International Conference, pages 823–828, Cary, NC, 1986. SAS Institute, Inc.

    Google Scholar 

  39. F. E. Harrell and K. L. Lee. Using logistic model calibration to assess the quality of probability predictions. Unpublished manuscript, 1987.

    Google Scholar 

  40. F. E. Harrell, K. L. Lee, R. M. Califf, D. B. Pryor, and R. A. Rosati. Regression modeling strategies for improved prognostic prediction. Stat Med, 3:143–152, 1984.

    Article  Google Scholar 

  41. D. P. Harrington and T. R. Fleming. A class of rank test procedures for censored survival data. Biometrika, 69:553–566, 1982.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  43. R. Henderson, M. Jones, and J. Stare. Accuracy of point predictions in survival analysis. Stat Med, 20:3083–3096, 2001.

    Article  Google Scholar 

  44. 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 

  45. I. Hertz-Picciotto and B. Rockhill. Validity and efficiency of approximation methods for tied survival times in Cox regression. Biometrics, 53:1151–1156, 1997.

    Article  MATH  Google Scholar 

  46. K. R. Hess. Assessing time-by-covariate interactions in proportional hazards regression models using cubic spline functions. Stat Med, 13:1045–1062, 1994.

    Article  Google Scholar 

  47. K. R. Hess. Graphical methods for assessing violations of the proportional hazards assumption in Cox regression. Stat Med, 14:1707–1723, 1995.

    Article  Google Scholar 

  48. T. Hielscher, M. Zucknick, W. Werft, and A. Benner. On the prognostic value of survival models with application to gene expression signatures. Stat Med, 29:818–829, 2010.

    Article  MathSciNet  Google Scholar 

  49. J. Huang and D. Harrington. Penalized partial likelihood regression for right-censored data with bootstrap selection of the penalty parameter. Biometrics, 58:781–791, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  50. J. D. Kalbfleisch and R. L. Prentice. Marginal likelihood based on Cox’s regression and life model. Biometrika, 60:267–278, 1973.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  53. 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 

  54. M. W. Kattan, G. Heller, and M. F. Brennan. A competing-risks nomogram for sarcoma-specific death following local recurrence. Stat Med, 22:3515–3525, 2003.

    Article  Google Scholar 

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

    Article  Google Scholar 

  56. J. T. Kent and J. O’Quigley. Measures of dependence for censored survival data. Biometrika, 75:525–534, 1988.

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  59. E. L. Korn and R. Simon. Measures of explained variation for survival data. Stat Med, 9:487–503, 1990.

    Article  Google Scholar 

  60. D. Kronborg and P. Aaby. Piecewise comparison of survival functions in stratified proportional hazards models. Biometrics, 46:375–380, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  61. J. M. Lachin and M. A. Foulkes. Evaluation of sample size and power for analyses of survival with allowance for nonuniform patient entry, losses to follow-up, noncompliance, and stratification. Biometrics, 42:507–519, 1986.

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  63. J. F. Lawless and Y. Yuan. Estimation of prediction error for survival models. Stat Med, 29:262–274, 2010.

    MathSciNet  Google Scholar 

  64. S. Lehr and M. Schemper. Parsimonious analysis of time-dependent effects in the Cox model. Stat Med, 26:2686–2698, 2007.

    Article  MathSciNet  Google Scholar 

  65. L. F. León and C. Tsai. Functional form diagnostics for Cox’s proportional hazards model. Biometrics, 60:75–84, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  66. D. Y. Lin and L. J. Wei. The robust inference for the Cox proportional hazards model. J Am Stat Assoc, 84:1074–1078, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  67. D. Y. Lin, L. J. Wei, and Z. Ying. Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika, 80:557–572, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  68. G. S. Maddala. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press, Cambridge, UK, 1983.

    Book  MATH  Google Scholar 

  69. L. Magee. R 2 measures based on Wald and likelihood ratio joint significance tests. Am Statistician, 44:250–253, 1990.

    Google Scholar 

  70. M. Mandel, N. Galae, and E. Simchen. Evaluating survival model performance: a graphical approach. Stat Med, 24:1933–1945, 2005.

    Article  Google Scholar 

  71. D. B. Mark, M. A. Hlatky, F. E. Harrell, K. L. Lee, R. M. Califf, and D. B. Pryor. Exercise treadmill score for predicting prognosis in coronary artery disease. Ann Int Med, 106:793–800, 1987.

    Article  Google Scholar 

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

    Google Scholar 

  73. M. May, P. Royston, M. Egger, A. C. Justice, and J. A. C. Sterne. Development and validation of a prognostic model for survival time data: application to prognosis of HIV positive patients treated with antiretroviral therapy. Stat Med, 23:2375–2398, 2004.

    Article  Google Scholar 

  74. L. R. Muenz. Comparing survival distributions: A review for nonstatisticians. II. Ca Invest, 1:537–545, 1983.

    Google Scholar 

  75. V. M. R. Muggeo and M. Tagliavia. A flexible approach to the crossing hazards problem. Stat Med, 29:1947–1957, 2010.

    Article  MathSciNet  Google Scholar 

  76. N. J. D. Nagelkerke. A note on a general definition of the coefficient of determination. Biometrika, 78:691–692, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  77. N. H. Ng’andu. An empirical comparison of statistical tests for assessing the proportional hazards assumption of Cox’s model. Stat Med, 16:611–626, 1997.

    Article  Google Scholar 

  78. J. O’Quigley, R. Xu, and J. Stare. Explained randomness in proportional hazards models. Stat Med, 24(3):479–489, 2005.

    Article  MathSciNet  Google Scholar 

  79. M. J. Pencina and R. B. D’Agostino. Overall C as a measure of discrimination in survival analysis: model specific population value and confidence interval estimation. Stat Med, 23:2109–2123, 2004.

    Article  Google Scholar 

  80. M. J. Pencina, R. B. D’Agostino, and L. Song. Quantifying discrimination of Framingham risk functions with different survival C statistics. Stat Med, 31(15):1543–1553, 2012.

    Article  MathSciNet  Google Scholar 

  81. A. Perperoglou, A. Keramopoullos, and H. C. van Houwelingen. Approaches in modelling long-term survival: An application to breast cancer. Stat Med, 26:2666–2685, 2007.

    Article  MathSciNet  Google Scholar 

  82. A. Perperoglou, S. le Cessie, and H. C. van Houwelingen. Reduced-rank hazard regression for modelling non-proportional hazards. Stat Med, 25:2831–2845, 2006.

    Article  MathSciNet  Google Scholar 

  83. B. Peterson and S. L. George. Sample size requirements and length of study for testing interaction in a 1 × k factorial design when time-to-failure is the outcome. Controlled Clin Trials, 14:511–522, 1993.

    Article  Google Scholar 

  84. A. N. Pettitt and I. Bin Daud. Investigating time dependence in Cox’s proportional hazards model. Appl Stat, 39:313–329, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  85. M. C. Pike. A method of analysis of certain class of experiments in carcinogenesis. Biometrics, 22:142–161, 1966.

    Article  Google Scholar 

  86. D. B. Pryor, F. E. Harrell, J. S. Rankin, K. L. Lee, L. H. Muhlbaier, H. N. Oldham, M. A. Hlatky, D. B. Mark, J. G. Reves, and R. M. Califf. The changing survival benefits of coronary revascularization over time. Circulation (Supplement V), 76:13–21, 1987.

    Google Scholar 

  87. H. Putter, M. Sasako, H. H. Hartgrink, C. J. H. van de Velde, and J. C. van Houwelingen. Long-term survival with non-proportional hazards: results from the Dutch Gastric Cancer Trial. Stat Med, 24:2807–2821, 2005.

    Article  MathSciNet  Google Scholar 

  88. C. Quantin, T. Moreau, B. Asselain, J. Maccaria, and J. Lellouch. A regression survival model for testing the proportional hazards assumption. Biometrics, 52:874–885, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  89. S. Sahoo and D. Sengupta. Some diagnostic plots and corrective adjustments for the proportional hazards regression model. J Comp Graph Stat, 20(2):375–394, 2011.

    Article  MathSciNet  Google Scholar 

  90. D. E. Schaubel, R. A. Wolfe, and R. M. Merion. Estimating the effect of a time-dependent treatment by levels of an internal time-dependent covariate: Application to the contrast between liver wait-list and posttransplant mortality. J Am Stat Assoc, 104(485):49–59, 2009.

    Article  MathSciNet  Google Scholar 

  91. M. Schemper. Analyses of associations with censored data by generalized Mantel and Breslow tests and generalized Kendall correlation. Biometrical J, 26:309–318, 1984.

    Article  MathSciNet  Google Scholar 

  92. M. Schemper. The explained variation in proportional hazards regression (correction in 81:631, 1994). Biometrika, 77:216–218, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  93. M. Schemper. Cox analysis of survival data with non-proportional hazard functions. The Statistician, 41:445–455, 1992.

    Article  Google Scholar 

  94. M. Schemper. Further results on the explained variation in proportional hazards regression. Biometrika, 79:202–204, 1992.

    Article  Google Scholar 

  95. M. Schemper. The relative importance of prognostic factors in studies of survival. Stat Med, 12:2377–2382, 1993.

    Article  Google Scholar 

  96. M. Schemper. Predictive accuracy and explained variation. Stat Med, 22:2299–2308, 2003.

    Article  Google Scholar 

  97. M. Schemper and R. Henderson. Predictive accuracy and explained variation in Cox regression. Biometrics, 56:249–255, 2000.

    Article  MATH  Google Scholar 

  98. M. Schemper and J. Stare. Explained variation in survival analysis. Stat Med, 15:1999–2012, 1996.

    Article  Google Scholar 

  99. M. Schmid and S. Potapov. A comparison of estimators to evaluate the discriminatory power of time-to-event models. Stat Med, 31(23):2588–2609, 2012.

    Article  MathSciNet  Google Scholar 

  100. D. Schoenfeld. Partial residuals for the proportional hazards regression model. Biometrika, 69:239–241, 1982.

    Article  Google Scholar 

  101. D. A. Schoenfeld. Sample size formulae for the proportional hazards regression model. Biometrics, 39:499–503, 1983.

    Article  MATH  Google Scholar 

  102. R. H. Somers. A new asymmetric measure of association for ordinal variables. Am Soc Rev, 27:799–811, 1962.

    Article  Google Scholar 

  103. D. M. Stablein, W. H. Carter, and J. W. Novak. Analysis of survival data with nonproportional hazard functions. Controlled Clin Trials, 2:149–159, 1981.

    Article  Google Scholar 

  104. H. T. Thaler. Nonparametric estimation of the hazard ratio. J Am Stat Assoc, 79:290–293, 1984.

    Article  Google Scholar 

  105. P. F. Thall and J. M. Lachin. Assessment of stratum-covariate interactions in Cox’s proportional hazards regression model. Stat Med, 5:73–83, 1986.

    Article  Google Scholar 

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

    Book  Google Scholar 

  107. 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 

  108. A. A. Tsiatis. A large sample study of Cox’s regression model. Ann Stat, 9:93–108, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  109. H. Uno, T. Cai, M. J. Pencina, R. B. D’Agostino, and L. J. Wei. On the C-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data. Stat Med, 30:1105–1117, 2011.

    MathSciNet  Google Scholar 

  110. M. G. Valsecchi, D. Silvestri, and P. Sasieni. Evaluation of long-term survival: Use of diagnostics and robust estimators with Cox’s proportional hazards model. Stat Med, 15:2763–2780, 1996.

    Article  Google Scholar 

  111. J. C. van Houwelingen and S. le Cessie. Predictive value of statistical models. Stat Med, 9:1303–1325, 1990.

    Article  Google Scholar 

  112. P. J. M. Verweij and H. C. van Houwelingen. Cross-validation in survival analysis. Stat Med, 12:2305–2314, 1993.

    Article  Google Scholar 

  113. P. J. M. Verweij and H. C. van Houwelingen. Time-dependent effects of fixed covariates in Cox regression. Biometrics, 51:1550–1556, 1995.

    Article  MATH  Google Scholar 

  114. A. Winnett and P. Sasieni. A note on scaled Schoenfeld residuals for the proportional hazards model. Biometrika, 88:565–571, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  115. A. Winnett and P. Sasieni. Iterated residuals and time-varying covariate effects in Cox regression. J Roy Stat Soc B, 65:473–488, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  116. D. M. Zucker. The efficiency of a weighted log-rank test under a percent error misspecification model for the log hazard ratio. Biometrics, 48:893–899, 1992.

    Article  MathSciNet  MATH  Google Scholar 

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    Frank E. Harrell Jr.

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Harrell, F.E. (2015). Cox Proportional Hazards Regression Model. In: Regression Modeling Strategies. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-19425-7_20

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