Lifetime Data Analysis

, Volume 13, Issue 3, pp 317–332 | Cite as

On proportional hazards assumption under the random effects models

  • Ronghui Xu
  • Anthony Gamst


The proportional hazards mixed-effects model (PHMM) was proposed to handle dependent survival data. Motivated by its application in genetic epidemiology, we study the interpretation of its parameter estimates under violations of the proportional hazards assumption. The estimated fixed effect turns out to be an averaged regression effect over time, while the estimated variance component could be unaffected, inflated or attenuated depending on whether the random effect is on the baseline hazard, and whether the non-proportional regression effect decreases or increases over time. Using the conditional distribution of the covariates we define the standardized covariate residuals, which can be used to check the proportional hazards assumption. The model checking technique is illustrated on a multi-center lung cancer trial.


Average regression effect Frailty Variance components EM algorithm Standardized covariate residual 


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  1. Chang IS and Hsiung CA (1996). An efficient estimator for proportional hazards models with frailties. Scand J Stat 23: 13–26 MATHMathSciNetGoogle Scholar
  2. Chang IS, Hsiung CA, Wang MC and Wen CC (2005). An asymptotic theory for the nonparametric maximum likelihood estimator in the Cox gene model. Bernoulli 11: 863–892 MATHCrossRefMathSciNetGoogle Scholar
  3. Cox DR (1972). Regression models and life tables (with discussion). J R Stat Soc, Ser B 34: 187–220 MATHGoogle Scholar
  4. Cox DR (1975). Partial likelihood. Biometrika 62: 269–276 MATHCrossRefMathSciNetGoogle Scholar
  5. Economou P and Caroni C (2005). Graphical tests for the assumption of gamma and inverse Gaussian frailty distribution. Lifetime Data Anal 11: 565–582 MATHCrossRefMathSciNetGoogle Scholar
  6. Gail MH, Wieand S and Piantadosi S (1984). Biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates. Biometrika 71: 431–444 MATHCrossRefMathSciNetGoogle Scholar
  7. Glidden DV (1999). Checking the adequacy of the gamma frailty model for multivariate failure time. Biometrika 86: 381–393 MATHCrossRefMathSciNetGoogle Scholar
  8. Glidden DV and Vittinghoff E (2004). Modelling clustered survival data from multicenter clinical trials. Stat Med 23: 369–388 CrossRefGoogle Scholar
  9. Gray R (1995). Test for variation over groups in survival data. J Am Stat Assoc 90: 198–203 MATHCrossRefGoogle Scholar
  10. Heagerty PJ and Zheng Y (2005). Survival model predictive accuracy and ROC curves. Biometrics 61: 92–105 MATHCrossRefMathSciNetGoogle Scholar
  11. Hougaard P (2000) Analysis of multivariate survival Data. Springer-VerlagGoogle Scholar
  12. Jiang J (1998). Asymptotic properties of the empirical BLUP and BLUE in mixed linear models. Stat Sin 8: 861–886 MATHGoogle Scholar
  13. Kosorok MR, Lee BL, Fine JP (2001) Semiparametric inference for proportional hazards frailty regression models. Technical report, Department of Biostatistics, University of Wisconsin.Google Scholar
  14. Kosorok MR, Lee BL and Fine JP (2004). Robust inference for proportional hazards univariate frailty regression models. Ann Statist 32: 1448–1491 MATHCrossRefMathSciNetGoogle Scholar
  15. Lagakos SW and Schoenfeld DA (1984). Properties of proportional-hazards score tests under misspecified regression models. Biometrics 40: 1037–1048 MATHCrossRefMathSciNetGoogle Scholar
  16. Lancaster T and Nickell S (1980). The analysis of re-employment probabilities for the unemployed. J R Stat Soc, Ser A 143(2): 141–165 MATHCrossRefGoogle Scholar
  17. Li H, Thompson EA and Wijsman EM (1998). Semiparametric estimation of major gene effects for age of onset. Gen Epidemiol 15: 279–298 CrossRefGoogle Scholar
  18. Liu I, Blacker DL, Xu R, Fitzmaurice G, Lyons MJ and Tsuang MT (2004). Genetic and environmental contributions to the development of alcohol dependence in male twins. Archives of General Psychiatry 61: 897–903 CrossRefGoogle Scholar
  19. Liu I, Blacker DL, Xu R, Fitzmaurice G, Tsuang MT and Lyons MJ (2004). Genetic and environmental contributions to age of onset of alcohol dependence symptoms in male twins. Addiction 99(11): 1403–1409 CrossRefGoogle Scholar
  20. Liu I, Xu R, Blacker DL, Fitzmaurice G, Lyons MJ and Tsuang MT (2005). The application of a proportional hazards model with random effects to twin data. Beha Gen 190(2): 781–789 CrossRefGoogle Scholar
  21. Ma R, Krewski D and Burnett RT (2003). Random effects Cox model: a Poisson modelling approach. Biometrika 90: 157–169 MATHCrossRefMathSciNetGoogle Scholar
  22. Murphy S (1994). Consistency in a proportional hazards model incorporating a random effect. Ann Stat 22: 712–731 MATHMathSciNetGoogle Scholar
  23. Murphy S (1995). Asymptotic theory for the frailty model. Ann of Stat 23: 182–198 MATHMathSciNetGoogle Scholar
  24. Murphy S and Vaart A (2000). On profile likelihood. J Am Stat Asso 95: 449–485 MATHCrossRefGoogle Scholar
  25. Murray DM, Varnell SP and Blitstein JL (2004). Design and analysis of group-randomized trials: a review of recent methodological developments. A J Public Health 94: 423–432 Google Scholar
  26. O’Quigley J (2003). Khmaladze-type graphical evaluation of the proportional hazards assumption. Biometrika 90: 577–584 CrossRefMathSciNetGoogle Scholar
  27. O’Quigley J and Stare J (2002). Proportional hazards models with frailties and random effects. Stat Med 21: 3219–3233 CrossRefGoogle Scholar
  28. O’Quigley J, Xu R (1998) Goodness-of-fit in survival analysis. In: Armitage P, Colton T (eds) Encyclopedia of Biostatistics, vol 2. Wiley, pp 1731–1745Google Scholar
  29. O’Quigley J and Xu R (2001). Explained variation in proportional hazards regression. In: Crowley, J (eds) Handbook of Statistics in Clinical Oncology, pp 397–410. Marcel Dekker, Inc., New York Google Scholar
  30. Parner E (1998). Asymptotic theory for the correlated gamma-frailty model. The Annals of Statistics 26: 183–214 MATHCrossRefMathSciNetGoogle Scholar
  31. Paik MC, Tsai WY and Ottman R (1994). Multivariate survival analysis using piecewise gamma frailty. Biometrics 50: 975–988 MATHCrossRefGoogle Scholar
  32. Ripatti S, Gatz M, Pedersen NL and Palmgren J (2003). Three-state frailty model for age at onset of dementia and death in Swedish twins. Gen Epidemiol 24: 139–149 CrossRefGoogle Scholar
  33. Ripatti S, Larsen K and Palmgren J (2002). Maximum likelihood inference for multivariate frailty models using an automated Monte Carlo EM algorithm. Lifetime Data Anal 8: 349–360 MATHCrossRefMathSciNetGoogle Scholar
  34. Ripatti S and Palmgren J (2000). Estimation of multivariate frailty models using penalized partial likelihood. Biometrics 56: 1016–1022 MATHCrossRefMathSciNetGoogle Scholar
  35. Shorack GR and Wellner JA (1986). Empirical Processes with Applications to Statistics. Wiley, New York Google Scholar
  36. Solomon PJ (1984). Effect of misspecification of regression models in the analysis of survival data. Biometrika 71: 291–298 MATHCrossRefMathSciNetGoogle Scholar
  37. Spiekerman CF and Lin DY (1996). Checking the marginal Cox model for correlated failure time data. Biometrika 83: 143–156 MATHCrossRefMathSciNetGoogle Scholar
  38. Struthers CA and Kalbfleisch JD (1986). Misspecified proportional hazard models. Biometrika 73: 363–369 MATHCrossRefMathSciNetGoogle Scholar
  39. Sylvester R, Collette L, Suciu S, Baron B, Legrand C, Gorlia T, Collins G, Coens C, Declerck L, Therasse P and Glabbeke M (2002). Statistical methodology of phase III cancer clinical trials: advances and future perspectives. Eur J Cancer 38: S162–S168 CrossRefGoogle Scholar
  40. Vaida F and Xu R (2000). Proportional hazards model with random effects. Stat Med 19: 3309–3324 CrossRefGoogle Scholar
  41. Viswanathan B and Manatunga AK (2001). Diagnostic plots for assessing the frailty distribution in multivariate survival data. Lifetime Data Anal 7: 143–155 MATHCrossRefMathSciNetGoogle Scholar
  42. Xu R (2004). Proportional hazards model with mixed effects: a review with applications to twin data. Metodoloski Zvezki: J Stat Soc Slovenia 1(1): 205–212 Google Scholar
  43. Xu R, Gamst A, Donohue M, Vaida F, Harrington DP (2006) Using profile likelihood for semiparametric model selection with application to proportional hazards mixed models. Harvard University Biostatistics Working Paper Series. Scholar
  44. Xu R and Harrington DP (2001). A semiparametric estimate of treatment effects with censored data.. Biometrics 57: 875–885 CrossRefMathSciNetGoogle Scholar
  45. Xu R and O’Quigley J (2000). Estimating average regression effect under non-proportional hazards. Biostatistics 1: 423–439 MATHCrossRefGoogle Scholar
  46. Xu R and O’Quigley J (2000). Proportional hazards estimate of the conditional survival function. J R Stat Soc Ser B 62: 667–680 MATHCrossRefMathSciNetGoogle Scholar
  47. Yau KKW (2001). Multilevel models for survival analysis with random effects. Biometrics 57: 96–102 CrossRefMathSciNetGoogle Scholar
  48. Yau KKW and McGilchrist CA (1998). ML and REML estimation in survival analysis with time dependent correlated frailty. Stat in Med 17: 1201–1214 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Division of Biostatistics and Bioinformatics, Department of Family and Preventive MedicineUniversity of CaliforniaSan DiegoUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSan DiegoUSA

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