Lifetime Data Analysis

, Volume 21, Issue 4, pp 579–593 | Cite as

Does Cox analysis of a randomized survival study yield a causal treatment effect?

  • Odd O. Aalen
  • Richard J. Cook
  • Kjetil Røysland
Article

Abstract

Statistical methods for survival analysis play a central role in the assessment of treatment effects in randomized clinical trials in cardiovascular disease, cancer, and many other fields. The most common approach to analysis involves fitting a Cox regression model including a treatment indicator, and basing inference on the large sample properties of the regression coefficient estimator. Despite the fact that treatment assignment is randomized, the hazard ratio is not a quantity which admits a causal interpretation in the case of unmodelled heterogeneity. This problem arises because the risk sets beyond the first event time are comprised of the subset of individuals who have not previously failed. The balance in the distribution of potential confounders between treatment arms is lost by this implicit conditioning, whether or not censoring is present. Thus while the Cox model may be used as a basis for valid tests of the null hypotheses of no treatment effect if robust variance estimates are used, modeling frameworks more compatible with causal reasoning may be preferrable in general for estimation.

Keywords

Causation Collapsible model Confounding Hazard function Survival data 

References

  1. Aalen OO (1989) A linear regression model for the analysis of life times. Stat Med 8(8):907–925CrossRefGoogle Scholar
  2. Aalen OO, Borgan Ø, Gjessing HK (2008) Survival and event history analysis: a process point of view. Springer, New YorkCrossRefGoogle Scholar
  3. Aalen OO, Røysland K, Gran JM, Kouyos R, Lange T (2014) Can we believe the DAGs? A comment on the relationship between causal DAGs and mechanisms. Stat Methods Med Res. doi:10.1177/0962280213520436
  4. Brown BM, Wang Y-G (2005) Standard errors and covariance matrices for smoothed rank estimators. Biometrika 92(1):149–158MATHMathSciNetCrossRefGoogle Scholar
  5. Cheng SC, Wei LJ, Ying Z (1995) Analysis of transformation models with censored data. Biometrika 82(4):835–845MATHMathSciNetCrossRefGoogle Scholar
  6. Cox DR (1972) Survival models and life tables (with discussion). J R Stat Soc 34:187–220MATHGoogle Scholar
  7. Cox DR, Oakes D (1984) Anal Surviv Data. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  8. Durham LK, Halloran ME, Longini IM, Manatunga AK (1999) Comparison of two smoothing methods for exploring waning vaccine effects. J R Stat Soc 48(3):395–407MATHCrossRefGoogle Scholar
  9. Flanders WD, Klein M (2007) Properties of 2 counterfactual effect definitions of a point exposure. Epidemiology 18(4):453–460CrossRefGoogle Scholar
  10. Ford I, Norrie J, Ahmadi S (1995) Model inconsistency, illustrated by the Cox proportional hazards model. Stat Med 14(8):735–746CrossRefGoogle Scholar
  11. Gould A, Lawless JF (1988) Consistency and efficiency of regression coefficient estimates in location-scale models. Biometrika 75(3):535–540MATHMathSciNetGoogle Scholar
  12. Greenland S (1996) Absence of confounding does not correspond to collapsibility of the rate ratio or rate difference. Epidemiology 7(5):498–501CrossRefGoogle Scholar
  13. Hauck WW, Anderson S, Marcus SM (1998) Should we adjust for covariates in nonlinear regression analyses of randomized trials? Controlled Clin Trials 19(3):249–256CrossRefGoogle Scholar
  14. Hernán MA (2010) The hazards of hazard ratios. Epidemiology 21(1):13–15CrossRefGoogle Scholar
  15. Hernán MA, Robins JM (2015) Causal inference. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  16. Hernán MA, Hernández-Díaz S, Robins JM (2004) A structural approach to selection bias. Epidemiology 15(5):615–625CrossRefGoogle Scholar
  17. Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. Wiley, HobokenMATHCrossRefGoogle Scholar
  18. Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observation. J Am Stat Assoc 53(282):457–481MATHMathSciNetCrossRefGoogle Scholar
  19. Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley, HobokenMATHGoogle Scholar
  20. Lin DY, Wei LJ (1989) The robust inference for the Cox proportional hazards model. J Am Stat Assoc 84(408):1074–1078MATHMathSciNetCrossRefGoogle Scholar
  21. Lin H, Li Y, Jiang L, Li G (2014) A semiparametric linear transformation model to estimate causal effects for survival data. Can J Stat 42(1):18–35MATHMathSciNetCrossRefGoogle Scholar
  22. Martinussen T, Vansteelandt S (2013) On collapsibility and confounding bias in Cox and Aalen regression models. Lifetime Data Anal 19(3):279–296MathSciNetCrossRefGoogle Scholar
  23. Pearl J (2009) Causality: models, reasoning, and inference, 2nd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  24. Robins J (1992) Estimation of the time-dependent accelerated failure time model in the presence of confounding factors. Biometrika 79(2):321–334MATHMathSciNetCrossRefGoogle Scholar
  25. Strohmaier S, Røysland K, Hoff R, Borgan Ø, Pedersen T, Aalen OO (2014) Dynamic path analysis—a useful tool to investigate mediation processes in clinical survival trials. SubmittedGoogle Scholar
  26. Struthers CA, Kalbfleisch JD (1986) Misspecified proportional hazards models. Biometrika 74(2):363–369MathSciNetCrossRefGoogle Scholar
  27. Wei LJ (1992) The accelerated failure time model: a useful alternative to the Cox regression model in survival analysis. Stat Med 11(14–15):1871–1879CrossRefGoogle Scholar
  28. Yusuf S, Wittes J, Probstfield J, Tyroler HA (1991) Analysis and interpretation of treatment effects in subgroups of patients in randomized clinical trials. J Am Med Assoc 266(1):93–98CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Odd O. Aalen
    • 1
  • Richard J. Cook
    • 2
  • Kjetil Røysland
    • 1
  1. 1.Department of Biostatistics, Institute of Basic Medical SciencesUniversity of OsloOsloNorway
  2. 2.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada

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