Skip to main content

Modeling restricted mean survival time under general censoring mechanisms

Abstract

Restricted mean survival time (RMST) is often of great clinical interest in practice. Several existing methods involve explicitly projecting out patient-specific survival curves using parameters estimated through Cox regression. However, it would often be preferable to directly model the restricted mean for convenience and to yield more directly interpretable covariate effects. We propose generalized estimating equation methods to model RMST as a function of baseline covariates. The proposed methods avoid potentially problematic distributional assumptions pertaining to restricted survival time. Unlike existing methods, we allow censoring to depend on both baseline and time-dependent factors. Large sample properties of the proposed estimators are derived and simulation studies are conducted to assess their finite sample performance. We apply the proposed methods to model RMST in the absence of liver transplantation among end-stage liver disease patients. This analysis requires accommodation for dependent censoring since pre-transplant mortality is dependently censored by the receipt of a liver transplant.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  1. Andersen PK, Hansen MG, Klein JP (2004) Regression analysis of restricted mean survival time based on pseudo-observations. Lifetime Data Anal 10(4):335–350

    MathSciNet  Article  MATH  Google Scholar 

  2. Andersen PK, Perme MP (2010) Pseudo-observations in survival analysis. Stat Methods Med Res 19(1):71–99

    MathSciNet  Article  Google Scholar 

  3. Andersen PK (2013) Decomposition of number of life years lost according to causes of death. Stat Med 32(30):5278–5285

    MathSciNet  Article  Google Scholar 

  4. Binder N, Gerds TA, Andersen PK (2014) Pseudo-observations for competing risks with covariate dependent censoring. Lifetime Data Anal 20(2):303–315

    MathSciNet  Article  MATH  Google Scholar 

  5. Breslow NE (1972) Contribution to the discussion of the paper by D. R. Cox. J R Stat Soc Ser B 34(2):216–217

    MathSciNet  Google Scholar 

  6. Chen P-Y, Tsiatis AA (2001) Causal inference on the difference of the restricted mean lifetime between two groups. Biometrics 57(4):1030–1038

    MathSciNet  Article  MATH  Google Scholar 

  7. Cox DR (1972) Regression models and life-tables. J R Stat Soc Ser B 34(2):187–220

    MathSciNet  MATH  Google Scholar 

  8. Cox DR (1975) Partial likelihood. Biometrika 62(2):269–276

    MathSciNet  Article  MATH  Google Scholar 

  9. Davison AC, Hinkley DV (1997) Bootstrap methods and their application, volume 1 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  10. Foutz RV (1977) On the unique consistent solution to the likelihood equations. J Am Stat Assoc 72(357):147–148

    MathSciNet  Article  MATH  Google Scholar 

  11. Gillen DL, Emerson SS (2007) Nontransitivity in a class of weighted logrank statistics under nonproportional hazards. Stat Probab Lett 77(2):123–130

    MathSciNet  Article  MATH  Google Scholar 

  12. Grand MK, Putter H (2015) Regression models for expected length of stay. Stat Med 35(7):1178–1192

    MathSciNet  Article  Google Scholar 

  13. Harrell FE, Lee KL, Mark DB (1996) Tutorial in biostatistics multivariable prognostic models: issues in developing models, evaluating assumptions and adequacy, and measuring and reducing errors. Stat Med 15(4):361–87

    Article  Google Scholar 

  14. Heagerty PJ, Zheng Y (2005) Survival model predictive accuracy and ROC curves. Biometrics 61(1):92–105

    MathSciNet  Article  MATH  Google Scholar 

  15. Kalbfleisch J, Prentice R (2002) The statistical analysis of failure time data. Wiley Series in Probability and Statistics, Wiley, London

    Book  MATH  Google Scholar 

  16. Kamath PS, Wiesner RH, Malinchoc M, Kremers W, Therneau TM, Kosberg CL, Damico G, Dickson ER, Kim WR (2001) A model to predict survival in patients with end-stage liver disease. Hepatology 33(2):464–470

    Article  Google Scholar 

  17. Robins J, Rotnitzky A (1992) Recovery of information and adjustment for dependent censoring using surrogate markers. In: Jewell N, Dietz K, Farewell B (eds) AIDS epidemiology. Birkhuser, Boston, pp 297–331

    Chapter  Google Scholar 

  18. Robins JM (1993) Information recovery and bias adjustment in proportional hazards regression analysis of randomized trials using surrogate markers. In: Proceedings of the biopharmaceutical section, American Statistical Association, vol 24, no 3. American Statistical Association

  19. Robins JM, Finkelstein DM (2000) Correcting for noncompliance and dependent censoring in an aids clinical trial with inverse probability of censoring weighted (IPCW) log-rank tests. Biometrics 56(3):779–788

    Article  MATH  Google Scholar 

  20. Schaubel DE, Wei G (2011) Double inverse-weighted estimation of cumulative treatment effects under nonproportional hazards and dependent censoring. Biometrics 67(1):29–38

    MathSciNet  Article  MATH  Google Scholar 

  21. Tian L, Cai T, Goetghebeur E, Wei L (2007) Model evaluation based on the sampling distribution of estimated absolute prediction error. Biometrika 94(2):297–311

    MathSciNet  Article  MATH  Google Scholar 

  22. Tian L, Zhao L, Wei L (2014) Predicting the restricted mean event time with the subjects baseline covariates in survival analysis. Biostatistics 15(2):222–233

    Article  Google Scholar 

  23. Uno H, Cai T, Pencina MJ, Dagostino RB, Wei L (2011) On the C-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data. Stat Med 30(10):1105–1117

    MathSciNet  Google Scholar 

  24. Van Houwelingen HC (2007) Dynamic prediction by landmarking in event history analysis. Scand J Stat 34(1):70–85

    MathSciNet  Article  MATH  Google Scholar 

  25. Van Houwelingen HC, Putter H (2015) Comparison of stopped cox regression with direct methods such as pseudo-values and binomial regression. Lifetime Data Anal 21(2):180–196

    MathSciNet  Article  MATH  Google Scholar 

  26. Wiesner RH, McDiarmid SV, Kamath PS, Edwards EB, Malinchoc M, Kremers WK, Krom RA, Kim WR (2001) MELD and PELD: application of survival models to liver allocation. Liver Transplant 7(7):567–580

    Article  Google Scholar 

  27. Wiesner R, Edwards E, Freeman R, Harper A, Kim R, Kamath P, Kremers W, Lake J, Howard T, Merion RM et al (2003) Model for end-stage liver disease (meld) and allocation of donor livers. Gastroenterology 124(1):91–96

    Article  Google Scholar 

  28. Xiang F, Murray S (2012) Restricted mean models for transplant benefit and urgency. Stat Med 31(6):561–576

    MathSciNet  Article  Google Scholar 

  29. Xu R, O’Quigley J (2000) Estimating average regression effect under non-proportional hazards. Biostatistics 1(4):423–439

    Article  MATH  Google Scholar 

  30. Zhang M, Schaubel DE (2011) Estimating differences in restricted mean lifetime using observational data subject to dependent censoring. Biometrics 67(3):740–749

    MathSciNet  Article  MATH  Google Scholar 

  31. Zhao L, Claggett B, Tian L, Uno H, Pfeffer MA, Solomon SD, Wei LJ (2016) On the restricted mean survival time curve in survival analysis. Biometrics 72(1):215–221

    MathSciNet  Article  MATH  Google Scholar 

  32. Zucker DM (1998) Restricted mean life with covariates: modification and extension of a useful survival analysis method. J Am Stat Assoc 93(442):702–709

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported in part by National Institutes of Health Grant 5R01 DK070869. Data analyzed in this report were supplied by the Minneapolis Medical Research Foundation as the contractor for the Scientific Registry of Transplant Recipients. The interpretation and reporting of these data are the responsibility of the authors and in no way should be seen as an official policy of or interpretation by the Scientific Registry of Transplant Recipients or the U.S. Government.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Douglas E. Schaubel.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 330 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Schaubel, D.E. Modeling restricted mean survival time under general censoring mechanisms. Lifetime Data Anal 24, 176–199 (2018). https://doi.org/10.1007/s10985-017-9391-6

Download citation

Keywords

  • Dependent censoring
  • Generalized linear model
  • Inverse weighting
  • Pre-treatment survival
  • Restricted mean lifetime
  • Transplantation