Lifetime Data Analysis

, Volume 19, Issue 2, pp 219–241 | Cite as

Semiparametric inference on the absolute risk reduction and the restricted mean survival difference



For time-to-event data, when the hazards are non-proportional, in addition to the hazard ratio, the absolute risk reduction and the restricted mean survival difference can be used to describe the time-dependent treatment effect. The absolute risk reduction measures the direct impact of the treatment on event rate or survival, and the restricted mean survival difference provides a way to evaluate the cumulative treatment effect. However, in the literature, available methods are limited for flexibly estimating these measures and making inference on them. In this article, point estimates, pointwise confidence intervals and simultaneous confidence bands of the absolute risk reduction and the restricted mean survival difference are established under a semiparametric model that can be used in a sufficiently wide range of applications. These methods are motivated by and illustrated for data from the Women’s Health Initiative estrogen plus progestin clinical trial.


Absolute risk reduction Clinical trial Non-proportional hazards Restricted mean survival Semiparametric analysis Simultaneous inference 



The author would like to thank the reviewers and the Guest Editor for helpful comments and suggestions, which led to an improved version of the manuscript.


  1. Bie O, Borgan O, Liestøl K (1987) Confidence intervals and confidence bands for the cumulative hazard rate function and their small-sample properties. Scand J Stat 14:221–233MATHGoogle Scholar
  2. Chen P, Tsiatis AA (2001) Causal inference on the difference of the restricted mean lifetime between two groups. Biometrics 57:1030–1038MathSciNetMATHCrossRefGoogle Scholar
  3. Cheng SC, Wei LJ, Ying Z (1997) Predicting survival probabilities with semiparametric transformation models. J Am Stat Assoc 92:227–235MathSciNetMATHCrossRefGoogle Scholar
  4. Cox DR (1972) Regression models and life-tables (with Discussion). J R Stat Soc B 34:187–220MATHGoogle Scholar
  5. Dabrowska DM, Doksum KA, Song J (1989) Graphical comparison of cumulative hazards for two populations. Biometrika 76:763–773MathSciNetMATHCrossRefGoogle Scholar
  6. Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. Wiley, New YorkMATHCrossRefGoogle Scholar
  7. Kaplan E, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53:457–481MathSciNetMATHCrossRefGoogle Scholar
  8. Lin DY, Wei LJ, Ying Z (1993) Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika 80:557–572MathSciNetMATHCrossRefGoogle Scholar
  9. Lin DY, Fleming TR, Wei LJ (1994) Confidence bands for survival curves under the proportional hazards model. Biometrika 81:73–81MathSciNetMATHCrossRefGoogle Scholar
  10. Manson JE, Hsia J, Johnson KC, Rossouw JE, Assaf AR, Lasser NL, Trevisan M, Black HR, Heckbert SR, Detrano R, Strickland OL, Wong ND, Crouse JR, Stein E, Cushman M, for the Women’S Health Initiative Investigators, (2003) Estrogen plus progestin and the risk of coronary heart disease. New Eng J Med 349:523–534Google Scholar
  11. McKeague IW, Zhao Y (2002) Simultaneous confidence bands for ratios of survival functions via empirical likelihood. Stat Probab Lett 60:405–415MathSciNetMATHCrossRefGoogle Scholar
  12. Nair VN (1984) Confidence bands for survival functions with censored data: a comparative study. Technometrics 26:265–275CrossRefGoogle Scholar
  13. Parzen MI, Wei LJ, Ying Z (1997) Simultaneous confidence intervals for the difference of two survival functions. Scand J Stat 24:309–314MathSciNetMATHCrossRefGoogle Scholar
  14. Peng L, Huang Y (2007) Survival analysis with temporal covariate effects. Biometrika 94:719–733MathSciNetMATHCrossRefGoogle Scholar
  15. Pollard D (1990) Empirical processes: theory and applications. Institute of Mathematical Statistics, Hayward, CAGoogle Scholar
  16. Prentice RL, Langer R, Stefanick ML, Howard BV, Pettinger M, Anderson G, Barad D, Curb JD, Kotchen J, Kuller L, Limacher M, Wactawski-Wende J, for the Women’S Health Initiative Investigators (2005) Combined postmenopausal hormone therapy and cardiovascular disease: toward resolving the discrepancy between observational studies and the women’s health initiative clinical trial. Am J Epidemiol 162:404–414Google Scholar
  17. Royston P, Parmar MK (2011) The use of restricted mean survival time to estimate the treatment effect in randomized clinical trials when the proportional hazards assumption is in doubt. Stat Med 19:2409–2421MathSciNetCrossRefGoogle Scholar
  18. Schaubel DE, Wei G (2011) Double inverse-weighted estimation of cumulative treatment effects under nonproportional hazards and dependent censoring. Biometrics 67:29–38MathSciNetMATHCrossRefGoogle Scholar
  19. Tian L, Zucker D, Wei LJ (2005) On the Cox model with time-varying regression coefficients. J Am Stat Assoc 100:172–183MathSciNetMATHCrossRefGoogle Scholar
  20. Tong X, Zhu C, Sun J (2007) Semiparametric regression analysis of two-sample current status data, with applications to tumorigenicity experiments. Can J Stat 35:575–584MathSciNetMATHCrossRefGoogle Scholar
  21. Writing Group for the Women’s Health Initiative Investigators (2002) Risks and benefits of estrogen plus progestin in healthy postmenopausal women: principal results from the women’s health initiative randomized controlled trial. J Am Med Assoc 288:321–333Google Scholar
  22. Yang S, Prentice RL (2005) Semiparametric analysis of short-term and long-term hazard ratios with two-sample survival data. Biometrika 92:1–17MathSciNetMATHCrossRefGoogle Scholar
  23. Yang S, Prentice RL (2011) Estimation of the 2-sample hazard ratio function using a semiparametric model. Biostatistics 12:354–368CrossRefGoogle Scholar
  24. Zucker DM (1998) Restricted mean life with covariates: modification and extension of a useful survival analysis method. J Am Stat Assoc 93:702–709MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York (outside the USA) 2013

Authors and Affiliations

  1. 1.Office of Biostatistics ResearchNational Heart, Lung, and Blood InstituteBethesdaUSA

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