Statistics in Biosciences

, Volume 11, Issue 3, pp 659–676 | Cite as

Estimating Attributable Life Expectancy Under the Proportional Mean Residual Life Model

  • Yixin Wang
  • Ying Qing ChenEmail author


In population-based health research, the so-called population attributable fraction is an important quantity that calculates the percentage of excess risk of morbidity and mortality associated with modifiable risk factors for a given population. While the concept of “risk” is usually measured by event probabilities, in practice it may be of a more direct interest to know the excess life expectancy associated with the modifiable risk factors instead, particularly when mortality is of the ultimate concern. In this paper, we thus propose to study a novel quantity, termed “attributable life expectancy,” to measure the population attributable fraction of life expectancy. We further develop a model-based approach for the attributable life expectancy under the Oakes–Dasu proportional mean residual life model, and establish its asymptotic properties for inferences. Numerical studies that include Monte-Carlo simulations and an actual analysis of the mortality associated with smoking cessation in an Asia Cohort Consortium are conducted to evaluate the performance of our proposed method.


Excess life expectancy Population research Residual life regression Time-to-event 



We would like to thank the Editor-in-Chief Professor Mei-Cheng Wang, associate editor, and two anonymous reviewers for their constructive comments that led to a significant improvement of this paper. This research was partially supported by NIH/NCI R01 CA172415 and NIH/NIMH R01 MH105857.

Supplementary material

12561_2019_9258_MOESM1_ESM.pdf (161 kb)
Supplementary Materials: The online supplementary materials include the proofs for Theorem 1. (pdf 162KB)


  1. 1.
    Basu S, Landis JR (1995) Model-based estimation of population attributable risk under cross-sectional sampling. Am J Epidemiol 142:1338–1343CrossRefGoogle Scholar
  2. 2.
    Benichou J (1991) Methods of adjustment for estimating the attributable risk in case-control studies: a review. Stat Med 10:1753–1773CrossRefGoogle Scholar
  3. 3.
    Benichou J (2000) Attributable risk. In: Gail MH, Benichou J (eds) Encyclopedia of epidemiologic methods. Wiley, Chichester, pp 50–63Google Scholar
  4. 4.
    Benichou J (2001) A review of adjusted estimates of attributable risk. Stat Methods Med Res 10:195–216CrossRefzbMATHGoogle Scholar
  5. 5.
    Benichou J, Gail MH (1990) Variance calculations and confidence-intervals for estimates of the attributable risk based on logistic-models. Biometrics 46:991–1003CrossRefGoogle Scholar
  6. 6.
    Breslow NE, Day NE (1987) Statistical methods in cancer research, vol 1: The analysis of case-control studies. International Agency for Research on Cancer Scientific Publications No. 32, LyonGoogle Scholar
  7. 7.
    Breslow NE, Day NE (1987) Statistical methods in cancer research, vol 2: The design and analysis of cohort studies. International Agency for Research on Cancer Scientific Publications No. 82, LyonGoogle Scholar
  8. 8.
    Bruzzi P, Green SB, Byar DP, Brinton LA, Schairer C (1985) Estimating the population attributable risk for multiple risk factors using case-control data. Am J Epidemiol 122:904–914CrossRefGoogle Scholar
  9. 9.
    Chen YQ (2007) Additive expectancy regression. J Am Stat Assoc 102:153–166CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen YQ, Cheng S (2005) Semiparametric regression analysis of mean residual lifetime with censored data. Biometrika 92:19–29CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Chen YQ, Cheng S (2006) Linear lifetime expectancy regression with censored data. Biometrika 93:303–313CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Chen YQ, Hu C, Wang Y (2006) Attributable risk function in the proportional hazards model for censored time-to-event. Biostatistics 7:515–529CrossRefzbMATHGoogle Scholar
  13. 13.
    Chen L, Lin DY, Zeng D (2010) Attributable fraction functions for censored event times. Biometrika 97:713–726CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Chiang CL (1960) A stochastic study of the lifetime table and its applications: I. Probability distributions of the biometric functions. Biometrics 16:618–635CrossRefzbMATHGoogle Scholar
  15. 15.
    Coughlin SS, Nass CC, Pickle LW, Trock B, Bunin G (1991) Regression methods for estimating attributable risk in population-based case-control studies: a comparison of additive and multiplicative models. Am J Epidemiol 133:305–313CrossRefGoogle Scholar
  16. 16.
    Drescher K, Schill W (1991) Attributable risk estimation from case-control data via logistic regression. Biometrics 47:1247–1256CrossRefGoogle Scholar
  17. 17.
    Fleming TR, Harrington DP (2011) Counting processes and survival analysis. Wiley, New YorkzbMATHGoogle Scholar
  18. 18.
    Gefeller O (1992) Comparison of adjusted attributable risk estimators. Stat Med 11:2083–2091CrossRefGoogle Scholar
  19. 19.
    Graubard BI, Fears T (2005) Standard errors for attributable risk for simple and complex sample designs. Biometrics 61:847–855CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Greenland S (1984) Bias in methods for deriving standardized morbidity ratio and attributable fraction estimates. Stat Med 3:131–141CrossRefGoogle Scholar
  21. 21.
    Greenland S, Drescher K (1993) Maximum likelihood estimation of the attributable fraction from logistic models. Biometrics 49:865–872CrossRefzbMATHGoogle Scholar
  22. 22.
    Greenland S, Robins JM (1988) Conceptual problems in the definition and interpretation of attributable fractions. J Epidemiol 128:1185–1197CrossRefGoogle Scholar
  23. 23.
    Kuritz SJ, Landis JR (1987) Attributable risk estimation from matched-pairs case-control data. Am J Epidemiol 125:324–328CrossRefGoogle Scholar
  24. 24.
    Kuritz SJ, Landis JR (1988) Summary attributable risk estimation from unmatched case-control data. Stat Med 7:507–517CrossRefzbMATHGoogle Scholar
  25. 25.
    Kuritz SJ, Landis JR (1988) Attributable risk estimation from matched case-control data. Biometrics 44:355–367CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Landis JR, Sharp TJ, Kuritz SJ, Koch G (2000) Mantel-Haenszel methods. In: Gail MH, Benichou J (eds) Encyclopedia of epidemiologic methods. Wiley, Chichester, pp 499–512Google Scholar
  27. 27.
    Leung HM, Kupper LL (1981) Comparison of confidence intervals for attributable risk. Biometrics 37:293–302CrossRefzbMATHGoogle Scholar
  28. 28.
    Levin ML (1953) The occurrence of lung cancer in man. ACTA Unio Int Contra Cancrum 9:531–541Google Scholar
  29. 29.
    Lin D, Fleming TR, Wei LJ (1994) Confidence bands for survival curves under the proportional hazards model. Biometrika 81:73–81CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Mantel N, Haenszel W (1959) Statistical aspects of the analysis of data from retrospective studies of disease. J Natl Cancer Inst 22:719–748Google Scholar
  31. 31.
    Oakes D, Dasu T (1990) A note on residual lifetime. Biometrika 77:409–410CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Pearl J (2000) Causality: models, reasoning, and inference. Cambridge University Press, CambridgezbMATHGoogle Scholar
  33. 33.
    Rubin DB (1978) Bayesian inference for causal effects: the role of randomization. Ann Stat 6:34–58CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Schmittlein DC, Morrison DG (1981) The median residual lifetime: a characterization theorem and an application. Oper Res 29:392–399CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Sun L, Song X, Zhang Z (2012) Mean residual lifetime models with time-dependent coefficients under right censoring. Biometrika 99:185–197CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Sun L, Zhang Z (2009) A class of transformed mean residual lifetime models with censored survival data. J Am Stat Assoc 104:803–815CrossRefzbMATHGoogle Scholar
  37. 37.
    Walter SD (1975) The distribution of Levin’s measure of attributable risk. Biometrika 62:371–374CrossRefzbMATHGoogle Scholar
  38. 38.
    Walter SD (1976) The estimation and interpretation of attributable risk in health research. Biometrics 32:829–849CrossRefzbMATHGoogle Scholar
  39. 39.
    Walter SD (1980) Prevention for multifactorial diseases. Am J Epidemiol 112:409–416CrossRefGoogle Scholar
  40. 40.
    Walter SD (1983) Effects of interaction, confounding and observational error on attributable risk estimation. Am J Epidemiol 117:598–604CrossRefGoogle Scholar
  41. 41.
    Whittemore AS (1982) Statistical methods for estimating attributable risk from retrospective data. Stat Med 1:229–243CrossRefGoogle Scholar
  42. 42.
    Whittemore AS (1983) Estimating attributable risk from case-control studies. Epidemiology 117:76–85Google Scholar
  43. 43.
    Zheng W, McLerran DF, Rolland BA et al (2011) Association between body-mass index and risk of death in more than 1 million Asians. N Engl J Med 364:719–729CrossRefGoogle Scholar
  44. 44.
    Zheng W, McLerran DF, Rolland BA et al (2014) Burden of total and cause-specific mortality related to tobacco smoking among adults aged \(\ge 45\) years in Asia: a pooled analysis of 21 cohorts. PLoS Med 11:1–15CrossRefGoogle Scholar

Copyright information

© International Chinese Statistical Association 2019

Authors and Affiliations

  1. 1.Fred Hutchinson Cancer Research CenterSeattleUSA

Personalised recommendations