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Proportional Mean Residual Life Model with Varying Coefficients for Length-Biased and Right-Censored Data

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Abstract

Length-biased data are encountered in many fields, including economics, engineering and epidemiological cohort studies. There are two main challenges in the analysis of such data: the assumption of independent censoring is violated and the assumed model for the underlying population is no longer satisfied for the observed data. In this paper, a proportional mean residual life varying-coefficient model for length-biased data is considered and a local pseudo likelihood method is proposed for estimating the coefficient functions in the model. Asymptotic properties are investigated for the proposed estimators. The finite sample performance of the proposed methodology is demonstrated by simulation studies. Finally, the method is applied to a real data set concerning the Academy Awards.

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References

  1. Addona, V., Wolfson, D.: A formal test for the stationarity of the incidence rate using data from a prevalent cohort study with follow-up. Lifetime Data Analysis, 12, 267–284 (2006)

    Article  MathSciNet  Google Scholar 

  2. Arnold, B. C., Strauss, D.: Bivariate distributions with exponential conditionals. Journal of the American Statistical Association, 83, 522–527 (1988)

    Article  MathSciNet  Google Scholar 

  3. Cai, J., Fan, J., Zhou, H., et al.: Hazard models with varying coefficients for multivariate failure time data. Annals of Statistics, 35, 324–354 (2007)

    Article  MathSciNet  Google Scholar 

  4. Chen, X. P., Shi, J. H., Zhou, Y.: Monotone rank estimation of transformation models with length-biased and right-censored data. Science China Mathematics, 58, 2055–2068 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Chen, Y. Q.: Additive expectancy regression. Journal of the American Statistical Association, 102(477), 153–166 (2007)

    Article  MathSciNet  Google Scholar 

  6. Chen, Y. Q., Cheng, S. C.: Semiparametric regression analysis of mean residual life with censored survival data. Biometrika, 92, 19–29 (2005)

    Article  MathSciNet  Google Scholar 

  7. Chen, Y. Q., Cheng, S. C.: Linear life expectancy regression with censored data. Biometrika, 93, 303–313 (2006)

    Article  MathSciNet  Google Scholar 

  8. Chen, Y. Q., Jewell, N. P., Lei, X., et al.: Semiparametric estimation of proportional mean residual life model in presence of censoring. Biometrics, 61, 170–178 (2005)

    Article  MathSciNet  Google Scholar 

  9. Cox, D. R.: Renewal theory. Encyclopedia of Statistical Sciences, 4, 281–302 (1962)

    Google Scholar 

  10. Fan, J., Farmen, M., Gijbels, I.: Local maximum likelihood estimation and inference. Journal of the Royal Statistical Society, 60, 591–608 (1998)

    Article  MathSciNet  Google Scholar 

  11. Fan, J., Lin, H., Zhou, Y.: Local partial-likelihood estimation for lifetime data. Annals of Statistics, 34, 290–325 (2006)

    Article  MathSciNet  Google Scholar 

  12. Hastie, T., Tibshirani, R.: Varying-coefficient models. Journal of the Royal Statistical Society, 55, 757–796 (1993)

    MathSciNet  MATH  Google Scholar 

  13. Huang, C., Qin, J.: Composite partial likelihood estimation under length-biased sampling, with application to a prevalent cohort study of dementia. Journal of the American Statistical Association, 107(499), 946–957 (2012)

    Article  MathSciNet  Google Scholar 

  14. Johansen, S.: An extension of Cox’s regression model. International Statistical Review, 51, 165–174 (1983)

    Article  MathSciNet  Google Scholar 

  15. Kvam, P.: Length bias in the measurements of carbon nanotubes. Technometrics, 50, 462–467 (2008)

    Article  MathSciNet  Google Scholar 

  16. Lagakos, S. W., Barraj, L. M., De Gruttola, V.: Nonparametric analysis of truncated survival data, with applications to AIDS. Biometrika, 75, 515–523 (1988)

    Article  MathSciNet  Google Scholar 

  17. Lancaster, T.: The Econometric Analysis of Transition Data, Cambridge University Press, Cambridge, 1992

    MATH  Google Scholar 

  18. Lin, C., Zhou, Y.: Analyzing right-censored and length-biased data with varying-coefficient transformation model. Journal of Multivariate Analysis, 130, 45–63 (2014)

    Article  MathSciNet  Google Scholar 

  19. Ma, H. J., Fan, C. Y., Zhou, Y.: Estimating equation methods for quantile regression with length-biased and right censored data (in Chinese). Scientia Sinica Mathematica, 45, 1981–2000 (2015)

    Article  Google Scholar 

  20. Maguluri, G., Zhang, C. H.: Estimation in the mean residual life regression model. Journal of the Royal Statistical Society, Series B (Methodological), 56, 477–489 (1994)

    Article  MathSciNet  Google Scholar 

  21. Ning, J., Qin, J., Asgharian, M., et al.: Empirical likelihood-based confidence intervals for length-biased data. Statistics in Medicine, 32, 2278–2291 (2013)

    Article  MathSciNet  Google Scholar 

  22. Oakes, D., Dasu, T.: A note on residual life. Biometrika, 77, 409–410 (1990)

    Article  MathSciNet  Google Scholar 

  23. Qin, J., Shen, Y.: Statistical methods for analyzing right-censored length-biased data under Cox model. Biometrics, 66, 382–392 (2010)

    Article  MathSciNet  Google Scholar 

  24. Redelmeier, D. A., Singh, S. M.: Survival in Academy Award-winning actors and actresses. Annals of Internal Medicine, 134, 955–962 (2001)

    Article  Google Scholar 

  25. Shen, Y., Ning, J., Qin, J.: Analyzing length-biased data with semiparametric transformation and accelerated failure time models. Journal of the American Statistical Association, 104, 487 (2009)

    Article  MathSciNet  Google Scholar 

  26. Spiekerman, C. F., Lin, D. Y.: Marginal regression models for multivariate failure time data. Journal of the American Statistical Association, 93, 1164–1175 (1998)

    Article  MathSciNet  Google Scholar 

  27. Sun, L. Q., Song, X. Y., Zhang, Z. G.: Mean residual life models with time-dependent coefficients under right censoring. Biometrika, 99, 185–197 (2012)

    Article  MathSciNet  Google Scholar 

  28. Sun, L., Zhang, Z.: A class of transformed mean residual life models with censored survival data. Journal of the American Statistical Association, 104, 803–815 (2009)

    Article  MathSciNet  Google Scholar 

  29. Sylvestre, M. P., Huszti, E., Hanley, J. A.: Do Oscar winners live longer than less successful peers? A reanalysis of the evidence. Annals of Internal Medicine, 145, 361–363 (2006)

    Article  Google Scholar 

  30. Wang, M. C.: Hazards regression analysis for length-biased data. Biometrika, 83, 343–354 (1996)

    Article  MathSciNet  Google Scholar 

  31. Wang, Y., Zhou, Z., Zhou, X. H., et al.: Nonparametric and semiparametric estimation of quantile residual lifetime for length-biased and right-censored data. Canadian Journal of Statistics, 45(2), 220–250 (2017)

    Article  MathSciNet  Google Scholar 

  32. Wolfson, C., Wolfson, D. B., Asgharian, M., et al.: A reevaluation of the duration of survival after the onset of dementia. N. Engl. J. Med., 344, 1111 (2001)

    Article  Google Scholar 

  33. Wolkewitz, Martin, Allignol, et al.: Two pitfalls in survival analyses of time-dependent exposure: A case study in a cohort of Oscar nominees. American Statistician, 64, 205–211 (2010)

    Article  MathSciNet  Google Scholar 

  34. Xu, D., Zhou, Y.: Local composite partial likelihood estimation for length-biased and right-censored data. Journal of Statistical Computation and Simulation, 89(14), 2661–2677 (2019)

    Article  MathSciNet  Google Scholar 

  35. Zelen, M.: Forward and backward recurrence times and length biased sampling: Age specific models. Lifetime Data Analysis, 10, 325–334 (2004)

    Article  MathSciNet  Google Scholar 

  36. Zelen, M., Feinleib, M.: On the theory of screening for chronic diseases. Biometrika, 56, 601–614 (1969)

    Article  MathSciNet  Google Scholar 

  37. Zhang, F., Chen, X., Zhou, Y.: Proportional hazards model with varying coefficients for length-biased data. Lifetime Data Analysis, 20(1), 132–157 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to thank the editor, the associate editor and two reviewers for their many helpful and useful comments and suggestions.

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Authors and Affiliations

  1. Key Laboratory of Applied Statistics of MOE and School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, P. R. China

    Da Xu

  2. Key Laboratory of Advanced Theory and Application in Statistics and Data Science, MOE, Academy of Statistics and Interdisciplinary Sciences, East China Normal University, Shanghai, 200062, P. R. China

    Yong Zhou

  3. Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100190, P. R. China

    Yong Zhou

Authors
  1. Da Xu
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  2. Yong Zhou
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Corresponding authors

Correspondence to Da Xu or Yong Zhou.

Additional information

Supported by the State Key Program of National Natural Science Foundation of China (Grant No. 71931004) and the State Key Program in the Major Research Plan of National Natural Science Foundation of China (Grant No. 91546202)

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Xu, D., Zhou, Y. Proportional Mean Residual Life Model with Varying Coefficients for Length-Biased and Right-Censored Data. Acta. Math. Sin.-English Ser. 36, 578–596 (2020). https://doi.org/10.1007/s10114-020-8079-0

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  • DOI: https://doi.org/10.1007/s10114-020-8079-0

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