Advertisement

Nonparametric Estimation of a Cumulative Hazard Function with Right Truncated Data

  • Xu Zhang
  • Yong Jiang
  • Yichuan Zhao
  • Haci Akcin
Chapter
Part of the ICSA Book Series in Statistics book series (ICSABSS)

Abstract

The reverse-time hazard was routinely evaluated or modeled under the context of right truncation. However, this quantity does not have a natural interpretation. Based on the relation between the reverse-time and forward-time hazards, we developed the nonparametric inference for the forward-time hazard. We studied a family of weighted tests for comparing the hazard function between two independent samples. We showed the weak convergence properties and conduct the simulation studies to investigate the practical performances of the proposed variance estimators and tests. Finally, we analyzed the data set about AIDS incubation time to illustrate estimation and two-sample tests about the cumulative hazard function.

References

  1. Aalen, O. O. (1978). Nonparametric inference for a family of counting processes. The Annals of Statistics, 6, 701–726.MathSciNetCrossRefGoogle Scholar
  2. Andersen, P. K., Borgan, Ø., Gill, R. D., & Keiding, N. (1993). Statistical models based on counting processes. New York: Springer.CrossRefGoogle Scholar
  3. Bilker, W. B., & Wang, M. C. (1996). A semiparametric extension of the Mann-Whitney test for randomly truncated data. Biometrika, 52, 10–20.CrossRefGoogle Scholar
  4. Chao, M. T., & Lo, S. H. (1988). Some representations of the non-parametric maximum likelihood estimators with truncated data. The Annals of Statistics, 16, 661–668.MathSciNetCrossRefGoogle Scholar
  5. Chi, Y., Tsai, W. Y., & Chiang, C. L. (2007). Testing the equality of two survival functions with right truncated data. Statistics in Medicine, 26, 812–827.MathSciNetCrossRefGoogle Scholar
  6. Finkelstein, D. M., Moore, D. F., & Schoenfeld, D. A. (1993). A proportional hazards model for truncated AIDS data. Biometrics, 49, 731–740.MathSciNetCrossRefGoogle Scholar
  7. Gross, S. T., & Huber-Carol, C. (1992). Regression models for truncated survival data. Scandinavian Journal of Statistics, 19, 193–213.MathSciNetzbMATHGoogle Scholar
  8. Kalbfleisch, J. D., & Lawless, J. F. (1989). Inference based on retrospective ascertainment: An analysis of the data on transfusion-related AIDS. Journal of the American Statistical Association, 84, 360–372.MathSciNetCrossRefGoogle Scholar
  9. Kalbfleisch, J. D., & Lawless, J. F. (1991). Regression models for right truncated data with applications to AIDS incubation times and reporting lags. Statistica Sinica, 1, 19–32.zbMATHGoogle Scholar
  10. Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53, 457–481.MathSciNetCrossRefGoogle Scholar
  11. Keiding, N., & Gill, R. D. (1990). Random truncation models and Markov process. The Annals of Statistics 18, 582–602.MathSciNetCrossRefGoogle Scholar
  12. Klein, J. P. (1991). Small sample moments of some estimators of the variance of the Kaplan-Meier and Nelson-Aalen estimators. Scandinavian Journal of Statistics, 18, 333–340.MathSciNetzbMATHGoogle Scholar
  13. Klein, J. P., & Moeschberger, M. L. (2003). Survival analysis: Techniques for censored and truncated data. New York: Springer.zbMATHGoogle Scholar
  14. Lagakos, S. W., Barraj, L. M., & Gruttola, V. (1988). Nonparametric analysis of truncated survival data with applications to AIDS. Biometrika, 75, 515–523.MathSciNetCrossRefGoogle Scholar
  15. Lin, D. Y., Fleming, T. R., & Wei, L. J. (1994). Confidence bands for survival curves under the proportional hazards model. Biometrika, 81, 73–81.MathSciNetCrossRefGoogle Scholar
  16. Lui, K. J., Lawrence, D. L., Morgan, W. M., Peterman, T. A., Haverkos, H. W., & Bregman, D. J. (1986). A model-based approach for estimating the mean incubation period of transfusion-associated acquired immunodeficiency syndrome. Proceedings of National Academy of Sciences USA, 83, 3051–3055.CrossRefGoogle Scholar
  17. Medley, G. F., Anderson, R. M., Cox, D.R., & Billiard, L. (1987). Incubation period of AIDS in patients infected via blood transfusion. Nature, 328, 719–721.CrossRefGoogle Scholar
  18. Nelson, W. (1969). Hazard plotting for incomplete failure data. Journal of Quality Technology, 1, 27–52.CrossRefGoogle Scholar
  19. Shen, P. (2010). A class of semiparametric rank-based tests for right-truncated data. Statistics and Probability Letters, 80, 1459–1466.MathSciNetCrossRefGoogle Scholar
  20. Tsai, W. Y. (1990). Testing the assumption of independence between truncated time and failure time. Biometrika, 77, 169–178.MathSciNetCrossRefGoogle Scholar
  21. Wang, M. C. (1989). A semiparametric model for randomly truncated data. Journal of the American Statistical Association, 84, 742–748.MathSciNetCrossRefGoogle Scholar
  22. Wang, M. C., Jewell, N. P., & Tsai, W. Y. (1986). Asymptotic properties of the product limit estimate under random truncation. The Annals of Statistics, 14, 1597–1605MathSciNetCrossRefGoogle Scholar
  23. Woodroofe, M. (1985). Estimating a distribution function with truncated data. The Annals of Statistics, 13, 163–177.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Xu Zhang
    • 1
  • Yong Jiang
    • 2
  • Yichuan Zhao
    • 3
  • Haci Akcin
    • 4
  1. 1.Center for Clinical and Translational SciencesUniversity of Texas Health Science CenterHoustonUSA
  2. 2.MetLife Inc.WhippanyUSA
  3. 3.Department of Mathematics and StatisticsGeorgia State UniversityAtlantaUSA
  4. 4.Department of Risk Management and InsuranceGeorgia State UniversityAtlantaUSA

Personalised recommendations