Versatile estimation in censored single-index hazards regression

  • Chin-Tsang Chiang
  • Shao-Hsuan Wang
  • Ming-Yueh Huang
Article

Abstract

One attractive advantage of the presented single-index hazards regression is that it can take into account possibly time-dependent covariates. In such a model formulation, the main theme of this research is to develop a theoretically valid and practically feasible estimation procedure for the index coefficients and the induced survival function. In particular, compared with the existing pseudo-likelihood approaches, our one proposes an automatic bandwidth selection and suppresses an influence of outliers. By making an effective use of the considered versatile survival process, we further reduce a substantial finite-sample bias in the Chambless-Diao type estimator of the most popular time-dependent accuracy summary. The asymptotic properties of estimators and data-driven bandwidths are also established under some suitable conditions. It is found in simulations that the proposed estimators and inference procedures exhibit quite satisfactory performances. Moreover, the general applicability of our methodology is illustrated by two empirical data.

Keywords

Accuracy measure Conditional survival function Cross-validation Kaplan–Meier estimator Pseudo-integrated least squares estimator Pseudo-maximum likelihood estimator Single-index hazards model U-statistic 

Notes

Acknowledgements

The research of the first author was partially supported by the National Science Council Grants 99-2118-M-002-003- and 100-2118-M-002-005-MY2 (Taiwan). We would also like to thank the associate editor and a reviewer for some constructive comments on this paper.

Supplementary material

10463_2017_600_MOESM1_ESM.pdf (178 kb)
Supplementary material 1 (pdf 178 KB)

References

  1. Bouaziz, O., Lopez, O. (2010). Conditional density estimation in a censored single-index regression model. Bermoulli, 16, 514–542.Google Scholar
  2. Buckley, J., James, I. (1979). Linear regression with censored data. Biometrika, 66, 429–436.Google Scholar
  3. Chambless, L. E., Diao, G. (2006). Estimation of time-dependent area under the ROC curve for long-term risk prediction. Statistics in Medicine, 25, 3474–3486.Google Scholar
  4. Chiang, C. T., Huang, M. Y. (2012). New estimation and inference procedures for a single-index conditional distribution model. Journal of Multivariate Analysis, 111, 271–285.Google Scholar
  5. Chiang, C. T., Huang, S. Y. (2009). Estimation for the optimal combination of markers without modeling the censoring distribution. Biometrics, 65, 152–158.Google Scholar
  6. Chiang, C. T., Hung, H. (2010). Nonparametric estimation for time-dependent AUC. Journal of Statistical Planning and Inference, 140, 1162–1174.Google Scholar
  7. Chiang, C. T., Huang, M. Y., Wang, S. H. (2016). Bias and variance reduction in nonparametric estimation of time-dependent accuracy measures. Statistics in Medicine, 35, 5247–5266.Google Scholar
  8. Cox, D. R. (1972). Regression models and life tables. Journal of the Royal Statistical Society, B34, 187–220.MathSciNetMATHGoogle Scholar
  9. Cox, D. R., Oakes, D. (1984). Analysis of survival data. London: Chapman and Hall.Google Scholar
  10. Fleming, T. R., & Harrington, D. P. (1991). Counting processes and survival analysis. New York: Wiley.Google Scholar
  11. Gørgens, T. (2004). Average derivatives for hazard functions. Econometric Theory, 20, 437–463.MathSciNetCrossRefMATHGoogle Scholar
  12. Gørgens, T. (2006). Semiparametric estimation of single-index hazard functions without proportional hazards. Journal of Econometrics, 9, 1–22.MathSciNetCrossRefMATHGoogle Scholar
  13. Härdle, W., Hall, P., Ichimura, H. (1993). Optimal smoothing in single-index models. Annals of Statistics, 21, 157–178.Google Scholar
  14. Heagerty, P. J., Lumley, T., Pepe, M. S. (2000). Time-dependent ROC curves for censored survival data and diagnostic marker. Biometrics, 56, 337–344.Google Scholar
  15. Huang, M. Y., Chiang, C. T. (2016). An effective semiparametric estimation approach for the sufficient dimension reduction model. Journal of the American Statistical Association. doi: 10.1080/01621459
  16. Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation single-index models. Journal of Econometrics, 58, 71–120.MathSciNetCrossRefMATHGoogle Scholar
  17. Khan, S., Tamer, E. (2007). Partial rank estimation of duration models with general forms of censoring. Journal of Econometrics, 136, 251–280.Google Scholar
  18. Kosorok, M. R. (2008). Introduction to empirical processes and semiparametric inference. New York: Springer.CrossRefMATHGoogle Scholar
  19. Lee, K. W. J., Hill, J. S., Walley, K. R., Frohlich, J. J. (2006). Relative value of multiple plasma biomarkers as risk factors for coronary artery disease and death in an angiography cohort. Canadian Medical Association Journal, 174, 461–466.Google Scholar
  20. Lopez, O., Patilea, V., Van Keilegom, I. (2013). Single index regression models in the presence of censoring depending on the covariates. Bernoulli, 19, 721–747.Google Scholar
  21. McIntosh, M. W., Pepe, M. S. (2002). Combining several screening tests: Optimality of the risk score. Biometrics, 58, 657–664.Google Scholar
  22. Strzalkowska-Kominiak, E., Cao, R. (2013). Maximum likelihood estimation for conditional distribution single-index models under censoring. Journal of Multivariate Analysis, 114, 74–98.Google Scholar
  23. Strzalkowska-Kominiak, E., Cao, R. (2014). Beran-based approach for single-index models under censoring. Computational Statistics, 29, 1243–1261.Google Scholar
  24. Zeng, D., Lin, D. Y. (2007). Efficient estimation for the accelerated failure time model. Journal of the American Statistical Association, 102, 1387–1396.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2017

Authors and Affiliations

  • Chin-Tsang Chiang
    • 1
  • Shao-Hsuan Wang
    • 1
  • Ming-Yueh Huang
    • 1
  1. 1.Institute of Applied Mathematical SciencesNational Taiwan UniversityTaipeiTaiwan, ROC

Personalised recommendations