Abstract
The partially linear additive hazards model has been proposed to study the interaction between some covariates and an exposure variable. In this paper, we extend it to the partially varying coefficient single-index additive hazard model where the high dimension covariates are collapsed to a single index, due to practical needs. Two sets of estimating equations were proposed to estimate the varying coefficient functions in the linear components: the link function for the single index and the single-index parameter vector separately. It was shown that the proposed local and global estimators are asymptotically normal. Simulation studies were conducted to examine the finite-sample performance of our method to compare the relative performance of our method with existing ones. A real data analysis was used to illustrate the proposed methods.
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References
Aalen, O.O. (1989). A linear regression model for the analysis of life times. Statistics in Medicine, 8, 907–925.
Breslow, N. E., Day, N. E. (1980). Statistical Methods in Cancer Research I, Volume I: The Analysis of Case-Control studies. IARC Scientific Publication 32, International Agency for Research on Cancer, Lyon.
Breslow, N. E., Day, N. E. (1987). Statistical Methods in Cancer Research, Volume II: The Design and Analysis of Cohort studies. IARC Scientific Publication 82, International Agency for Research on Cancer, Lyon.
Cai, Z., Sun, Y. (2003). Local linear estimation for time-dependent coefficients in Cox’s regression models. Scandinavian Journal of Statistics, 30, 93–111.
Carroll, R., Fan, J., Gijbels, I., Wand, M. P. (1997). Generalized partially linear single-index models. Journal of the American Statistical Association, 92, 477–489.
Chen, K. N., Tong, X. W. (2010). Varying coefficient transformation models with censored data. Biometrika, 97, 969–976.
Chen, K. N., Jin, Z. Z., Ying, Z. L. (2002). Semiparametric analysis of transformation models with censored data. Biometrika, 89, 659–668.
Cox, D. R. (1972). Regression models and life-tables (with discussion). Journal of the Royal Statistical Society Series B, 34, 187–220.
Cui, X., Hardle, W., Zhu, L. X. (2011). The EFM approach for single-index models. The Annals of Statistics, 39, 1658–1688.
Fan, J., Lin, H. Z., Zhou, Y. (2006). Local partial likelihood estimation for lifetime data. The Annals of Statistics, 34, 290–325.
Fleming, T. R., Harrington, D. P. (1991). Counting Processes and Survival Analysis. New York: Wiley.
Huffer, F. W., McKeague, I.W. (1991). Weighted least squares estimation for Aalen’s additive risk model. Journal of the American Statistical Association, 86, 114–129.
Jin, Z., Ying, Z., Wei, L. L. (2001). A simple resampling method by perturbing the minimand. Biometrka, 88, 381–390.
Li, H., Yin, G.S., Zhou, Y. (2007). Local likelihood with time-varying additive hazards model. The Canadian Journal of Statistics, 35, 321–337.
Liang, H., Liu, X., Li, R. Z., Tsai, C. L. (2010). Estimation and testing for partially linear single-index models. The Annals of Statistics, 38, 3811–3836.
Lin, D. Y., Ying, Z. L. (1994). Semiparametric analysis of the additive risk model. Biometrika, 81, 61–71.
Lu, W. B., Zhang, H. H. (2010). On estimation of partially linear transformation models. Journal of the American Statistical Association, 105, 683–691.
McKeague, I. W., Sasieni, P. D. (1994). A partly parametric additive risk model. Biometrika, 81, 501–514.
Peng, L. M., Huang, Y. J. (2008). Survival analysis with quantile regression models. Journal of the American Statistical Association, 103, 637–649.
Tian, L., Zucker, D., Wei, L. J. (2005). On the Cox model with time varying regression coefficients. Journal of the American Statistical Association, 100, 172–183.
van der vaart, Wellner, J. (1996). Weak Convergence and Empirical Processes. New York: Springer.
Xia, Y., Hardle, W. (2006). Semi-parametric estimation of partially linear single-index models. Journal of Multivariate Analysis, 97, 1162–1184.
Yin, G. S., Li, H., Zeng, D. L. (2008). Partially linear additive hazards regression with varying coefficients. Journal of the American Statistical Association, 103, 1200–1213.
Yu, Y., Ruppert, D. (2002). Penalized spline estimation for partially linear single-index models. Journal of the American Statistical Association, 97, 1042–1054.
Wang, W., Wang, J. L., Wang, Q.H. (2009). Proportional hazards regression with unknown link function. Institute of Mathematical Statistics, 57, 47–66.
Wang, J. L., Xue, L. G., Zhu, L. X., Chong, Y. S. (2010). Estimation for a partial-linear single-index model. The Annals of Statistics, 38, 246–274.
Acknowledgments
Wang’s research was supported by the National Science Fund for Distinguished Young Scholars in China (10725106), the National Natural Science Foundation of China (General program 11171331 and Key program 11331011), a grant from the Key Lab of Random Complex Structure and Data Science, CAS and Natural Science Foundation of SZU.
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Wang, X., Wang, Q. & Zhou, XH.A. Partially varying coefficient single-index additive hazard models. Ann Inst Stat Math 67, 817–841 (2015). https://doi.org/10.1007/s10463-014-0484-7
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DOI: https://doi.org/10.1007/s10463-014-0484-7