The Annals of Statistics

Local partial-likelihood estimation for lifetime data

Jianqing Fan, Huazhen Lin, and Yong Zhou

Full-text: Open access


This paper considers a proportional hazards model, which allows one to examine the extent to which covariates interact nonlinearly with an exposure variable, for analysis of lifetime data. A local partial-likelihood technique is proposed to estimate nonlinear interactions. Asymptotic normality of the proposed estimator is established. The baseline hazard function, the bias and the variance of the local likelihood estimator are consistently estimated. In addition, a one-step local partial-likelihood estimator is presented to facilitate the computation of the proposed procedure and is demonstrated to be as efficient as the fully iterated local partial-likelihood estimator. Furthermore, a penalized local likelihood estimator is proposed to select important risk variables in the model. Numerical examples are used to illustrate the effectiveness of the proposed procedures.

Article information

Ann. Statist., Volume 34, Number 1 (2006), 290-325.

First available in Project Euclid: 2 May 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62N01: Censored data models 62N02: Estimation

Local partial likelihood one-step estimation varying coefficient proportional hazards model variable selection


Fan, Jianqing; Lin, Huazhen; Zhou, Yong. Local partial-likelihood estimation for lifetime data. Ann. Statist. 34 (2006), no. 1, 290--325. doi:10.1214/009053605000000796.

Export citation


  • Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
  • Andersen, P. K. and Gill, R. D. (1982). Cox's regression model for counting processes: A large sample study. Ann. Statist. \bf10 1100–1120.
  • Antoniadis, A. and Fan, J. (2001). Regularization of wavelet approximations (with discussion). J. Amer. Statist. Assoc. 96 939–967.
  • Bjerve, S., Doksum, K. A. and Yandell, B. S. (1985). Uniform confidence bounds for regression based on a simple moving average. Scand. J. Statist. 12 159–169.
  • Breiman, L. (1995). Better subset regression using the nonnegative garrote. Technometrics 37 373–384.
  • Breslow, N. (1972). Discussion of “Regression models and life-tables,” by D. R. Cox. J. Roy. Statist. Soc. Ser. B 34 216–217.
  • Brumback, B. and Rice, J. A. (1998). Smoothing spline models for the analysis of nested and crossed samples of curves (with discussion). J. Amer. Statist. Assoc. \bf93 961–994.
  • Cai, Z., Fan, J. and Li, R. (2000). Efficient estimation and inferences for varying-coefficient models. J. Amer. Statist. Assoc. 95 888–902.
  • Cai, Z., Fan, J. and Yao, Q. (2000). Functional-coefficient regression models for nonlinear time series. J. Amer. Statist. Assoc. \bf95 941–956.
  • Cai, Z. and Sun, Y. (2003). L ocal linear estimation for time-dependent coefficients in Cox's regression models. Scand. J. Statist. 30 93–111.
  • Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. J. Amer. Statist. Assoc. 92 477–489.
  • Carroll, R. J., Ruppert, D. and Welsh, A. H. (1998). Local estimating equations. J. Amer. Statist. Assoc. 93 214–227.
  • Chen, R. and Tsay, R. S. (1993). Functional-coefficient autoregressive models. J. Amer. Statist. Assoc. \bf88 298–308.
  • Cleveland, W. S., Grosse, E. and Shyu, W. M. (1992). Local regression models. In Statistical Models in S (J. M. Chambers and T. J. Hastie, eds.) 309–376. Wadsworth and Brooks/Cole, Pacific Grove, CA.
  • Fan, J. and Chen, J. (1999). One-step local quasi-likelihood estimation. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 927–943.
  • Fan, J., Gijbels, I. and King, M. (1997). Local likelihood and local partial likelihood in hazard regression. Ann. Statist. 25 1661–1690.
  • Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348–1360.
  • Fan, J. and Li, R. (2002). Variable selection for Cox's proportional hazards model and frailty model. Ann. Statist. \bf30 74–99.
  • Fan, J. and Zhang, W. (1999). Statistical estimation in varying coefficient models. Ann. Statist. \bf27 1491–1518.
  • Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. Wiley, New York.
  • Gamerman, D. (1998). Markov chain Monte Carlo for dynamic generalized linear models. Biometrika \bf85 215–227.
  • Härdle, W. (1989). Asymptotic maximal deviation of $M$-smoothers. J. Multivariate Anal. 29 163–179.
  • Hastie, T. and Tibshirani, R. (1990). Exploring the nature of covariate effects in the proportional hazards model. Biometrics \bf46 1005–1016.
  • Hastie, T. J. and Tibshirani, R. J. (1993). Varying-coefficient models (with discussion). J. Roy. Statist. Soc. Ser. B. 55 757–796.
  • Hoover, D. R., Rice, J. A., Wu, C. O. and Yang, L.-P. (1998). Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85 809–822.
  • Johnston, G. J. (1982). Probabilities of maximal deviations for nonparametric regression function estimates. J. Multivariate Anal. \bf12 402–414.
  • Martinussen, T., Scheike, T. H. and Skovgaard, I. M. (2000). Efficient estimation of fixed and time-varying covariate effects in multiplicative intensity models. Unpublished manuscript.
  • Marzec, L. and Marzec, P. (1997). On fitting Cox's regression model with time-dependent coefficients. Biometrika \bf84 901–908.
  • Morris, C. N., Norton, E. C. and Zhou, X. H. (1994). Parametric duration analysis of nursing home usage. In Case Studies in Biometry (N. Lange, L. Ryan, L. Billard, D. Brillinger, L. Conquest and J. Greenhouse, eds.) 231–248. Wiley, New York.
  • Murphy, S. A. (1993). Testing for a time dependent coefficient in Cox's regression model. Scand. J. Statist. \bf20 35–50.
  • Murphy, S. A. and Sen, P. K. (1991). Time-dependent coefficients in a Cox-type regression model. Stochastic Process. Appl. 39 153–180.
  • Tian, L., Zucker, D. and Wei, L. J. (2002). On the Cox model with time-varying regression coefficients. Working paper, Dept. Biostatistics, Harvard Univ.
  • Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B. \bf58 267–288.
  • Tibshirani, R. J. (1997). The lasso method for variable selection in the Cox model. Statistics in Medicine 16 385–395.
  • Wu, C. O. and Chiang, C.-T. (2000). Kernel smoothing on varying coefficient models with longitudinal dependent variable. Statist. Sinica \bf10 433–456.
  • Wu, C. O., Chiang, C.-T. and Hoover, D. R. (1998). Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data. J. Amer. Statist. Assoc. \bf93 1388–1402.
  • Zucker, D. M. and Karr, A. F. (1990). Nonparametric survival analysis with time-dependent covariate effects: A penalized partial likelihood approach. Ann. Statist. 18 329–353.