On the Cox model with time-varying regression coefficients

Lu Tian*, David Zucker, L. J. Wei

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

170 Scopus citations


In the analysis of censored failure time observations, the standard Cox proportional hazards model assumes that the regression coefficients are time invariant. Often, these parameters vary over time, and the temporal covariate effects on the failure time are of great interest. In this article, following previous work of Cai and Sun, we propose a simple estimation procedure for the Cox model with time-varying coefficients based on a kernel-weighted partial likelihood approach. We construct pointwise and simultaneous confidence intervals for the regression parameters over a properly chosen time interval via a simple resampling technique. We derive a prediction method for future patients' survival with any specific set of covariates. Building on the estimates for the time-varying coefficients, we also consider the mixed case and present an estimation procedure for time-independent parameters in the model. Furthermore, we show how to use an integrated function of the estimate for a specific regression coefficient to examine the adequacy of proportional hazards assumption for the corresponding covariate graphically and numerically. All of the proposals are illustrated extensively with a well-known study from the Mayo Clinic.

Original languageAmerican English
Pages (from-to)172-183
Number of pages12
JournalJournal of the American Statistical Association
Issue number469
StatePublished - Mar 2005

Bibliographical note

Funding Information:
Lu Tian is Assistant Professor, Department of Preventive Medicine, Feinberg School of Medicine, Northwestern University, Chicago, IL 60611 (E-mail: lutian@northwestern.edu). David Zucker is Associate Professor, Department of Statistics, Hebrew University, Mount Scopus, Jerusalem 91905, Israel (E-mail: mszucker@mscc.huji.ac.il). L. J. Wei is Professor, Department of Biostatistics, Harvard University, Boston, MA 02115 (E-mail: wei@sdac.harvard.edu). This work was supported in part by grants from the National Institutes of Health. The authors are grateful to the referees, the associate editor, and the editor for insightful comments on the manuscript.


  • Confidence band
  • Kernel estimation
  • Martingale
  • Model checking and selection
  • Partial likelihood
  • Prediction
  • Survival analysis


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