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Biometrika Advance Access originally published online on May 23, 2007
Biometrika 2007 94(3):691-703; doi:10.1093/biomet/asm037
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Copyright © 2007 Biometrika Trust

Articles

Adaptive Lasso for Cox's proportional hazards model

Hao Helen Zhang and Wenbin Lu

Department of Statistics, North Carolina State University, Raleigh, North Carolina 27695-8203, U.S.A.

hzhang{at}stat.ncsu.edu

lu{at}stat.ncsu.edu

Received for publication 1 January 2006. Revision received 1 October 2006.
  Abstract

We investigate the variable selection problem for Cox's proportional hazards model, and propose a unified model selection and estimation procedure with desired theoretical properties and computational convenience. The new method is based on a penalized log partial likelihood with the adaptively weighted L1 penalty on regression coefficients, providing what we call the adaptive Lasso estimator. The method incorporates different penalties for different coefficients: unimportant variables receive larger penalties than important ones, so that important variables tend to be retained in the selection process, whereas unimportant variables are more likely to be dropped. Theoretical properties, such as consistency and rate of convergence of the estimator, are studied. We also show that, with proper choice of regularization parameters, the proposed estimator has the oracle properties. The convex optimization nature of the method leads to an efficient algorithm. Both simulated and real examples show that the method performs competitively.

Key Words: Adaptive Lasso • Lasso • Penalized partial likelihood • Proportional hazards model • Variable selection


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