Adaptive-LASSO for Cox’s Proportional Hazards Model

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, and is named adaptive-LASSO (ALASSO) estimator. Instead of applying the same penalty to all the coefficients as other shrinkage methods, the ALASSO advocates different penalties for different coefficients: unimportant variables receive larger penalties than important variables. In this way, important variables can be protectively preserved in the model selection process, while unimportant ones are shrunk more towards zero and thus more likely to be dropped from the model. We study the consistency and rate of convergence of the proposed estimator. Further, with proper choice of regularization parameters, we have shown that the ALASSO perform as well as the oracle procedure in variable selection; namely, it works as well as if the correct submodel were known. Another advantage of the ALASSO is its convex optimization form and convenience in implementation. Simulated and real examples show that the ALASSO estimator compares favorably with the LASSO. Some key words: Adaptive LASSO (ALASSO), LASSO, Penalized partial likelihood, Proportional