Abstract
Assuming Cox’s regression model, we consider penalized likelihood approaches to conduct variable selection under nested case-control sampling or case-cohort sampling. Penalized non-parametric maximum likelihood estimate (PNPMLE) are characterized by self-consistency equations derived from score functions, which form the basis of the algorithm to compute PNPMLE. Consistency, asymptotic normality and oracle properties of the PNPMLE, the sparsity property of the penalty, and a consistent estimate of the asymptotic variance, based on observed profile likelihood, are established. A cross-validation method is used to choose the tuning parameter within a family of penalty function. Simulation studies indicate that the numerical performance of PNPMLE is satisfactory and that LASSO performs best when cohort size is small and SCAD performs best when cohort size is large and may eventually perform as well as the oracle estimator, resembling the findings when i.i.d. sampling is considered. This method is also illustrated in a real dataset.