Regularization for Cox's proportional hazards model with NP-dimensionality

TL;DR

This paper develops and analyzes nonconcave penalized methods, like SCAD, for high-dimensional Cox models with censored data, demonstrating oracle properties and improved model selection in genomics applications.

Contribution

It establishes strong oracle properties for nonconcave penalties in NP-dimensional Cox models with censoring, reducing conditions needed for consistent model selection.

Findings

Nonconcave penalties improve model selection consistency.

The proposed algorithm effectively finds solution paths.

Simulation and gene study results validate the methods.

Abstract

High throughput genetic sequencing arrays with thousands of measurements per sample and a great amount of related censored clinical data have increased demanding need for better measurement specific model selection. In this paper we establish strong oracle properties of nonconcave penalized methods for nonpolynomial (NP) dimensional data with censoring in the framework of Cox's proportional hazards model. A class of folded-concave penalties are employed and both LASSO and SCAD are discussed specifically. We unveil the question under which dimensionality and correlation restrictions can an oracle estimator be constructed and grasped. It is demonstrated that nonconcave penalties lead to significant reduction of the "irrepresentable condition" needed for LASSO model selection consistency. The large deviation result for martingales, bearing interests of its own, is developed for characterizing the strong oracle property. Moreover, the nonconcave regularized estimator, is shown to achieve asymptotically the information bound of the oracle estimator. A coordinate-wise algorithm is developed for finding the grid of solution paths for penalized hazard regression problems, and its performance is evaluated on simulated and gene association study examples.