A surrogate ℓ₀ sparse Cox's regression with applications to sparse high-dimensional massive sample size time-to-event data

Sparse high-dimensional massive sample size (sHDMSS) time-to-event data present multiple challenges to quantitative researchers as most current sparse survival regression methods and software will grind to a halt and become practically inoperable. This paper develops a scalable ℓ₀-based sparse Cox regression tool for right-censored time-to-event data that easily takes advantage of existing high performance implementation of ℓ₂-penalized regression method for sHDMSS time-to-event data. Specifically, we extend the ℓ₀-based broken adaptive ridge (BAR) methodology to the Cox model, which involves repeatedly performing reweighted ℓ₂-penalized regression. We rigorously show that the resulting estimator for the Cox model is selection consistent, oracle for parameter estimation, and has a grouping property for highly correlated covariates. Furthermore, we implement our BAR method in an R package for sHDMSS time-to-event data by leveraging existing efficient algorithms for massive ℓ₂-penalized Cox regression. We evaluate the BAR Cox regression method by extensive simulations and illustrate its application on an sHDMSS time-to-event data from the National Trauma Data Bank with hundreds of thousands of observations and tens of thousands sparsely represented covariates.