A surrogateℓ₀sparse Cox's regression with applications to sparse high‐dimensional massive sample size time‐to‐event data

Abstract

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.

Extreme learning machine Cox model for high‐dimensional survival analysis

“…More recently, a novel sparse Cox regression via broken adaptive ridge (CoxBAR) is developed using L 0 -based iteratively reweighted L 2 -penalized Cox regression. 32 The CoxBAR estimation of β starts with an initial Cox ridge regression estimator trueβ^(0)=argminβ{−2l(β)+ξntrue∑j=1pnβj2}, which is updated iteratively by a reweighed L 2 -penalized Cox regression estimator trueβ^(k)=argminβ{−2l(β)+λntrue∑j=1pnβj2(β^j(k−1))2}, k≥1, where ξ n and λ n are nonnegative penalization tuning parameters. The CoxBAR estimator is defined as β^=limk→∞trueβ^(k). The above CoxBAR estimator inherits some appealing properties of both L 0 - and L 2 -penalized regressions, such as an oracle property for selection and estimation and a grouping property for highly correlated covariates.…”

“…Moreover, the CoxBAR estimator with λ = ln(n) is insensitive to the choice of ξ n . 32 These properties suggest that the CoxBAR estimator can be a very competitive candidate for Cox models with common penalty terms in computation-intensive setting such as high dimensional or large-scale survival data.…”

“…Let trueβ˜(0) be a Cox ridge estimator (the ridge penalty parameter need not to be optimal and can be simply set to ln( n )). 32 Estimate β by maximizing g (β) via a simple Newton-Raphson procedure 38 for a small or moderate n or a sophisticated optimization algorithm 6 for a large n .…”

“…31 Third, different from backpropagation approaches that consider MLP networks or deep networks as a black box, ELM handles SLFNs as a white box and can train hidden neurons layer by layer if multiple layers are necessary. Together with ELM, we adopt a recently proposed L 0 -based penalization method, named CoxBAR, 32 for parameter estimation. In contrast to most regularization methods that require data-driven selection of the regularization tuning parameter(s), the CoxBAR method uses pre-specified tuning parameters and thus is computationally efficient.…”

Sampling‐based estimation for massive survival data with additive hazards model

“…Only a limited number of articles have studied the topics on massive survival data. For example, Kawaguchi et al 12 developed a new scalable sparse Cox regression method for high‐dimensional survival data with massive sample sizes. Wang et al 13 proposed an efficient “divide and conquer” algorithm to fit sparse Cox regression with massive datasets.…”