[Paper Review] Confidence intervals for high-dimensional Cox models

Yi Yu, Jelena Bradić+1 more|arXiv (Cornell University)|Mar 3, 2018

Statistical Methods and Inference23 references16 citations

TL;DR

This paper proposes a debiasing method for high-dimensional Cox models with p ≫ n, constructing asymptotically valid confidence intervals for regression coefficients using a modified CLIME estimator for the sparse precision matrix and a one-step debiased estimator. The key contribution is a theoretical guarantee of asymptotic normality for linear functionals of the debiased estimator under mild regularity conditions, even with infinite time support and non-constant risk sets.

ABSTRACT

The purpose of this paper is to construct confidence intervals for the regression coefficients in high-dimensional Cox proportional hazards regression models where the number of covariates may be larger than the sample size. Our debiased estimator construction is similar to those in Zhang and Zhang (2014) and van de Geer et al. (2014), but the time-dependent covariates and censored risk sets introduce considerable additional challenges. Our theoretical results, which provide conditions under which our confidence intervals are asymptotically valid, are supported by extensive numerical experiments.

Motivation & Objective

• To construct valid confidence intervals for regression coefficients in high-dimensional Cox proportional hazards models where the number of covariates p exceeds the sample size n.
• To address the challenge of time-dependent covariates and censored risk sets in high-dimensional survival analysis, which complicate standard inference.
• To develop a debiased estimator that achieves asymptotic normality for linear functionals c⊤β, enabling valid inference in ultrahigh-dimensional settings (p = o(exp(n^a)) for all a > 0).
• To overcome theoretical obstacles such as weak convergence assumptions and infinite time support, using a novel truncation argument and modified precision matrix estimation.
• To provide a practical and theoretically grounded method for inference in high-dimensional survival data, particularly relevant in biomedicine and genomics.

Proposed method

• Uses the Lasso-penalized partial likelihood estimator as an initial sparse estimator β̂ for high-dimensional regression coefficients.
• Constructs a sparse precision matrix estimator bΘ via a modified CLIME approach to approximate the inverse of the negative Hessian of the log-partial likelihood.
• Applies a one-step debiasing correction: β̃ = β̂ + bΘ × ˙ℓ(β̂), where ˙ℓ(β̂) is the score function at β̂, to reduce bias in the initial estimator.
• Employs a novel truncation argument to handle infinite time support and non-constant risk sets, avoiding strong assumptions on weak convergence of covariance processes.
• Uses a modified CLIME estimator that accounts for mean-shifted design matrices arising from tilting weights in the Cox model, preserving necessary independence structure.
• Establishes asymptotic normality of c⊤β̃ via concentration inequalities and martingale arguments, controlling the deviation of empirical processes from their mean.

Experimental results

Research questions

• RQ1 Can valid confidence intervals be constructed for regression coefficients in high-dimensional Cox models with p ≫ n?
• RQ2 How can the debiasing approach be adapted to the time-dependent, censored nature of survival data, particularly with non-constant risk sets?
• RQ3 What conditions ensure the asymptotic normality of the debiased estimator in the presence of infinite time support and mean-shifted covariates?
• RQ4 Can the theoretical framework avoid the strong weak convergence assumptions inherent in classical martingale central limit theorems?
• RQ5 How does the performance of the method depend on sample size and sparsity, particularly for signal versus noise variables?

Key findings

• The proposed debiased estimator β̃ is asymptotically normal for any fixed linear functional c⊤β, with variance-covariance structure estimated via the modified CLIME precision matrix.
• The method achieves asymptotic validity of confidence intervals under mild conditions, including p = o(exp(n^a)) for all a > 0, covering the ultrahigh-dimensional regime.
• Numerical experiments show that valid p-values and confidence intervals can be obtained for noise variables with relatively small sample sizes, while signal variables require larger samples for good coverage.
• Theoretical analysis establishes that the estimation error of the debiased estimator is Op(√(log(np)/n)) under appropriate sparsity and eigenvalue conditions.
• The truncation argument successfully handles infinite time support and non-constant risk sets, allowing the theory to apply beyond the bounded time horizon setting.
• The modified CLIME estimator enables consistent estimation of the precision matrix even when the design matrix is mean-shifted, overcoming a key challenge in the Cox model.