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A Modern Theory for High-dimensional Cox Regression Models

Hanxuan Ye, Xianyang Zhang, Huijuan Zhou

TL;DR

This work analyzes the maximum partial likelihood estimator (MPLE) for high-dimensional Cox regression in which $p$ grows proportionally with $n$. Using the Convex Gaussian Min-max Theorem (CGMT), it establishes a sharp phase transition for MPLE existence and derives a new asymptotic theory that precisely characterizes MPLE error and the asymptotic distribution of Wald-type statistics under $p/n\to\delta\in(0,1)$. The results reveal that classical MPLE inference is invalid in this regime and provide a rigorous framework for exact high-dimensional behavior, including finite-sample checks and practical guidance for hypothesis testing. The findings have broad implications for survival analysis in high-dimensional settings and extend CGMT-based analysis to non-separable partial-likelihood objectives.

Abstract

The proportional hazards model has been extensively used in many fields such as biomedicine to estimate and perform statistical significance testing on the effects of covariates influencing the survival time of patients. The classical theory of maximum partial-likelihood estimation (MPLE) is used by most software packages to produce inference, e.g., the coxph function in R and the PHREG procedure in SAS. In this paper, we investigate the asymptotic behavior of the MPLE in the regime in which the number of parameters p is of the same order as the number of samples n. The main results are (i) existence of the MPLE undergoes a sharp 'phase transition'; (ii) the classical MPLE theory leads to invalid inference in the high-dimensional regime. We show that the asymptotic behavior of the MPLE is governed by a new asymptotic theory. These findings are further corroborated through numerical studies. The main technical tool in our proofs is the Convex Gaussian Min-max Theorem (CGMT), which has not been previously used in the analysis of partial likelihood. Our results thus extend the scope of CGMT and shed new light on the use of CGMT for examining the existence of MPLE and non-separable objective functions.

A Modern Theory for High-dimensional Cox Regression Models

TL;DR

This work analyzes the maximum partial likelihood estimator (MPLE) for high-dimensional Cox regression in which grows proportionally with . Using the Convex Gaussian Min-max Theorem (CGMT), it establishes a sharp phase transition for MPLE existence and derives a new asymptotic theory that precisely characterizes MPLE error and the asymptotic distribution of Wald-type statistics under . The results reveal that classical MPLE inference is invalid in this regime and provide a rigorous framework for exact high-dimensional behavior, including finite-sample checks and practical guidance for hypothesis testing. The findings have broad implications for survival analysis in high-dimensional settings and extend CGMT-based analysis to non-separable partial-likelihood objectives.

Abstract

The proportional hazards model has been extensively used in many fields such as biomedicine to estimate and perform statistical significance testing on the effects of covariates influencing the survival time of patients. The classical theory of maximum partial-likelihood estimation (MPLE) is used by most software packages to produce inference, e.g., the coxph function in R and the PHREG procedure in SAS. In this paper, we investigate the asymptotic behavior of the MPLE in the regime in which the number of parameters p is of the same order as the number of samples n. The main results are (i) existence of the MPLE undergoes a sharp 'phase transition'; (ii) the classical MPLE theory leads to invalid inference in the high-dimensional regime. We show that the asymptotic behavior of the MPLE is governed by a new asymptotic theory. These findings are further corroborated through numerical studies. The main technical tool in our proofs is the Convex Gaussian Min-max Theorem (CGMT), which has not been previously used in the analysis of partial likelihood. Our results thus extend the scope of CGMT and shed new light on the use of CGMT for examining the existence of MPLE and non-separable objective functions.
Paper Structure (20 sections, 8 theorems, 146 equations, 7 figures)

This paper contains 20 sections, 8 theorems, 146 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Motivation
  4. Prior works and our contribution
  5. Technical tools
  6. Preliminaries
  7. Basic setup
  8. Failures of classical large sample theories
  9. Existence of the MPLE
  10. Phase transition boundary curve
  11. Checking the existence of MPLE in finite sample
  12. A New Asymptotic Theory
  13. Error analysis
  14. Asymptotic distributions
  15. Conclusion
  16. ...and 5 more sections

Key Result

Theorem 3.1

Define the quantities where $\widetilde{{\bf q}}=(q_{11},\dots,q_{n1})^\top$ and ${\bf h}\sim N(0,\mathbf{I}_n)$ is independent of $\{(y_i,q_{i1})\}^{n}_{i=1}$. The MPLE exists (with probability tending to one) if $\delta<h_L(\lambda_0,\kappa,P_\mathcal{C})$ and the MLE does not exist (with probability tending to one) if

Figures (7)

  • Figure 1: The biasness of the MPLE.
  • Figure 2: Comparison between the Fisher std. and true std. The red line represents the Fisher std. The blue histogram depicts the empirical distribution of the std's of the 400 null coefficients.
  • Figure 3: Invalid inferences based on the classical theory for MPLE.
  • Figure 4: Theoretical transition boundary and the empirical probability that the MPLE exists.
  • Figure 5: Comparison between $(\hat{a},\hat{b})$ and $(a^*,b^*)$ for various values of $\delta$ and $\kappa$, where $n=500$ and the number of replications is 100.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Theorem 3.1
  • Remark 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.1
  • Theorem 4.3
  • Definition S1.1: GMT admissible sequence ThrampoulidisRegularized2015
  • Theorem S1.2: CGMT ThrampoulidisRegularized2015
  • Proposition S2.1
  • Proposition S2.2: Polar cone theorem
  • ...and 2 more