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Square root Cox's survival analysis by the fittest linear and neural networks model

Maxime van Cutsem, Sylvain Sardy

TL;DR

This work challenges standard feature selection in Cox proportional hazards models by introducing a square-root form of the partial likelihood, a quantile universal threshold for penalty tuning, and a non-convex penalty that reduces shrinkage on true signals. The approach applies to linear predictors and extends to neural networks, enabling sparse, interpretable models even in high dimensions and nonlinear settings. Through phase-transition analyses on simulated data and multiple real datasets, the authors demonstrate superior exact support recovery with low false discovery rates, while maintaining competitive predictive performance. The methodology is demonstrated on the Primary Biliary Cirrhosis study, where QUT-based penalties yield sparse, stable selections and robust predictive capacity, with code available for replication.

Abstract

We revisit Cox's proportional hazard models and LASSO in the aim of improving feature selection in survival analysis. Unlike traditional methods relying on cross-validation or BIC, the penalty parameter $λ$ is directly tuned for feature selection and is asymptotically pivotal thanks to taking the square root of Cox's partial likelihood. Substantially improving over both cross-validation LASSO and BIC subset selection, our approach has a phase transition on the probability of retrieving all and only the good features, like in compressed sensing. The method can be employed by linear models but also by artificial neural networks.

Square root Cox's survival analysis by the fittest linear and neural networks model

TL;DR

This work challenges standard feature selection in Cox proportional hazards models by introducing a square-root form of the partial likelihood, a quantile universal threshold for penalty tuning, and a non-convex penalty that reduces shrinkage on true signals. The approach applies to linear predictors and extends to neural networks, enabling sparse, interpretable models even in high dimensions and nonlinear settings. Through phase-transition analyses on simulated data and multiple real datasets, the authors demonstrate superior exact support recovery with low false discovery rates, while maintaining competitive predictive performance. The methodology is demonstrated on the Primary Biliary Cirrhosis study, where QUT-based penalties yield sparse, stable selections and robust predictive capacity, with code available for replication.

Abstract

We revisit Cox's proportional hazard models and LASSO in the aim of improving feature selection in survival analysis. Unlike traditional methods relying on cross-validation or BIC, the penalty parameter is directly tuned for feature selection and is asymptotically pivotal thanks to taking the square root of Cox's partial likelihood. Substantially improving over both cross-validation LASSO and BIC subset selection, our approach has a phase transition on the probability of retrieving all and only the good features, like in compressed sensing. The method can be employed by linear models but also by artificial neural networks.
Paper Structure (14 sections, 2 theorems, 17 equations, 4 figures, 4 tables)

This paper contains 14 sections, 2 theorems, 17 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Cox model
  3. Model selection for survival analysis
  4. Our Proposal
  5. Selection of $\lambda$
  6. Extension to neural networks
  7. Optimization scheme
  8. Evaluation of the method by phase transition
  9. Linear models
  10. Nonlinear models
  11. Real survival data experiments
  12. Primary Biliary Cirrhosis Study
  13. Conclusions
  14. Acknowledgement

Key Result

Proposition 1

Assuming a linear learner $\mu_{{\boldsymbol \theta}}({\bf x})={\bf x}^{\rm T}{\boldsymbol \theta}$ and $P(\boldsymbol{\theta})=\|\boldsymbol{\theta}\|_1$ or $P(\boldsymbol{\theta})=P_\nu(\boldsymbol{\theta})$ as in eq:non convex pen, there exist finite values of the penalty $\lambda$ such that a mi

Figures (4)

  • Figure 1: Linear Monte-Carlo simulation results in terms of probability of exact support recovery, C-index, true positive rate and false discovery rate as a function of the sparsity index $s$.
  • Figure 2: Nonlinear Monte-Carlo simulation results in terms of probability of exact support recovery, C-index, true positive rate and false discovery rate as a function of the sparsity index $s$.
  • Figure 3: Trade-off between feature selection and predictive performance for different learners. Boxplots display the number of selected variables ($\hat{s}$, left panel) and the concordance index (C-index, right panel) for five datasets (linewise). Method abbreviations are as follows: LASSO_Q (our method with $\ell_1$ penalization), HLASSO_Q (our method with non convex $P_\nu$ penalty), LASSO_CV (LASSO tuned via cross-validation), AIC and BIC (models selected via Akaike and Bayesian information criteria), HLASSO_Q_net (neural network version of HLasso_Q), and Lasso_Net (LassoNet neural architecture). Both neural network architectures use a single hidden layer with 20 neurons. The vertical dashed line separates linear models (left) from neural network-based models (right).
  • Figure 4: Model comparison and feature effects for the Primary Biliary Cirrhosis dataset. The top row displays baseline cumulative hazard (left) and baseline survival (middle) for both the full model and our HarderLASSO method. The remaining panels plot the estimated effects of the retained predictors on patient survival probability over time.

Theorems & Definitions (4)

  • Proposition 1: Zero-thresholding function
  • Definition 1: QUT
  • Proposition 2
  • proof