Scalable nonlinear Cox modeling via random Fourier features with analytic uncertainty

Area of Science:

• Biostatistics
• Machine Learning
• Survival Analysis

Background:

• The Cox proportional hazards model assumes linearity, limiting its ability to model complex biomedical risk structures like U-shaped associations.
• Existing flexible kernel-based methods have high computational costs, restricting their use in large cohorts.
• Many non-linear machine learning approaches lack analytical uncertainty measures for individual predictions.

Purpose of the Study:

• To introduce a scalable Random Fourier Features-based Cox regression (RFF-Cox) approach for modeling non-linear risk relationships.
• To enable uncertainty quantification and formal inference, including covariate-level interpretation and interaction detection.
• To provide a computationally efficient alternative to existing methods for large-scale survival analysis.

Main Methods:

• Developed RFF-Cox by mapping stationary kernels to a finite-dimensional feature space, reducing computational complexity.
• Estimated model parameters using Newton-Raphson on a ridge-regularized partial likelihood with automatic bandwidth optimization.
• Quantified uncertainty via the Fisher information matrix and a multivariate Delta method, enabling Taylor-expansion-based inference.

Main Results:

• RFF-Cox accurately modeled non-linear associations (RMSE: 0.137 vs. 0.314) and recovered U-shaped risk functions, while performing comparably to classical Cox under linearity.
• Detected a significant SBP × Smoking interaction (estimate -1.093, p < 0.001) with good analytical 95% confidence interval coverage (81.9% in non-linear scenarios).
• Demonstrated competitive discriminatory power and computational efficiency against Random Survival Forests and XGBoost, with favorable calibration on the METABRIC dataset.

Conclusions:

• RFF-Cox offers a practical survival analysis framework that handles non-linear relationships efficiently.
• The method provides formal inference tools, including covariate-level interpretation and interaction detection, surpassing limitations of tree-based or deep learning survival models.
• RFF-Cox degenerates to the classical Cox model under linearity, ensuring broad applicability.