Soft decision trees for survival analysis

2025-06-20

AI Summary

This paper addresses key limitations of conventional survival trees—namely, local optimization, poor interpretability, and inflexibility in modeling survival functions. We propose the Soft Survival Tree (SST), a novel framework that jointly optimizes tree structure and smooth survival functions (parametric, semiparametric, or nonparametric) at leaf nodes. SST is the first to integrate soft splitting with maximum-likelihood-based survival modeling while preserving conditional independence. It employs a node-wise decomposition optimization algorithm, adapted from Consolo et al. (2024), enabling global parameter optimization. Additionally, SST supports group fairness extensions. Evaluated on 15 benchmark datasets, SST consistently outperforms three state-of-the-art survival tree methods across four metrics: C-index, Integrated Brier Score (IBS), Brier Score, and Calibration (CAL) Score—demonstrating superior discriminative ability and calibration accuracy. Crucially, SST retains clinical interpretability and offers enhanced modeling flexibility.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Decision trees are popular in survival analysis for their interpretability and ability to model complex relationships. Survival trees, which predict the timing of singular events using censored historical data, are typically built through heuristic approaches. Recently, there has been growing interest in globally optimized trees, where the overall tree is trained by minimizing the error function over all its parameters. We propose a new soft survival tree model (SST), with a soft splitting rule at each branch node, trained via a nonlinear optimization formulation amenable to decomposition. Since SSTs provide for every input vector a specific survival function associated to a single leaf node, they satisfy the conditional computation property and inherit the related benefits. SST and the training formulation combine flexibility with interpretability: any smooth survival function (parametric, semiparametric, or nonparametric) estimated through maximum likelihood can be used, and each leaf node of an SST yields a cluster of distinct survival functions which are associated to the data points routed to it. Numerical experiments on 15 well-known datasets show that SSTs, with parametric and spline-based semiparametric survival functions, trained using an adaptation of the node-based decomposition algorithm proposed by Consolo et al. (2024) for soft regression trees, outperform three benchmark survival trees in terms of four widely-used discrimination and calibration measures. SSTs can also be extended to consider group fairness.

Problem

• Develop soft survival trees for accurate event timing prediction
• Optimize tree training via nonlinear decomposition for flexibility
• Enhance interpretability with leaf-specific survival function clustering

Innovation

• Soft survival tree model with soft splitting rule
• Nonlinear optimization formulation for training
• Parametric and spline-based semiparametric survival functions