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Functional Decomposition and Shapley Interactions for Interpreting Survival Models

Sophie Hanna Langbein, Hubert Baniecki, Fabian Fumagalli, Niklas Koenen, Marvin N. Wright, Julia Herbinger

TL;DR

This work introduces Survival Functional Decomposition (SurvFD), a principled approach for analyzing feature interactions in machine learning survival models, and proposes SurvSHAP-IQ, which extends Shapley interactions to time-indexed functions, providing a practical estimator for higher-order, time-dependent interactions.

Abstract

Hazard and survival functions are natural, interpretable targets in time-to-event prediction, but their inherent non-additivity fundamentally limits standard additive explanation methods. We introduce Survival Functional Decomposition (SurvFD), a principled approach for analyzing feature interactions in machine learning survival models. By decomposing higher-order effects into time-dependent and time-independent components, SurvFD offers a previously unrecognized perspective on survival explanations, explicitly characterizing when and why additive explanations fail. Building on this theoretical decomposition, we propose SurvSHAP-IQ, which extends Shapley interactions to time-indexed functions, providing a practical estimator for higher-order, time-dependent interactions. Together, SurvFD and SurvSHAP-IQ establish an interaction- and time-aware interpretability approach for survival modeling, with broad applicability across time-to-event prediction tasks.

Functional Decomposition and Shapley Interactions for Interpreting Survival Models

TL;DR

This work introduces Survival Functional Decomposition (SurvFD), a principled approach for analyzing feature interactions in machine learning survival models, and proposes SurvSHAP-IQ, which extends Shapley interactions to time-indexed functions, providing a practical estimator for higher-order, time-dependent interactions.

Abstract

Hazard and survival functions are natural, interpretable targets in time-to-event prediction, but their inherent non-additivity fundamentally limits standard additive explanation methods. We introduce Survival Functional Decomposition (SurvFD), a principled approach for analyzing feature interactions in machine learning survival models. By decomposing higher-order effects into time-dependent and time-independent components, SurvFD offers a previously unrecognized perspective on survival explanations, explicitly characterizing when and why additive explanations fail. Building on this theoretical decomposition, we propose SurvSHAP-IQ, which extends Shapley interactions to time-indexed functions, providing a practical estimator for higher-order, time-dependent interactions. Together, SurvFD and SurvSHAP-IQ establish an interaction- and time-aware interpretability approach for survival modeling, with broad applicability across time-to-event prediction tasks.
Paper Structure (35 sections, 5 theorems, 85 equations, 19 figures, 5 tables)

This paper contains 35 sections, 5 theorems, 85 equations, 19 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Survival Analysis
  4. Functional Decomposition
  5. Shapley Values for Survival Models
  6. Methodology
  7. Ground-Truth Assumptions
  8. Functional Decomposition for Survival (SurvFD)
  9. SurvFD with independent features
  10. SurvFD with dependent features
  11. Shapley Interactions for Survival (SurvSHAP-IQ)
  12. Empirical Validation
  13. Experiments with Simulated Data and Risk Scores
  14. Experiments with Real-world Applications
  15. Conclusion and Limitations
  16. ...and 20 more sections

Key Result

Theorem 3.2

Let $\log h(t|\bm{x})$ be defined as in Eq. eq:log_hazard with ground-truth sets $\mathcal{I}_d$ and $\mathcal{I}_{id}$. Assume that features are mutually independent. If either (i)$G(t|\bm{x})$ is linear in $\bm{x}$ including interactions, or (ii)$G(t|\bm{x})$ is an additive main effect model, then

Figures (19)

  • Figure 1: SurvFD and SurvSHAP-IQ facilitate the interpretation of interactions in survival models. (images: Flaticon.com)
  • Figure 2: Ten simulation scenarios.
  • Figure 3: Exact SurvSHAP-IQ attribution curves for a selected observation computed on the ground-truth log-hazard (left), hazard (middle), and survival function (right). The difference between individual ground-truth values and the dataset average is plotted as a grey dashed line. The full results are shown in Figures \ref{['fig:sim_loghazard_gt']}-\ref{['fig:sim_survival_cox']}.
  • Figure 4: Exact SurvSHAP-IQ attribution curves for a selected observation computed on the predicted survival functions of CoxPH (left) and GBSA (right). For complete results see Fig. \ref{['fig:sim_survival_gbsa']} & \ref{['fig:sim_survival_cox']}.
  • Figure 5: SurvSHAP-IQ attributions for a multi-modal survival deep learning model predicting a patient's (ID: TCGA-A7-A13D) survival probability (left) and probability mass function (right) at $t = 4.24$ years. Node size represents individual feature effects, edges indicate pairwise interaction effects between patches of histopathological WSI and clinical features.
  • ...and 14 more figures

Theorems & Definitions (17)

  • Definition 2.1: Hazard function
  • Definition 2.2: Survival function
  • Definition 3.1: SurvFD
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Proposition 3.5
  • Theorem 3.6
  • Definition 3.7: SurvSHAP-IQ decomposition
  • proof : Proof using assumption (i)
  • ...and 7 more