Functional Decomposition and Shapley Interactions for Interpreting Survival Models

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

stat.MLFeb 18, 2026

Functional Decomposition and Shapley Interactions for Interpreting Survival Models

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

Organizations: Leibniz Institute for Prevention Research and Epidemiology – BIPS · Faculty of Mathematics and Computer Science, University of Bremen · University of Warsaw · Centre for Credible AI, Warsaw University of Technology ·

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.

Figures & tables

Figure 1: SurvFD and SurvSHAP-IQ facilitate the interpretation of interactions in survival models. (images: Flaticon.com)

Panel of Figure 2 — 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 B.2 - B.5 .

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. B.4 & B.5 .

Panel of Figure 5 — 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.

Panel of Figure 6 — Figure 6: Explanation of a GBSA model predicting the survival of patients diagnosed with uveal melanoma. Top: Shapley attributions (order 1) for multiple observations (curves) with different feature values (in color). Middle: SurvSHAP-IQ individual effects (order 2). We observe how adding interaction terms decreases the in-group variance of attributions. Bottom: SurvSHAP-IQ interaction effects (order 2), where color denotes four groups of observations. The remaining attribution and interaction effects are shown in Figure B.11 .

Figure 7: Error of different SurvSHAP-IQ approximators as a function of budget for the Bergamaschi task. Extended results for all tasks are in Figure B.12 .

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

#	$G(t\|\bm{x})$
( 1 )	$x_{1}\beta_{1}+x_{2}\beta_{2}+x_{3}\beta_{3}$ (linear $G(t\|\bm{x})$ , TI)
( 2 )	$x_{1}\beta_{1}\log(t+1)+x_{2}\beta_{2}+x_{3}\beta_{3}$ (linear $G(t\|\bm{x})$ , TD main)
( 3 )	$x_{1}\beta_{1}+x_{2}\beta_{2}+x_{3}\beta_{3}+x_{1}x_{3}\beta_{13}$ (linear $G(t\|\bm{x})$ , TI, interaction)
( 4 )	$x_{1}\beta_{1}\log(t+1)+x_{2}\beta_{2}+x_{3}\beta_{3}+x_{1}x_{3}\beta_{13}$ (linear $G(t\|\bm{x})$ , TD main, interaction)
( 5 )	$x_{1}\beta_{1}+x_{2}\beta_{2}+x_{3}\beta_{3}+x_{1}x_{3}\beta_{13}\log(t+1)$ (linear $G(t\|\bm{x})$ , TD interaction)
( 6 )	$\beta_{1}x_{1}^{2}+\beta_{2}\frac{2}{\pi}\arctan(0.7x_{2})+\beta_{3}x_{3}$ (generalized additive $G(t\|\bm{x})$ , TI)

Appendix

Table B.1 : Overview of the ten simulation scenarios.

Model	Scenario	IBS $\downarrow$	C-Index $\uparrow$
CoxPH	( 1 ) linear $G(t\|\bm{x})$ TI, no interactions	0.143	0.759
	( 2 ) linear $G(t\|\bm{x})$ TD main, no interactions	0.118	0.788
	( 3 ) linear $G(t\|\bm{x})$ TI, interactions	0.173	0.725
	( 4 ) linear $G(t\|\bm{x})$ TD main, interactions	0.172	0.707
	( 5 ) linear $G(t\|\bm{x})$ TD interactions	0.202	0.673
	( 6 ) generalized additive $G(t\|\bm{x})$ TI, no interactions	0.151	0.673

Appendix

Table B.2 : Comparison of model performance across ten simulation scenarios for CoxPH and GBSA using Integrated Brier Score (IBS) and Concordance Index (C-Index). Lower IBS and higher C-Index indicate better performance.

Figure B.1 : Exact SurvSHAP-IQ decomposition attribution curves for a selected observation

[-1.2650,2.4162,-0.6436]

computed on the ground-truth hazard functions of all ten simulation scenarios. The plots show feature- and interaction-specific hazard attributions (colored curves, Y-axis) over time (X-axis). The reference curve (differences in individual hazard and mean hazard) is overlaid in a dashed grey line.

Figure B.2 : Exact SurvSHAP-IQ decomposition curves for a selected observation

[-1.2650,2.4162,-0.6436]

computed on the ground-truth log-hazard functions of all ten simulation scenarios. The plots show feature- and interaction-specific log-hazard attributions (colored curves, Y-axis) over time (X-axis). The reference curve (differences in individual log-hazard and mean log-hazard) is overlaid in a dashed grey line.

Figure B.3 : Exact SurvSHAP-IQ decomposition curves for a selected observation

[-1.2650,2.4162,-0.6436]

computed on the approximated ground-truth survival functions of all ten simulation scenarios. The plots show feature- and interaction-specific survival attributions (colored curves, Y-axis) over time (X-axis). The reference curve (differences in individual survival and mean survival) is overlaid in a dashed grey line.

Figure B.4 : Exact SurvSHAP-IQ decomposition curves for a selected observation

[-1.2650,2.4162,-0.6436]

computed on the predicted survival functions of fitted CoxPH for all ten simulation scenarios. The plots show feature- and interaction-specific survival attributions (colored curves, Y-axis) over time (X-axis). The reference curve (differences in individual survival and mean survival) is overlaid in a dashed grey line.

Figure B.5 : Exact SurvSHAP-IQ decomposition curves for a selected observation

[-1.2650,2.4162,-0.6436]

computed on the predicted survival functions of fitted GBSA models for all ten simulation scenarios. The plots show feature- and interaction-specific survival attributions (colored curves, Y-axis) over time (X-axis). The reference curve (differences in individual survival and mean survival) is overlaid in a dashed grey line.

Scenario	$h(t\|\bm{x})$	$\log(h(t\|\bm{x}))$	$S(t\|\bm{x})$	$\hat{S}_{CoxPH}(t\|\bm{x})$	$\hat{S}_{GBSA}(t\|\bm{x})$
( 1 ) linear $G(t\|\bm{x})$ TI, no inter.	$<$ 0.00001	$<$ 0.00001	$<$ 0.00001	0.00315	0.00552
( 2 ) linear $G(t\|\bm{x})$ TD main, no inter.	$<$ 0.00001	$<$ 0.00001	0.00200	0.00416	0.00425
( 3 ) linear $G(t\|\bm{x})$ TI, inter.	$<$ 0.00001	$<$ 0.00001	$<$ 0.00001	0.00317	0.00180
( 4 ) linear $G(t\|\bm{x})$ TD main, inter.	$<$ 0.00001	$<$ 0.00001	0.00160	0.00408	0.00216
( 5 ) linear $G(t\|\bm{x})$ TD inter.	$<$ 0.00001	$<$ 0.00001	0.00215	0.00258	0.00349
( 6 ) general additive $G(t\|\bm{x})$ TI, no inter.	$<$ 0.00001	$<$ 0.00001	$<$ 0.00001	0.00157	0.01061

Appendix

Table B.3 : Average local accuracy over time comparing ground-truth hazard, log-hazard, and survival function with predicted survival from CoxPH and GBSA models, evaluated on ten simulated scenarios. Values are low, indicating high accuracy of the decomposition.

Figure B.6 : Local accuracy curves over time (X-axis) computed for the ground-truth (GT) hazard, log-hazard, survival and predicted survival functions of CoxPH and GBSA models (colored curves, Y-axis) for all observations from the 10 simulation scenarios. Local accuracy is lower for ground-truth functions than for model prediction functions.

Scenario	$\rho$	$x_{1}$	$x_{2}$	$x_{3}$	$x_{1}*x_{2}$	$x_{1}*x_{3}$	$x_{2}*x_{3}$
(1)	0.0	$-0.5167(0.0073)$	$-1.9000(0.0040)$	$0.3750(0.0053)$	$0.0015(0.0026)$	$0.0015(0.0026)$	$0.0015(0.0026)$
	0.2	$-0.5245(0.0079)$	$-1.9449(0.0024)$	$0.3619(0.0042)$	$0.0040(0.0028)$	$0.0040(0.0028)$	$0.0040(0.0028)$
	0.5	$-0.5061(0.0089)$	$-1.9581(0.0012)$	$0.3860(0.0029)$	$0.0069(0.0022)$	$0.0069(0.0022)$	$0.0069(0.0022)$
	0.9	$-0.4979(0.0137)$	$-1.9305(0.0003)$	$0.4067(0.0008)$	$-0.0065(0.0024)$	$-0.0065(0.0024)$	$-0.0065(0.0024)$
(2)	0.0	$-1.2849(0.3242)$	$-1.9084(0.0103)$	$0.3709(0.0107)$	$0.0042(0.0051)$	$0.0042(0.0051)$	$0.0042(0.0051)$
	0.2	$-1.2688(0.2982)$	$-1.9437(0.0090)$	$0.3580(0.0100)$	$0.0010(0.0047)$	$0.0010(0.0047)$	$0.0010(0.0047)$

Appendix

Table B.4: Time-wise mean and standard deviation of the exact SurvSHAP-IQ decomposition results for a selected observation

[-1.2650,2.4162,-0.6436]

computed on the ground-truth log-hazard functions for all ten simulation scenarios for four different correlation levels (

\rho=0,0.2,0.5,0.9

). Each cell reports the mean over time of the SurvSHAP-IQ attributions, with the standard deviation given in parentheses.

Figure B.7 : Exact SurvSHAP-IQ decomposition curves for a selected observation

[-1.2650,\,2.4162,\,-0.6436]

computed on the ground-truth log-hazard functions. Left: Interventional SurvSHAP-IQ. Right: Conditional SurvSHAP-IQ. Colored curves show feature- and interaction-specific log-hazard attributions, dashed grey shows the individual–mean log-hazard difference.

Panel of Figure 19 — Figure B.8 : Explanation of an RSF model predicting survival in patients treated for the HIV-1 infection. Top left: Shapley value attributions (order 1) for multiple observations (curves) with different feature values (in color). Bottom left: SurvSHAP-IQ individual effects (order 2). We observe that these are smoother compared to attribution curves, indicating distincter relationships with feature values. Right: SurvSHAP-IQ interaction effects (order 2). We observe a significant interaction between the karnof and cd4 features, which is ‘hidden’ in the Shapley value simplified explanation. The remaining interaction effects are shown in Figure B.9 .

Panel of Figure 20 — Figure B.9 : Extended Figure B.8 . The remaining interaction effects in an explanation of an RSF model predicting the survival in patients treated for the HIV-1 infection.

Panel of Figure 21 — Figure B.10 : Explanation of an RSF model predicting breast cancer survival. Top: SurvSHAP-IQ individual effects (order 2) for four most important gene expressions. Bottom: SurvSHAP-IQ interaction effects (order 2) for four most important gene interactions.

Panel of Figure 22 — Figure B.11 : Extended Figure 6 with a narrowed Y-axis scale for readability. The remaining attribution and interaction effects in an explanation of a GBSA model predicting the survival of patients diagnosed with uveal melanoma, a type of eye cancer.

Figure B.12 : Extended Figure 7 . Mean absolute error of different SurvSHAP-IQ approximators as a function of budget for the three survival tasks.

Feature	Levels	Type	Reference
Age	–	Continuous	–
Lymph Nodes	Lymph nodes examined	Count	–
Surgery	lumpectomy, simple mastectomy, other	Categorical	Radical mast.
Menopause	Indeterminate, pre-menopausal	Categorical	Post-menopausal
T-Stage	T1, T3, T4, TX	Categorical	T2
N-Stage	N1, N2, N3, NX	Categorical	N0

Appendix

Table B.5 : Description of the clinical features of the TCGA-BRCA dataset. Dummy-encoded variables use the listed reference category as baseline.

Feature Attribution

Feature attribution estimates which input features or internal components contribute to a model’s predictions.

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DynSHAP: Towards Explainable Dynamic Survival Analysis

Nastasya Anokhina, Jonas Jürß, Pietro Liò

University of Cambridge

Feature Attribution SHAP Feature Attribution Survival Analysis

Open paper AskPDF arXiv

Deep learning models for dynamic survival analysis (DSA) achieve strong predictive performance by incorporating longitudinal patient data, but their black box nature limits clinical trust and adoption. Existing explainability methods cannot handle longitudinal, irregular inputs and functional survival outputs simultaneously, which limits their usability in DSA. We propose DynSHAP, a SHAP framework suited specifically for dynamic survival analysis. It extends common marginal SHAP estimators to this setting by treating time--feature pairs as players in the Shapley game. We further introduce Temporal DynSHAP, which learns linear dependencies in features over time and uses conditional sampling to address them in explanations. When applied to synthetic data with known ground-truth attributions, Temporal DynSHAP recovers temporally dependent features more accurately than marginal estimators for a given state-of-the-art model. Applied to two real-world clinical datasets and two DSA architectures, DynSHAP produces attributions faithful to model learning, allowing medical experts to see which patient information drove the prediction and when.

#	$G(t\|\bm{x})$
( 1 )	$x_{1}\beta_{1}+x_{2}\beta_{2}+x_{3}\beta_{3}$ (linear $G(t\|\bm{x})$ , TI)
( 2 )	$x_{1}\beta_{1}\log(t+1)+x_{2}\beta_{2}+x_{3}\beta_{3}$ (linear $G(t\|\bm{x})$ , TD main)
( 3 )	$x_{1}\beta_{1}+x_{2}\beta_{2}+x_{3}\beta_{3}+x_{1}x_{3}\beta_{13}$ (linear $G(t\|\bm{x})$ , TI, interaction)
( 4 )	$x_{1}\beta_{1}\log(t+1)+x_{2}\beta_{2}+x_{3}\beta_{3}+x_{1}x_{3}\beta_{13}$ (linear $G(t\|\bm{x})$ , TD main, interaction)
( 5 )	$x_{1}\beta_{1}+x_{2}\beta_{2}+x_{3}\beta_{3}+x_{1}x_{3}\beta_{13}\log(t+1)$ (linear $G(t\|\bm{x})$ , TD interaction)
( 6 )	$\beta_{1}x_{1}^{2}+\beta_{2}\frac{2}{\pi}\arctan(0.7x_{2})+\beta_{3}x_{3}$ (generalized additive $G(t\|\bm{x})$ , TI)

Abstract

Feature Attribution

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