Understanding whether effects vary across subgroups, patient characteristics, or co-exposures is critical for clinical and public health decisions. Such interactions are important in survival analysis, e.g., between genetic and environmental factors (Minelli et al., 2011), obesity and treatment effects (Hanai et al., 2014;Jensen et al., 2008), or age and tumor markers (Julkunen & Rousu, 2025;Nielsen & Grønbaek, 2008;Stehlik et al., 2010). Survival machine learning models can automatically capture complex, non-linear, and timevarying interactions without prior specification (Barnwal et al., 2022;Ishwaran et al., 2008;Wiegrebe et al., 2024), but 1 Leibniz Institute for Prevention Research and Epidemiology -BIPS 2 Faculty of Mathematics and Computer Science, University of Bremen 3 University of Warsaw 4 Centre for Credible AI, Warsaw University of Technology 5 LMU Munich, MCML 6 Bielefeld University. Correspondence to: Marvin N. Wright
. their opacity limits interpretability and clinical utility (Baniecki et al., 2025b;Langbein et al., 2025).A principled way to formalize feature interactions is functional decomposition (FD), which additively separates prediction functions into main and interaction effects (Hooker, 2004;2007;Owen, 2013;Stone, 1994). In survival analysis, however, FD is more challenging because predictions are intrinsically time-dependent. A key limitation is that standard FD does not distinguish between time-independent and time-dependent effects. Moreover, although additive decomposition is natural on the log-hazard scale, transformations to interpretable scales, such as hazard or survival functions, induce additional time-dependent effects and interactions not present on the log-hazard scale. This motivates a principled FD approach for understanding time-dependence and interactions in survival models and the need to quantify interactions across scales and time.
Recently, Shapley-based interaction indices have been used to estimate interactions based on FD (Bordt & von Luxburg, 2023;Fumagalli et al., 2023;Grabisch & Roubens, 1999;Sundararajan et al., 2020;Tsai et al., 2023), but are largely restricted to scalar outcomes, leaving survival-specific decompositions and their estimation underdeveloped. This prevents a systematic and theoretically grounded quantification of feature interactions in time-to-event modeling. We now review existing approaches and their limitations.
Related work. In statistical survival models, interactions are assessed via (1) subgroup analyses (effect modification, VanderWeele, 2009) or (2) explicit interaction (product) terms. Their interpretation is model-specific: interactions represent departures from multiplicativity in CoxPH models and from additivity in additive hazard models (Aalen, 1980;Rod et al., 2012). In contrast, interaction analysis in machine learning survival models is limited. Existing local, modelagnostic explanation methods-such as SurvLIME (Kovalev et al., 2020), SurvSHAP(t) (Krzyziński et al., 2023), JointLIME (Chen et al., 2024b), and GradSHAP(t) (Langbein et al., 2025)-do not capture interactions, leaving their detection and quantification largely unaddressed.
So far, FD has not been widely adopted in survival analysis. Huang et al. (2000) use functional ANOVA decomposition (Hooker, 2004) on the log-hazard, yielding flexible nonlinear, time-dependent effects via splines. Mercadier & Ressel (2021) extend the Hoeffding-Sobol decomposition to homogeneous co-survival functions, decomposing joint survival for multiple events into main and interaction effects. Yet, none of these approaches provide a principled decomposition of survival models across time and prediction scales.
Contributions. Our work advances the literature in three ways: (1) SurvFD. We formalize functional decomposition for survival models (SurvFD), recovering additive, anyorder interaction effects, split into time-dependent and timeindependent components. We characterize SurvFD across log-hazard (Thms. 3.2 & 3.3), hazard, and survival functions (Prop. 3.5,Cor. 3.4) and its behavior under feature dependencies (Thm. 3.6). (2) SurvSHAP-IQ. Building on SurvFD, we extend Shapley interaction quantification to survival models. SurvSHAP-IQ provides estimates of higher-order interactions that can be visualized to interpret time-dependent survival predictions. (3) Empirical validation. We show that SurvSHAP-IQ accurately recovers interaction effects across several simulated prediction functions while satisfying local accuracy. We further demonstrate its utility for interpreting cancer survival models on multiple real-world datasets including multi-modal data.
This section gives the necessary background on survival analysis, functional decomposition, and Shapley-based interpretation of survival models, as foundations for SurvFD theory and its practical implementation in SurvSHAP-IQ.
General notation. Let D = {(x (i) , y (i) , δ (i) ) : i = 1, . . . , n} denote a s