SurvSHAP-IQ: Interaction-Aware Survival Explanations

• SurvSHAP-IQ is an interaction-aware framework that quantifies time-dependent feature interactions in survival models for hazard and survival outputs.
• It integrates Survival Functional Decomposition to isolate time-dependent versus time-independent effects, addressing non-additivity induced by exponential transforms.
• The framework provides precise visualizations and local accuracy metrics, ensuring faithful interpretation even under complex interaction structures.

Searching arXiv for the primary paper and closely related methods mentioned in the provided data. SurvSHAP-IQ is an interaction-aware explanation framework for survival models that extends Shapley interactions to time-indexed prediction targets such as the log-hazard, hazard, survival function, and the discrete-time probability mass function. It is introduced together with Survival Functional Decomposition (SurvFD) in “Functional Decomposition and Shapley Interactions for Interpreting Survival Models” (Langbein et al., 18 Feb 2026). The central motivation is that hazard and survival functions are natural explanatory targets in time-to-event prediction, yet their non-additivity over features causes standard additive explanation methods to fail when they are applied directly to time-indexed survival outputs. SurvSHAP-IQ addresses this by quantifying higher-order interactions as functions of time and by linking these interaction attributions to a functional decomposition of survival prediction functions (Langbein et al., 18 Feb 2026).

1. Survival-model setting and the source of non-additivity

The framework is defined for survival data of the form

$$D = \{(x^{(i)}, y^{(i)}, \delta^{(i)}) : i = 1, \dots, n\},$$

where $x^{(i)} \in \mathbb{R}^p$ is the feature vector for instance $i$, $y^{(i)} = \min(T^{(i)}, C^{(i)})$ is the observed time, $T^{(i)}$ is the event time, $C^{(i)}$ is the censoring time, and $\delta^{(i)} \in \{0,1\}$ indicates whether the event was observed. The formulation considers a single event and excludes competing risks (Langbein et al., 18 Feb 2026).

Two standard survival targets are central. The hazard function is

$$h(t \mid x) = \lim_{\Delta t \to 0} \frac{P(t < T < t + \Delta t \mid T \ge t, x)}{\Delta t},$$

and a multiplicative hazard model has the form

$h(t \mid x) = h_0(t)\cdot \exp(G(t \mid x)),$

with baseline hazard $h_0(t)$ and risk score $x^{(i)} \in \mathbb{R}^p$0. The survival function is

$x^{(i)} \in \mathbb{R}^p$1

These definitions make clear that even when the risk score is additive on the log scale, the exponential and integral transforms induce multiplicative and non-linear combinations of features on the hazard and survival scales (Langbein et al., 18 Feb 2026).

This is the technical basis for the claim that standard additive explanation methods are fundamentally limited in this setting. Even if $x^{(i)} \in \mathbb{R}^p$2 is additive in features, hazard and survival explanations inherit transformation-induced interactions. A direct implication is that a decomposition that is adequate for scalar outputs or for additive latent scores need not remain faithful on the clinically interpretable prediction scales $x^{(i)} \in \mathbb{R}^p$3 and $x^{(i)} \in \mathbb{R}^p$4 (Langbein et al., 18 Feb 2026). This suggests that survival explainability requires simultaneous treatment of time dependence and interaction structure rather than a straightforward transplantation of standard SHAP-style decompositions.

2. Survival Functional Decomposition

SurvFD provides the theoretical backbone for SurvSHAP-IQ. The starting point is a generalized additive representation of the risk score, where the power set $x^{(i)} \in \mathbb{R}^p$5 is partitioned into time-dependent subsets $x^{(i)} \in \mathbb{R}^p$6 and time-independent subsets $x^{(i)} \in \mathbb{R}^p$7 such that $x^{(i)} \in \mathbb{R}^p$8 and $x^{(i)} \in \mathbb{R}^p$9. The model is written as

$i$0

and, with $i$1,

$i$2

SurvFD then decomposes a square-integrable prediction function $i$3 as

$i$4

where the pure effects are defined through the inclusion–exclusion principle:

$i$5

This is a functional ANOVA / Hoeffding decomposition for time-indexed prediction functions (Langbein et al., 18 Feb 2026).

A key aspect of SurvFD is that the reference distribution determines the interpretation. Marginal FD integrates over $i$6 and is described as “true to the model,” whereas conditional FD integrates over $i$7 and is described as “true to the data” (Langbein et al., 18 Feb 2026). This distinction becomes important when features are dependent.

The paper’s theoretical results characterize when additive explanations fail. Under independence, Theorem 3.2 states that if $i$8 is linear in $i$9, including interactions, or additive in main effects, then SurvFD on $y^{(i)} = \min(T^{(i)}, C^{(i)})$0 recovers the ground-truth time-dependent and time-independent partitions (Langbein et al., 18 Feb 2026). However, Theorem 3.3 shows that with a single time-dependent interaction set $y^{(i)} = \min(T^{(i)}, C^{(i)})$1, downward propagation can occur: lower-order subsets $y^{(i)} = \min(T^{(i)}, C^{(i)})$2 may appear time-dependent in SurvFD even if they are time-independent in the ground truth. Corollary 3.4 further states that on the hazard and survival scales, both downward and upward propagation can occur, so subsets and supersets of $y^{(i)} = \min(T^{(i)}, C^{(i)})$3 may appear time-dependent despite belonging to $y^{(i)} = \min(T^{(i)}, C^{(i)})$4 in the ground truth. Proposition 3.5 sharpens the point by showing that even a linear Cox proportional hazards model,

$y^{(i)} = \min(T^{(i)}, C^{(i)})$5

exhibits interaction effects in SurvFD on the hazard and survival scales because of the exponential and integral transforms (Langbein et al., 18 Feb 2026).

For dependent features, Theorem 3.6 states that if a feature $y^{(i)} = \min(T^{(i)}, C^{(i)})$6 has no direct effect on $y^{(i)} = \min(T^{(i)}, C^{(i)})$7 but is dependent on a time-dependent feature $y^{(i)} = \min(T^{(i)}, C^{(i)})$8, then the marginal SurvFD component of $y^{(i)} = \min(T^{(i)}, C^{(i)})$9 for $T^{(i)}$0 is zero, while the conditional component is non-zero and can be time-dependent (Langbein et al., 18 Feb 2026). A plausible implication is that conditional explanations can legitimately report induced time-dependent effects for variables that are not causally or structurally active in the fitted risk score.

3. Definition of SurvSHAP-IQ

SurvSHAP-IQ extends SHAP-style interaction quantification from scalar outputs to time-indexed survival outputs. Its starting point is a time-dependent value function on coalitions $T^{(i)}$1. For survival, the paper gives

$T^{(i)}$2

The time-dependent SHAP value for feature $T^{(i)}$3 is then

$T^{(i)}$4

with

$T^{(i)}$5

This is the time-dependent single-feature setting previously associated with SurvSHAP(t), which quantifies time-varying marginal effects but not interactions (Langbein et al., 18 Feb 2026).

For pairwise interactions, SurvSHAP-IQ generalizes the Grabisch–Roubens Shapley interaction index to time-indexed functions:

$T^{(i)}$6

Time aggregation can be performed through

$T^{(i)}$7

or by a discrete average over a selected time grid (Langbein et al., 18 Feb 2026).

The framework then extends to any-order interactions through n-Shapley values, following the construction in Bordt and von Luxburg. For any subset $T^{(i)}$8, the discrete derivative is

$T^{(i)}$9

SurvSHAP-IQ adopts n-Shapley values so that the top-order coefficients coincide exactly with the Shapley interaction index, and the case $C^{(i)}$0 reduces to standard SHAP (Langbein et al., 18 Feb 2026).

Definition 3.7 gives the order-$C^{(i)}$1 additive decomposition

$C^{(i)}$2

where $C^{(i)}$3 terms are individual effects and $C^{(i)}$4 terms are interaction effects up to order $C^{(i)}$5 (Langbein et al., 18 Feb 2026). The relationship to SurvFD is explicit: with

$C^{(i)}$6

the Möbius transform of $C^{(i)}$7 aligns with SurvFD pure effects, and computing n-Shapley values up to full order recovers these pure effects exactly. At truncated order $C^{(i)}$8, higher-order Möbius components are redistributed into lower orders, yielding an axiomatic interaction estimator (Langbein et al., 18 Feb 2026).

The suffix “IQ” denotes “Interaction Quantification,” consistent with SHAP-IQ terminology (Langbein et al., 18 Feb 2026).

4. Estimation procedure and computational properties

SurvSHAP-IQ supports prediction targets

$C^{(i)}$9

Coalition values are computed for each coalition $\delta^{(i)} \in \{0,1\}$0 and time point $\delta^{(i)} \in \{0,1\}$1 as

$\delta^{(i)} \in \{0,1\}$2

where the expectation is taken either over the interventional reference distribution $\delta^{(i)} \in \{0,1\}$3 or the observational reference distribution $\delta^{(i)} \in \{0,1\}$4 (Langbein et al., 18 Feb 2026).

The explanation procedure operates on model predictions rather than on raw event data. Any handling of censoring, risk sets, or partial likelihood is delegated to model training. The explanation stage itself does not require censoring adjustments because it decomposes predictions produced by the trained model (Langbein et al., 18 Feb 2026).

For continuous-time settings, the method evaluates explanations on a time grid $\delta^{(i)} \in \{0,1\}$5, which may be regular or event-quantile based, and may approximate survival numerically from the hazard if necessary. For discrete-time models such as DeepHit, the method works directly with the model-provided discrete time points and PMF or survival outputs. Optional smoothing such as Savitzky–Golay may be used for visualization only; the explanations are still computed per time point (Langbein et al., 18 Feb 2026).

The exact method is exponential in the number of features, which the paper states is feasible for $\delta^{(i)} \in \{0,1\}$6 and modest time grids. Several scalable approximators are listed as extended to survival settings: Monte Carlo sampling of coalitions, permutation-based estimators, stratified SVARM-IQ, and regression-based KernelSHAP-IQ. Among these, KernelSHAP-IQ is reported as the most faithful in experiments, with the caveat that it requires a sampling budget exceeding the number of interaction effects (Langbein et al., 18 Feb 2026).

The paper also defines a time-dependent local accuracy metric. With the order-$\delta^{(i)} \in \{0,1\}$7 approximation

$\delta^{(i)} \in \{0,1\}$8

the normalized time-local error is

$\delta^{(i)} \in \{0,1\}$9

According to the reported results, exact decompositions on the log-hazard and hazard scales yield near-zero $$h(t \mid x) = \lim_{\Delta t \to 0} \frac{P(t < T < t + \Delta t \mid T \ge t, x)}{\Delta t},$$0, whereas survival approximations via hazard yield small but non-zero error (Langbein et al., 18 Feb 2026).

For the interventional regression-based estimator, the pseudocode proceeds timepoint by timepoint. It samples a budget $$h(t \mid x) = \lim_{\Delta t \to 0} \frac{P(t < T < t + \Delta t \mid T \ge t, x)}{\Delta t},$$1 of coalitions, computes the coalition value for each sampled set using a reference dataset, constructs a SHAP-IQ linear-model basis row with indicators for all subsets $$h(t \mid x) = \lim_{\Delta t \to 0} \frac{P(t < T < t + \Delta t \mid T \ge t, x)}{\Delta t},$$2 satisfying $$h(t \mid x) = \lim_{\Delta t \to 0} \frac{P(t < T < t + \Delta t \mid T \ge t, x)}{\Delta t},$$3, assigns kernel weights, and solves a weighted least-squares problem. The resulting coefficients are the interaction curves $$h(t \mid x) = \lim_{\Delta t \to 0} \frac{P(t < T < t + \Delta t \mid T \ge t, x)}{\Delta t},$$4 (Langbein et al., 18 Feb 2026).

The complexity formulas are stated explicitly. Exact SurvSHAP-IQ has time complexity

$$h(t \mid x) = \lim_{\Delta t \to 0} \frac{P(t < T < t + \Delta t \mid T \ge t, x)}{\Delta t},$$5

and memory complexity

$$h(t \mid x) = \lim_{\Delta t \to 0} \frac{P(t < T < t + \Delta t \mid T \ge t, x)}{\Delta t},$$6

The regression-based approximation has time complexity

$$h(t \mid x) = \lim_{\Delta t \to 0} \frac{P(t < T < t + \Delta t \mid T \ge t, x)}{\Delta t},$$7

and memory complexity

$$h(t \mid x) = \lim_{\Delta t \to 0} \frac{P(t < T < t + \Delta t \mid T \ge t, x)}{\Delta t},$$8

These expressions make clear that interaction order $$h(t \mid x) = \lim_{\Delta t \to 0} \frac{P(t < T < t + \Delta t \mid T \ge t, x)}{\Delta t},$$9, sampling budget $h(t \mid x) = h_0(t)\cdot \exp(G(t \mid x)),$0, and grid size $h(t \mid x) = h_0(t)\cdot \exp(G(t \mid x)),$1 jointly determine feasibility (Langbein et al., 18 Feb 2026).

5. Interpretation, visualization, and empirical behavior

The recommended visual outputs are time series of $h(t \mid x) = h_0(t)\cdot \exp(G(t \mid x)),$2 and $h(t \mid x) = h_0(t)\cdot \exp(G(t \mid x)),$3, heatmaps over feature-by-time, aggregated scores such as

$h(t \mid x) = h_0(t)\cdot \exp(G(t \mid x)),$4

or

$h(t \mid x) = h_0(t)\cdot \exp(G(t \mid x)),$5

and explicit separation of constant lines from time-varying curves to distinguish time-independent from time-dependent components (Langbein et al., 18 Feb 2026).

The reported synthetic examples illustrate the interaction phenomena predicted by the theory. A time-dependent main effect such as $h(t \mid x) = h_0(t)\cdot \exp(G(t \mid x)),$6 produces a time-varying first-order curve on the log-hazard scale, whereas interactions can remain constant on the log-hazard scale but become time-varying on hazard or survival scales. A time-dependent interaction such as $h(t \mid x) = h_0(t)\cdot \exp(G(t \mid x)),$7 can leak into lower-order effects on the log-hazard scale through downward propagation, and into supersets on hazard and survival scales through upward propagation (Langbein et al., 18 Feb 2026).

The real-data examples cover several model families and domains. On the HIV ACTG dataset with a random survival forest and features karnof, cd4, priorzdv, and age, a strong pairwise interaction between karnof and cd4 is reported, and second-order curves reduce variance relative to first-order curves while revealing structure not captured by single-feature SHAP (Langbein et al., 18 Feb 2026). On a uveal melanoma dataset using GBSA, the paper reports that age, max tumor diameter, and mitotic rate dominate first-order effects, while pairwise interactions reveal redundancy and negative synergy, including diameter $h(t \mid x) = h_0(t)\cdot \exp(G(t \mid x)),$8 mitotic rate; the validation C-index is given as 0.758 (Langbein et al., 18 Feb 2026). On TCGA-BRCA with a multimodal DeepHit model using 6 image patches and 8 clinical features, the reported findings include within-modality interactions among top-attention patches, a positive interaction between N0 and lumpectomy for survival, and sign flips in PMF interactions consistent with the PMF–survival relationship (Langbein et al., 18 Feb 2026).

These examples support the broader methodological point that interactions are prediction-scale dependent. A plausible implication is that reporting only one scale, especially only survival or only hazard, can conceal whether the observed interaction is structurally present in the latent risk score or induced by the transformation from risk score to clinically interpretable output.

6. Evaluation results, assumptions, and limitations

The paper reports simulations across 10 scenarios with $h(t \mid x) = h_0(t)\cdot \exp(G(t \mid x)),$9, using independent features unless stated otherwise. The simulated risk scores include linear and generalized additive forms with time-dependent main and interaction terms. CoxPH models with main effects are compared against Gradient Boosting Survival Analysis. Table B.2 is summarized as showing that GBSA generally attains higher C-index and lower IBS than CoxPH except in purely linear/no-interaction scenarios; one example given is scenario (8), where CoxPH has C-index 0.655 and IBS 0.162, while GBSA has C-index 0.697 and IBS 0.149 (Langbein et al., 18 Feb 2026).

The local-accuracy results are also explicit. For ground-truth decompositions, the reported error is less than $h_0(t)$0 for log-hazard and hazard and at most 0.002 for survival. For predicted survival, the reported error is at most 0.015 across models and scenarios, which the paper interprets as near-perfect decomposition quality (Langbein et al., 18 Feb 2026).

For dependent features, the reported experiments align with Theorem 3.6: marginal and conditional SurvFD differ, and conditioning on correlated features induces time-dependent effects and interactions for non-influential variables under conditional FD, whereas interventional FD remains “true to model” (Langbein et al., 18 Feb 2026).

The real-world datasets and summary outcomes reported in the paper are as follows:

Dataset / model	Reported setting	Reported finding
actg / RSF	$h_0(t)$1, 4 features, OOB C-index 0.723	SurvSHAP-IQ recovers karnof $h_0(t)$2 cd4
Uveal melanoma / GBSA	train $h_0(t)$3, val $h_0(t)$4, 100 trees, depth 4, validation C-index 0.758	nuanced second-order interactions among clinical variables
nki70 / RSF	$h_0(t)$5, 76 features	no pairwise interaction surpasses first-order effects
TCGA-BRCA / DeepHit	990 patients	notable interactions within image modality and among favorable clinical factors

An approximator benchmark on Bergamaschi, smarto, support2, and php104K8a is summarized as showing that regression-based KernelSHAP-IQ is the most faithful, while instability can occur when the sampling budget is smaller than the number of interaction effects (Langbein et al., 18 Feb 2026).

The method is developed under several assumptions. The main SurvFD theory is characterized under multiplicative hazards and, for Theorems 3.2 and 3.3, under feature independence. Dependent features require careful selection of reference distribution. The predictive model must supply reliable $h_0(t)$6 over time, and coalition values are computed through marginal or conditional expectations over a reference dataset (Langbein et al., 18 Feb 2026).

The limitations are also stated directly. Exact computation is exponential in $h_0(t)$7; approximators require thoughtful budget choices. Sparse time grids may miss transient interactions, while dense grids increase cost. Heavy censoring affects training and calibration of the predictive model, and explanations inherit those model biases. Interpretation itself requires caution: transformation-induced interactions on the hazard or survival scales need not correspond to the ground-truth structure on the log scale, and conditional explanations can attribute induced effects to non-influential variables because they incorporate correlations (Langbein et al., 18 Feb 2026).

7. Relation to prior survival explainers and practical use

Within the paper’s framing, SurvSHAP-IQ is positioned relative to prior survival explainers. SurvSHAP(t) provides time-dependent single-feature SHAP values but does not quantify interactions. SurvLIME, JointLIME, and GradSHAP(t) offer local explanations without explicit interaction quantification. SurvSHAP-IQ is described as the first Shapley interaction framework explicitly adapted to time-indexed survival outputs, providing any-order interaction curves over time together with axiomatic guarantees and local accuracy (Langbein et al., 18 Feb 2026). The theoretical bridge to functional ANOVA is supplied by SurvFD, while the bridge to cooperative game theory is supplied by n-Shapley values and the Möbius transform (Langbein et al., 18 Feb 2026).

The implementation ecosystem named in the paper includes shapiq for SHAP-IQ and SurvSHAP-IQ approximations, scikit-survival for RSF, GBSA, and CoxPH, and pycox for DeepHit and neural survival models (Langbein et al., 18 Feb 2026). Configuration choices include the time weighting function $h_0(t)$8, interaction order $h_0(t)$9 with default emphasis on pairwise interactions, the reference distribution, and the coalition sampling budget $x^{(i)} \in \mathbb{R}^p$00 with SHAP-inspired kernel weights for stability (Langbein et al., 18 Feb 2026).

The practical guidance in the paper can be summarized along four axes. First, the choice of prediction scale matters: log-hazard is recommended when alignment with additive ground-truth structure is important, while hazard or survival may be preferable for clinical interpretability despite transformation-induced interactions. Second, interventional explanations are appropriate when the aim is to stay “true to the model,” whereas conditional explanations are appropriate when the aim is to stay “true to the data” in correlated settings. Third, the time grid should match clinically relevant horizons and should be sufficiently dense where risk changes sharply. Fourth, pairwise interactions often provide the main interpretive gain, with higher-order interactions reserved for cases where both dimensionality and budget permit them (Langbein et al., 18 Feb 2026).

The paper also outlines several future directions: uncertainty quantification for interaction curves through confidence bands, counterfactual survival explanations and policy evaluation using interaction-aware functional targets, scalable higher-order estimators and visualization tools, and subgroup analyses combining SurvSHAP-IQ with stratified survival modeling (Langbein et al., 18 Feb 2026). These are presented as prospective extensions rather than established properties.

In sum, SurvSHAP-IQ formalizes time-dependent interaction attribution for survival prediction functions and embeds it in a broader decomposition theory that explains when additive explanations succeed and when they fail. Its main significance lies in showing that survival explainability is not merely a matter of attaching SHAP values to a time axis: the prediction scale, the reference distribution, and the interaction order each alter what is being explained and what kinds of effects can appear in the explanation (Langbein et al., 18 Feb 2026).