Survival Functional Decomposition (SurvFD)

• SurvFD is a methodological pattern that decomposes survival functions into interpretable components such as causal pathways, observable regressions, and geometric or stochastic factors.
• It enables rigorous analysis through diverse frameworks, including causal fairness, global survival stacking, and functional decomposition for model interpretability.
• The approach enhances insights into survival disparities, personalized predictions, and multi-event trajectories by aligning decomposition methods with specific assumptions and applications.

Searching arXiv for the cited SurvFD-related papers to ground the article in current preprints. Survival Functional Decomposition (SurvFD) denotes a family of decomposition strategies for survival or time-to-event objects rather than a single universally standardized method. Across current arXiv usage, the phrase is associated with causal pathway decompositions of survival disparities, decompositions of conditional survival estimation into observable regressions, time-indexed functional decompositions of survival-model outputs, multi-event decompositions of clinical functional decline, stochastic radial–angular factorizations of multivariate survival laws, and viability-based partitions of dynamical state space by survival outcome (Plecko, 12 May 2026, Wolock et al., 2022, Langbein et al., 18 Feb 2026, Lillelund et al., 2 Jun 2025, Mai et al., 2020, McShaffrey et al., 16 May 2026). This suggests that the common thread is not a single estimator or theorem, but the use of a survival functional as the object to be factorized into interpretable components.

1. Scope and principal formulations

The term is used for distinct decompositions of different mathematical objects. In the causal-fairness setting, the object is a disparity in interventional survival over time. In personalized survival estimation, it is the conditional survival curve $S(t\mid x)$ itself, represented through observable regressions. In model interpretability, it is a function-valued prediction such as hazard or survival, decomposed into time-dependent and time-independent pure effects. In ALS functional decline, the decomposition is across clinically meaningful event types. In $\ell_p$-symmetric survival theory, it is a stochastic factorization of a multivariate survival law. In viability theory, it is a state-space partition induced by time-to-death.

Paper	Object decomposed	Components
"Causal Fairness for Survival Analysis" (Plecko, 12 May 2026)	Survival disparity	Direct, indirect, spurious
"A framework for leveraging machine learning tools to estimate personalized survival curves" (Wolock et al., 2022)	$S(t\mid x)$, $\Lambda(t\mid x)$	Observable regressions $F_\delta, G_\delta, \pi$
"Functional Decomposition and Shapley Interactions for Interpreting Survival Models" (Langbein et al., 18 Feb 2026)	$h(t\mid x)$, $S(t\mid x)$, $\log h(t\mid x)$	Time-dependent and time-independent pure effects
"A meaningful prediction of functional decline in amyotrophic lateral sclerosis based on multi-event survival analysis" (Lillelund et al., 2 Jun 2025)	ALS trajectory	Five function-specific ISDs
"Stochastic decomposition for $\ell_p$-norm symmetric survival functions on the positive orthant" (Mai et al., 2020)	$\ell_p$-symmetric survival law	$\ell_p$0, $\ell_p$1, $\ell_p$2
"Viability Space Decomposition: A geometric partition of survival outcomes in single- and multi-agent systems" (McShaffrey et al., 16 May 2026)	Survival outcome over state space	Mortality, ordering, collapse manifolds

This heterogeneity matters methodologically. Some formulations are identification results, some are causal estimands, some are representation theorems, and some are geometric classifications. Treating them as interchangeable would collapse distinctions that are explicit in the literature.

2. Causal pathway decomposition of survival disparities

In "Causal Fairness for Survival Analysis," SurvFD is a causal, time-resolved decomposition of disparities in time-to-event outcomes under censoring (Plecko, 12 May 2026). The setup uses true event time $\ell_p$3, censoring time $\ell_p$4, observed follow-up $\ell_p$5, and event indicator $\ell_p$6. The protected attribute is $\ell_p$7, pre-exposure covariates are $\ell_p$8, and post-exposure mediators are $\ell_p$9. The observational conditional survival is

$S(t\mid x)$0

while interventional survival under $S(t\mid x)$1 is

$S(t\mid x)$2

The primary fairness target is the time-resolved disparity

$S(t\mid x)$3

SurvFD reduces the time-to-event problem to a deterministic functional

$S(t\mid x)$4

and then applies path-specific counterfactuals. On the difference scale, the disparity is decomposed as

$S(t\mid x)$5

with

$S(t\mid x)$6

$S(t\mid x)$7

$S(t\mid x)$8

The direct pathway is $S(t\mid x)$9, the indirect pathway is $\Lambda(t\mid x)$0, and the spurious pathway is $\Lambda(t\mid x)$1, interpreted as distributional shift in $\Lambda(t\mid x)$2 and selection or collider effects consistent with the assumed DAG.

Identification rests on a Standard Fairness Model with assumptions on censoring and confounding. Under non-informative censoring, $\Lambda(t\mid x)$3. Under competing risks, $\Lambda(t\mid x)$4. Under informative censoring, the paper drops $\Lambda(t\mid x)$5 and uses an Archimedean copula

$\Lambda(t\mid x)$6

with Kendall’s $\Lambda(t\mid x)$7 as sensitivity parameter. The Causal Reduction Theorem states that if $\Lambda(t\mid x)$8 maps identifiable $\Lambda(t\mid x)$9 to $F_\delta, G_\delta, \pi$0, then $F_\delta, G_\delta, \pi$1 is a deterministic function of $F_\delta, G_\delta, \pi$2, so path-specific counterfactuals of $F_\delta, G_\delta, \pi$3 are identified under the same assumptions.

Two estimators are developed. The model-based plug-in estimator combines a flexible survival model with importance ratios, and the efficient influence-function estimator uses cross-fitting and is doubly robust: it is consistent if either the survival and censoring models are correct or the propensity models $F_\delta, G_\delta, \pi$4 and $F_\delta, G_\delta, \pi$5 are correct. The framework also supports RMST and competing-risk CIF targets.

The ICU application uses the ANZICS Adult Patient Database, Australia (Jan 2023–Jul 2024), with $F_\delta, G_\delta, \pi$6 ICU admissions and $F_\delta, G_\delta, \pi$7 Indigenous admissions. For death, the raw survival difference is positive for the majority during the first $F_\delta, G_\delta, \pi$8 weeks and then decreases; by 180 days, the point estimate flips sign but CIs include zero after $F_\delta, G_\delta, \pi$9 days. The direct component is negative and stabilizes after $h(t\mid x)$0 days, the indirect component is negative and grows in magnitude over time, and the spurious component is positive and increasing. For readmission under informative censoring, sensitivity analyses over $h(t\mid x)$1 suggest higher minority burden on direct and mediated paths, with a spurious component in the opposite direction. The central substantive point is that near-zero net disparity can mask large pathway-specific components.

3. Decomposing conditional survival estimation into observable regressions

In "A framework for leveraging machine learning tools to estimate personalized survival curves," SurvFD refers to a decomposition of the conditional survival function into observable regression functions that do not require explicit modeling of censoring or truncation mechanisms (Wolock et al., 2022). The target is

$h(t\mid x)$2

with observed data $h(t\mid x)$3, where $h(t\mid x)$4, $h(t\mid x)$5, and $h(t\mid x)$6 is delayed entry. Under Assumption A, $h(t\mid x)$7, and positivity $h(t\mid x)$8, the paper defines the observable regressions

$h(t\mid x)$9

$S(t\mid x)$0

$S(t\mid x)$1

These identify the cumulative hazard through

$S(t\mid x)$2

The survival function is then obtained by the product integral

$S(t\mid x)$3

or by the exponential mapping in continuous time.

This decomposition motivates "global survival stacking." The procedure fits binary regressions for $S(t\mid x)$4, for $S(t\mid x)$5 and $S(t\mid x)$6 via pooled regressions of $S(t\mid x)$7 within $S(t\mid x)$8-strata, and for $S(t\mid x)$9 and $\log h(t\mid x)$0 via pooled regressions of $\log h(t\mid x)$1 among those at risk with $\log h(t\mid x)$2. After isotonization, the estimated hazard increments are combined on a grid and mapped either by a product integral approximation or by

$\log h(t\mid x)$3

The framework is fully general with respect to event-time distributions, since product integrals allow continuous, discrete, or mixed laws. It also extends to a residual censoring assumption and to retrospective sampling through time reversal. Empirically, global survival stacking showed strong performance across prospective right-censored, left-truncated and right-censored, retrospective right-truncated, proportional hazards, and discrete-time scenarios. It was often competitive with or better than local survival stacking, survival Super Learner, LTRC forests, and Cox or GAM Cox in non-PH settings, while correctly specified Cox was moderately better in the PH scenario. On FLCHAIN, GBSG, METABRIC, NWTCO, and SUPPORT, global stacking was consistently competitive and typically better than naïve regression ignoring censoring. The STEP HIV vaccine trial application further demonstrated stratified personalized survival estimation at 1- and 2-year landmarks.

4. Functional decomposition and interaction structure in survival-model explanations

In "Functional Decomposition and Shapley Interactions for Interpreting Survival Models," SurvFD is a formal decomposition of time-indexed survival prediction functions into pure effects that may be time-dependent or time-independent (Langbein et al., 18 Feb 2026). The starting point is the multiplicative hazard model

$\log h(t\mid x)$4

with survival

$\log h(t\mid x)$5

The paper’s motivation is that additive explanations are natural on the log-hazard scale,

$\log h(t\mid x)$6

but the transformations from $\log h(t\mid x)$7 to $\log h(t\mid x)$8 and then to $\log h(t\mid x)$9 are non-linear and multiplicative, so higher-order and time-dependent interactions can emerge even when $\ell_p$0 is additive.

SurvFD generalizes functional decomposition to a function $\ell_p$1 and separates pure effects into time-dependent and time-independent parts. Under mutual independence of features, Theorem 3.2 shows that on the log-hazard scale SurvFD recovers ground-truth effects when $\ell_p$2 is either linear in $\ell_p$3 including interactions or an additive main-effect model. Theorem 3.3 shows downward propagation but no upward propagation of time-dependence on the log-hazard scale: subsets of a time-dependent interaction may appear time-dependent, but supersets do not become time-dependent. Corollary 3.4 shows that on hazard and survival scales both downward and upward propagation can occur. Proposition 3.5 shows that even a linear CoxPH model with $\ell_p$4 exhibits interaction effects in the SurvFD of hazard and survival.

These structural results are coupled to SurvSHAP-IQ, which extends Shapley interactions to time-indexed outputs. With value function

$\ell_p$5

the time-dependent Shapley value is

$\ell_p$6

and higher-order terms are defined through discrete derivatives and Shapley interaction indices. Exact computation is exponential in $\ell_p$7, so the paper evaluates Monte Carlo, permutation-based, SVARM stratified sampling, and regression-based approximators.

The empirical analysis uses ten simulated scenarios and applications to an RSF on ACTG HIV-1, a GBSA model on uveal melanoma, and a multimodal DeepHit model on TCGA-BRCA. Simulation results confirm the theoretical claims: exact attribution curves recover ground-truth behavior on the log-hazard scale, while hazard and survival explanations display induced interactions. On model predictions, GBSA often outperformed CoxPH in nonlinear settings, and local accuracy remained close to exact decomposition quality. The paper’s broader claim is narrow but consequential: additive explanation methods can fail on hazard and survival scales for reasons intrinsic to the survival functional itself.

5. Multi-event, stochastic, and geometric realizations

The ALS study "A meaningful prediction of functional decline in amyotrophic lateral sclerosis based on multi-event survival analysis" presents its method as instantiating a SurvFD framework in which an overall functional trajectory is decomposed into five function-specific individual survival distributions: Speaking, Swallowing, Handwriting, Walking, and Breathing (Dyspnea) (Lillelund et al., 2 Jun 2025). Using the PRO-ACT database, restricted to $\ell_p$8 ALS patients, the study predicts each event over a fixed horizon of 500 days after baseline. For each function $\ell_p$9, it learns $\ell_p$0, $\ell_p$1, and $\ell_p$2 with CoxPH, RSF, DeepSurv, MTLR, or MENSA. Under conditional independence, an overall intact-function survival can be constructed as

$\ell_p$3

Evaluation uses Harrell’s C-index, Uno’s C-index, IPCW Brier score and IBS, mMAE, and D-calibration. Across all five functions, covariate-based models achieved lower mMAE than the population-level Kaplan–Meier estimator. RSF achieved perfect D-calibration (10/10) across all events and seeds; speech and swallowing showed the highest C-indices; and counterfactual analyses indicated considerably shorter predicted times to speech and swallowing decline for bulbar-onset ALS than for limb-onset ALS. The paper also reports little to no impact of Riluzole on predicted functional decline within 500 days.

In the $\ell_p$4-norm symmetric setting, "Stochastic decomposition for $\ell_p$5-norm symmetric survival functions on the positive orthant" gives a stochastic representation of any law with survival function $\ell_p$6 (Mai et al., 2020). The decomposition is

$\ell_p$7

where $\ell_p$8 is uniform on the standard simplex $\ell_p$9, $\ell_p$00 is a universal within-unit-ball factor depending only on $\ell_p$01, and $\ell_p$02 is the outer radial variable determined by the Williamson–$\ell_p$03 transform

$\ell_p$04

The law of $\ell_p$05 is a finite mixture of Beta distributions plus an atom at $\ell_p$06, with weights obtained recursively. The representation yields an exact simulation algorithm for arbitrary $\ell_p$07-symmetric survival laws and, through a Poisson point construction, for max-infinitely divisible laws with $\ell_p$08-symmetric exponent measures. Within this family, the minimal-association law is obtained by setting $\ell_p$09, with minimal Kendall’s tau

$\ell_p$10

"Viability Space Decomposition: A geometric partition of survival outcomes in single- and multi-agent systems" does not use SurvFD as the authors’ term, but it directly operationalizes a survival-functional decomposition through viability theory (McShaffrey et al., 16 May 2026). The survival functional is the exit time

$\ell_p$11

with $\ell_p$12 for asymptotic viability and $\ell_p$13 outside the viability set. The induced partition distinguishes asymptotically viable and transiently viable sets and is organized by mortality manifolds, ordering manifolds, and, in multi-agent systems, collapse manifolds. Mortality manifolds are traced backward from tangency points on the viability boundary and separate $\ell_p$14 from $\ell_p$15. Ordering manifolds are traced backward from points where multiple guard conditions are simultaneously satisfied and separate neighboring transiently viable sets with different causes of death. Collapse manifolds pull back lower-dimensional post-collapse organizing structures to the higher-dimensional domain. In the single-cell, behaving-cell, and two-cell examples, these manifolds complete a viability portrait that classical attractors and separatrices do not supply when death occurs in the transient.

6. Assumptions, limitations, and unifying interpretation

Each SurvFD formulation is tightly coupled to its assumptions. The causal-fairness framework requires consistency, SUTVA, positivity, lack of unmeasured confounding for the relevant paths encoded by the Standard Fairness Model, and either non-informative censoring or a copula-based sensitivity analysis for informative censoring (Plecko, 12 May 2026). The observable-regression framework for personalized survival curves requires $\ell_p$16 or the alternative residual censoring assumptions, together with overlap conditions for $\ell_p$17 (Wolock et al., 2022). The interpretability framework assumes square-integrability and, for its clearest recovery results, mutual independence of features; its exact interaction estimators are exponential in $\ell_p$18, and marginal versus conditional reference distributions lead to different semantics (Langbein et al., 18 Feb 2026). The ALS instantiation assumes non-informative censoring conditional on baseline covariates, no time-varying covariates within the core analysis, and either conditional independence across functions or a shared representation that still optimizes a product of per-event likelihoods (Lillelund et al., 2 Jun 2025). The $\ell_p$19-symmetric stochastic factorization requires $\ell_p$20 symmetry and a $\ell_p$21-monotone generator $\ell_p$22, while sampling $\ell_p$23 can require model-specific inversion (Mai et al., 2020). VSD assumes smooth dynamics inside a piecewise smooth viability set and faces the combinatorial growth of hybrid domains, which scale as $\ell_p$24 in the multi-agent construction (McShaffrey et al., 16 May 2026).

A common misconception would be to treat SurvFD as a single named algorithm. The literature does not support that reading. Rather, the phrase indexes a class of decompositions whose shared feature is the selection of a survival functional—interventional disparity, conditional survival curve, hazard or survival prediction, multi-event trajectory, multivariate survival law, or exit-time functional—and the partition of that object into components with specific causal, statistical, functional, stochastic, or geometric meanings. This suggests that SurvFD is best understood as a methodological pattern: survival analysis becomes more interpretable when the target functional is decomposed into parts aligned with mechanisms, observables, features, events, radial–angular factors, or viability outcomes.