Privacy-Protected Causal Survival Analysis Under Distribution Shift
Authors: Yi Liu
Alexander W. Levis
Ke Zhu
Shu Yang
Peter B. Gilbert
Larry Han
Venue: ICLR 2026 Poster (2026)
Keywords: Time-to-event outcome
Conditional distribution shifts
Semiparametric efficiency theory
Federated learning
Primary area: causal reasoning

Abstract: Causal inference across multiple data sources can improve the generalizability and reproducibility of scientific findings. However, for time-to-event outcomes, data integration methods remain underdeveloped, especially when populations are heterogeneous and privacy constraints prevent direct data pooling. We propose a federated learning method for estimating target site-specific causal effects in multi-source survival settings. Our approach dynamically re-weights source contributions to correct for distributional shifts, while preserving privacy. Leveraging semiparametric efficiency theory under a site-specific exchangeability assumption}, data-adaptive weighting and flexible machine learning, the method achieves double robustness, and it improves efficiency {if at least one source site provides a consistent estimate. Through simulations and two real data applications: (i) multi-site randomized trials of monoclonal antibodies for HIV-1 prevention among cisgender men and transgender persons in the United States, Brazil, Peru, and Switzerland, as well as women in sub-Saharan Africa, and (ii) an analysis of sex disparities across biomarker groups for all-cause mortality using the ``flchain'' dataset, we demonstrate the validity, efficiency gains, and practical utility of the approach. Our findings highlight the promise of federated methods for efficient, privacy-preserving causal survival analysis under distribution shift.

Published as a conference paper at ICLR 2026
PRIVACY-PROTECTED CAUSAL SURVIVAL ANALYSIS UNDER DISTRIBUTION SHIFT
Yi Liu', Alexander W. Levis Ke Zhu', Shu Yang', Peter B. Gilbert', Larry Han
1 North Carolina State University
University of Pennsylvania
3 Fred Hutchinson Cancer Center
Northeastern University
ABSTRACT
Causal inference across multiple data sources can improve the generalizability and reproducibility of scientific findings. However, for time-to-event outcomes, data integration methods remain underdeveloped, especially when populations are heterogeneous and privacy constraints prevent direct data pooling. We propose federated learning method for estimating target site-specific causal effects in multi-source survival settings. Our approach dynamically re- re-weights source contri- butions to correct for distributional shifts, while preserving privacy. Leveraging semiparametric efficiency theory under a site-specific exchangeability assump- tion, data-adaptive weighting and flexible machine learning, the method achieves double robustness, and it improves efficiency if at least one source site provides a consistent estimate. Through simulations and two real data applications: (i) multi-site randomized trials of monoclonal antibodies for HIV-1 prevention among cisgender men and transgender persons in the United States, Brazil, Peru, and Switzerland, well as women in uub-Saharan Africa, and (ii) analysis of sex disparities across biomarker groups for all-cause mortality using the "flchain" dataset, we demonstrate the validity, efficiency gains, and practical utility of the approach. Our findings highlight the promise of federated methods for efficient, privacy-preserving causal survival analysis under distribution shift.
INTRODUCTION
Data fusion matters in applications with time-to-event outcomes, such as clinical studies of progres- sion or readmission. Integrating survival data across sources can improve efficiency, particularly for rare events, but naïve pooling can be invalid under shifts in covariates, outcomes, or censoring. In addition, time-stamped event histories are considered identifiable information under General Data Protection Regulation (GDPR) and Health Insurance Portability and Accountability Act (HIPAA) regulations, limiting cross- trossinstiitttinn data sharing. Federated learning provides a practical alternative by enabling collaboration through aggregate-level statisties rather than raw survival trajectories.
We consider multiple right-censored survival datasets each with two treatment groups, with restrictions on data sharing, and possible heterogeneity in covariates, outcomes, and censoring. Our goal is to estimate the survival function for a given target site while borrowing information from the additional source sites in a federated learning-based approach. We use adaptive weighting strategy that aims to asymptotically exclude biased sources while retaining efficiency gains when at least one source provides a consistent estimate (see Corollary 2.11
Related work. A growing literature studies data fusion for causal inference (Yang & Ding 2019
Li & Luedtkee 2023 Colnet al. 2024), including recent advances in federated data fusion where
full data sharing across sites is not permitted (Han et al.) 2025 2024| 2023) Xiong et al. 2023 Rémi
Li et al. 2023 Makhija et al., 20244] Almodóvar et al.) (024). Most of these works Tocus on continuous, ordinal, or binary outcomes and do not address time-to-event data. Archetti
al. 2025
al. 2023) have begun examining federated survival settings but they focus on data generation and
simulation frameworks rather than estimation and inference.
Existing extensions survival outcome data often rely on restrictive assumptions. For example, the Cox proportional hazards model imposes a log-linear hazard structure (Hernán 2010 Han 2023
"Corresooniing author: Larry Ham; l han@northeaatere

---

Published as a conference paper at ICLR 2026
Nagpal et al. 2023), or common conditional outcome distribution (CCOD) assumption across sites (Lee et al. 2022; Cao et al. 2024; Wen et al. 2025) may fail under heterogeneous data distribu- tions. Violations of these assumptions yield biased estimates and inference. Related meta-analysis approaches, which aggregate site-specific estimators using inverse-variance weighting, possibly after density-ratio correction, also implicitly require such conditional homogeneity assumptions across sites (DerSimonian & Laird, 1986, Mariin-artúnnz & Sánchez-Mect 2010). In addition,
privacy-preserving methods avoiding sharing raw data across sites for survival outcomes remain scarce (Jia et al. 202T). Recent work such as FedECA (Ogier du Terrall et al., (225) develops federated external control arms for single-arm trials, but this setting differs from the more general multi-source integration problem.
Regarding estimation, time-to-event data are typically analyzed within single-site studies using nonparametric methods such as the Kaplan-Meier estimator (Kaplan & Meier 1958). With covariate- rich data, semiparametric extensions such as the Cox model (Cox) 1972; & Liu 2005| [Bull & Spiegelhalter 1997), doubly robust estimators (Bai et al. 2013) standard. In addition, targeted maximum likelihood estimation (TMLE) can improve the famite-ample performance of doubly robust estimators (van der Laan & Rubin 2006; Díaz et al. 2019), and the collaborative TMLE (C-TMLE) further enhances double/muitiple robustness to model misspecification (Stitelman & van der Laan 20TO). More recently. WWeslling al. (2024) integrated double machine learning (Chernozhukov et al. 2018 to flexibly estimate nuisance functions in survival analysis (Wolock et al. [202]] Cui et al.
2023 van der Laan al.. (207).. However, these methods remain focused on single-study contexts and do not address how to combine survival data across multiple sources.
Contributions. Recognizing that pooling is often infeasible and that CCOD may not hold, we develop federated estimator with adaptive site weighting that accommodates both continuous- and discrete-time outcomes. Our approach leverages influence function theory to construct site- specific estimators based only on local summary statistics, combined through a constrained convex optimization that upweights informative sites and downweights or excludes biased ones. We establish consistency, asymptotic normality, and conditions under which method improves efficiency over onlget-only analysis. By integrating cross- fitttsniiigggg (Chernozhukov et al. 2018) and ensemble learning (Diaz al.] 2019 Díaz 22200] /Westling et al., 2024] van der Laan al. (007). estimator avoids restrictive assumptions while retaining fast convergence rates.
We validate the method through extensive Monte Carlo simulation studies and two real applications: (i) multi-site randomized trials of monoclonal antibodies for HIV-1 prevention among cisgender men and transgender persons in the United States, Brazil, Peru, and Switzerland, as well as women in sub-Saharan Africa, and (ii) an analysis of sex disparities in all-cause mortality using the fichain in dataset in the surviva val R package, stratified into biomarker-defines groups Together, these examples highlight the potential of federated methods to enable efficient, privacy-preservie causal inference for time-to-event outcomes in realistic multi-source settings.
METHODOLOGY
2.1 PROBLEM SETUP AND TARGET ESTIMAND
Observed data. Consider K studies, each of which may be randomized or observational. For each participant, we observe baseline covariates X, a binary treatment E (0, 1}, and right-censored outcomes. Let T(a) and ((a) denote the potential event and censoring times under treatment
E [0, 1}. observed event and censoring times are T = (T + (1 A C ACO1) + (1 A)
By the stable unit treatment value assumption (SUTVA) (Rosenbaum & Rubin 1983) the
With right censoring, however, we only observe Y = min C*) and A = I(T C
Denote copy of the independent and identically distributed (ii.d.) data by O. The observed data across all sites are then given by (O, = (x, Y,, A,, R,) i = I,..., (0. 1.... K }} denotes the site, with R = indicating the target site and R 1..... K - 1 the external sources.
n), where R e
Target estimand. Throughout, P denotes the population-level probability under the true data- generating process, and with a subscript "n", [(O) = n Σ f(O.) denotes the empirical average. Our goal is to estimate the treatment-specific survival function in the target population over
2

---

Published as a conference paper at ICLR 2026
finite horizon oc: (t, a) P(T
= a a {0, 1), E [0,
This function gives the probability that a target-site individual on treatment a (a for treated, 0 for control) survives beyond time
Conditional survival functions. For each site k, define the conditional survival function S* (t X) (T' > A a, X, R = k). To simultancously accommodate continuous- and discrete- time outcomes, we use the product integral representation (Gill & Johansen, 1990):
(t a. X) = π (1 (du | x));
(0.f
where (t) a, X) is the conditional cumulative hazard function. This notation unifies both discrete and continuous- time survival models, because in discrete time the product integral becomes the standard discrete product II. and in continuous time becomes exp(-A<( a, x..
We impose three standard assumptions for causal survival analysis:
Assumption 2.1 (Unconfoundedness). A T(a) X. R and A ((e) X, R.
Assumption 2.2 TTreatment-specfffc non-informative censoring). C(a) 1L T(,) A = a, X, R. Assumption 2.3 (Positivity). There exists such that P(R k) 1/n, and for almost all X, min
(a X), G (t a. X)) 2 1/m, min (t a. a. > 0.
A O.....
Here n. (a X) = P P a X, R k) is the site-specific propensity score for treatment = a, and (* a. X) = P t A= X, R k) the conditional survival function of censoring. Each treatment and censoring mechanism has non-vanishin vanishinn probability, and each site contributes non-negligible fraction of participants. These quantities are referred to nuisance functions, auxiliary components that are not of primary scientific interest but are essential for estimating the target parameter
2.2 SINGLE-SITE ESTIMATION
Auxiliary process. For later use, define
T(Y & 1, 1)
MAY
A x)
(O; G*) 5*(Y a (Y a
(I)
Sk a, (" a, x)
where A Y = min residual (part of an augmentation term) in doubly robust estimators for right-censored data.
(t, Y). This functional plays role as an inverse prohability-weighted mean-zero
Efficient influence function (EIF). When using only target-site data (R = 0), the nonparametric EIF of (1,
(1, a) given & [0, 7] and E {0, 13 is given by (Westling et al. 2024):
I(R = 0) P. (O:P) P(R
K I(A=)) a) (a X)
H4, (O; 5 50, G9) 5 (t| - a)]
Here, P in (O; indicates that the EIF depends on nuisance functions under the true data distribution. In other words, 00,00 (O, pro (O; 5 G T0). Furthermore, EIF evaluated with estimated nuisance functions. This should not be confused with the empirical average P, introduced earlier. The same convention applies all other EIFs throughout the paper.
IP to denote the
The EIF pi (O; P) highlights two components: (i) an anchor term that 5t a, -x 0 a). which anchors estimation through the conditional survival function under an outcome model by using target data; and (ii) an augmentation term-the weighted part involving ( Go) and (a X), which adjusts for censoring and treatment assignment. Furthermore, the weighting term 0) selects target-site observations, and [[AA a)/x" X) restricts to units with treatment A = a while reweighting them to represent the full target population.
I(R - 0)/
Target-only estimator. Motivated by the EIF, we define (t, a) as the solution to the estimating equation
0 = P. P. (O;
Under regularity conditions, (t, a) is regular and asymptotically linear (RAL) and achieves the semiparametric efficiency bound uniformly over / [0, when only target-site data are available.
3

---

Published as a conference paper at ICLR 2026
2.3 THE CCOD ASSUMPTION
When multiple data sources are available, precision can be improved by data fusion. A common simplifying assumption is that conditional survival functions are identical across sites given covariates.
Assumption 2.4 (Common conditional outcome distribution).
T( 1L R X for E {0, 10,
Assumption 2.4 implies that (t a, X) = S(t a, x) P(T t A a, X) for all k, while still allowing shifts in the covariate distribution X across sites, i.e., adjusted for covariates, the event-time distribution no longer depends on site.
Figure iillustrates the data structure through directed acyelic graph (DAG), depicting the relation- ships among covariates X, treatment A, site indicator R event time T, and censoring time C, and compares scenarios with and without the CCOD assumption.
X
(a) CCOD holds
(b) CCOD possibly violated
Figure 1: Data structures under and without CCOD. (a) Under CCOD, site and event time are conditionally independent given treatment A and covariates X. (b) When CCOD is possibly violated, indicated by the red dashed arrow, R and T may not be conditionally independent.
2.4 FEDERATED ESTIMATION UNDER DISTRIBUTION SHIFTS AND PRIVACY
Motivation. In many settings, pooling individual-level data across sites is infeasible due to privacy constraints, At the same time, CCOD may fail, so naïve pooling is invalid. Still, some sites may provide information that improves estimation for the target population. We propose federated method that adaptively re-weights source sites using only summary-level information.
2.4.1 LOCAL SETE-LEVEL ESTIMATION
For each source we posit "site-k CCOD" assumption, S*t(t a, X) 5t a, X) almost surely, in order to derive the EIF. This assumption is only for formulating site-level estimators; its violation will be tested and corrected by adaptive weighting in Section 2.4.2)
Theorem 2.5. For each l E {0. 1,... K 1}, (t, a) is pathwise differentiable parameter given at) E [0, r] x {0, Under the site-k k CCOD assumption, the semiparametric EIF is given by
Pt,a
=k.0
(O;IP)
I(R 0)
I(R k)
I(A = a) H4 (O; S* T X)
P(R = 0) (t a, (t,
w (X)S* x,
P(R k)
Anchoring term using target data
Augmented torm using source R k) a density ratio comparing covariate distributions
where k.U (X) P(X R O)/P(X|
between the target site and source site k.
We prove Theorem 2.5 in Appendix E.
With the derived EIF, each site computes source-site
estimator @t: by solving P. P.
K.0 (O.P))
RAL property. A central result of this paper is the regular and asymptotically linear (RAL) property of the local estimator (L, stated in the following theorem. An estimator is RAL if it can be written as an i.i.d. average of influence functions plus a negligible remainder. This property allows the central limit theorem to be applied to obtain its asymptotic normal distribution. Below, k,0
we use (mi go who Gk .kk ) to denote the probability limits the estimated nuisance functions
(*th 00.00 Gh S*k) for estimating ( a) using data from source site k. These limits may differ from nuisance truths ( who ".K S.)).
Theorem 2.6. Under Conditions
E.
in Appendix E. with (# who G Sk (0, P " for a) [0, x {0,
(Tr t.o, G*,S" (n(00e( (1, 0 a))
4

---

Published as a conference paper at ICLR 2026
We prove Theorem 2.6 in Appendix E.I but summarize the regularity conditions here. Condition E.1 requires nuisance estimators to converge to well-defined limits: Condition E.2 bounds these limits and their estimates from extreme values (0 or infinity); and Condition E.3 controls product-type errors. They ensure that 0% (t, a) converges some well-defined limits and asymptotically normal.
Theoretical novelty. Theorem 2.6 establishes the semiparametric efficiency bound under the site-k CCOD assumption in 2.5] and the source-site estimators attain this bound under this assumption. These results add theoretical novelty to prior work on continuous outcomes (Han et al., 2025 The interactions between the density ratio and the other nuisance functions also represent previously unexplored theoretical components.
Remark 2.7
(Density ratio model). To estimate the density ratio while respecting data-sharing constraints, a common approach is to adopt the exponential tilt model detailed in Han et al. (2025): y vb (X)), where Yg is the model parameter and (b(-) is set of basis functions of
(x)) exp(y) the covariates. A simple choice is (X) = X for a linear component, and higher-order terms can be added to capture non- ao-eiiiaissies in estimating = To estimate each Y via maximum likelihood, only the target-site sample mean of t (X) needs to be shared with the source sites. In addition to this model, more flexible nonparametric machine learning approaches may be used, but these typically require sharing covariance matrices and/or other higher dimensional summaries. Thus, greater model flexibility comes at the cost of sharing more information. Finally, while the = model may be misspecified, this does not necessarily invalidate our framework or estimators. As noted in Theorem 2.8 our estimator is doubly robust: under Condition E.3 errors in estimating = influence the final estimator only through product-type ttyp term that enters the second-order remainder term.
Theorem 2.8 (Double robustness). For consistency of (t, a) under site-k CCOD, is not necessary that (Te wh: G Sxx=== (** w G ) in Theorem 2.6 must hold. Instead, at any single time point I, if either (i) the conditional survival model S*: or (ii) other nuisance functions Gk, t and wk.0 are correctly specified, (1, is consistent.
more technical version of Theorem 2.8
(Remark
E.4 and its proof are presented in Appendix E.I
2.4.2 AGGREGATION ACROSS SITES
Data-adaptive weighting. We define the site-specific discrepancy measure 2h.0
(t,a
(t, a) and the weight vector nr, = (Th..> Th.......hh... f--penalized convex optimization problem: we minimize Q(Tr,a), where
Mha -).a-a). To aggregate information, we solve an
[
}1
Q( =Pp
Gro (0:1))
Σ
PaaFra
* -=000 (0:
o.00 In (tt,,
(x,,t,a)
(2)
subject to v²t 2 0 and E T/. = 1; A is a tuning parameter that controls the bias-variance trade-off and is chosen by -ross-validation.
Interpretation. The objective function balances two goals: aligning site-level EIFs with the target distribution and excluding sites that would induce bias (possibly due site-k CCOD violation). The quadratic term ensures that sites well-aligned with the target survival distribution contribute more to the estimation, while the & penalty induces sparsity by driving the weights of misaligned sites exactly to zero. This contrasts with an & penalty, which merely shrinks weights without fully removing them. As result, the procedure asymptotically includes only the informative sources.
Federated estimator. The final estimator is obtained as a weighted average of the estimated local survival eurves: @itdd a) ΣK=1 nh (t, a). The variance of gred (t, a) can be estimated from its influence function, with the explicit formula given in Appendix E.2
Remark 2.9. We summarize the procedure of the federated method in Algorithm I and illustrate its flow in Figure 2 Implementation details can be found in Appendix D including for the cross- foss-fiiting procedure for nuisance fitting. Figure emphasizes that our approach follows a federated learning paradigm (McMahan et al.) 2017). Importantly, all steps require only summary-level transmission. never raw participant data. Source sites receive only the target-site 50 model parameters and summary statistics for the density-ratio model, and the leading analysis center receives only EIFs. This contrasts
5

---

Published as a conference paper at ICLR 2026
Target-site 5 model parameters and summary statistics for estimating RD
Target Site
Source Site 1
Source 2
Source
(t,
(t,
0220 (t,a)
0 (t,a)
Leading Analysis Center
n argmillg... (1ta)
0lod (t,a)
Figure 2: Algorithm flow. Target-site 5 model parameters and summary statistics for estimating #,0 are transmitted to source sites; each site estimates its survival functions locally; EIFs are sent and aggregated in a leading analysis center to compute federated weights by minimizing the Q(-).
Algorithm 1 Federated Learning for Multi-Source Causal Survival Analysis.
Input: Multi-source right-censored data (O, (X,, A, Y, A,, R.), 1,... n), time horizon > 0; fine time grid {0, c, 2c..... . for [0, with a small e >> O; and the number of disjoint folds into which the data are split, M.
2: Output: Estimated treatment-specine sppecffcc survival curves (iad (t, a) and its estimated variance teo
for a E {0, 1} and t E {0, c, 2c, .c.......
3: for (t, {0, ..,,,
f0, do
4: Estimate the EIFs via an M-fold local cross-fitting (see full detail in Algorithm 2
5: Obtain local estimates (t, a) as solutions of 0 Pn, (O; P)), for 0,..., K 1.
6: Obtain the site- specific discrepancy measure (difference of the target and source estimators) as (O; P)
Xn,ta P.
Pr.a
=k00
G.. (O:P) for k 1, 1....
1.
7: Solve for treatment- and time-specific weights Mi,- = (ft nh
k-1) that minimizes Mh
Q( = Pn { (
K-1
}
27
K
mh Prb (0, 1
1
Int (2
k=1
subject to < v/. S 1, for all k E I.... K 1} and K- parameter chosen by vross-validation centrally the leading analysis center; additional communication between sites is required.
Th. 1. and À is a tuning
8: end for
9: Return:
K-1 Σ Th. (1, (1, and Dfedt for a) E (0, 2e, 10,1).
(1, a)
t: Died computed based on the influence function of @t,l a) (see Remark E.5 in Appendix E.2
full decentralized learning (Lian et al. 2017 where there no central aggregator and sites interact directly to reach consensus. Our method also differs from meta-analysis (Borenstein et al.) 202T), which relies only on coarse population- summaries (such information is insufficient in setting) and often targets the pooled population.

---

Published as a conference paper at ICLR 2026
2.4.3 THEORETICAL PROPERTIES
We now summarize the asymptotic results and efficiency gain of the federated estimator; proofss are in Appendices [.. and E.2 Below, we say source site k has "site-k consistency" if it provides (t,
consistent estimator of (t.a) a), and we refer to the collection of all such Ic as the oracle selection set.
Theorem 2.10 (Asymptotic distribution). If regularity conditions for local estimates (Conditions E.I
E.3 in Appendix E.1 and the adaptive weights ir. recover the oracle selection set, then 0% (t, a), at each (1, E [0, x {0, 1J, has asymptotic distribution
V /
Vio { (t, - (1, -a
where Vid is an snffueence-uucciion-ooo consistent estimator for the underlying asymptotic variance of Glad (t, 1 a) (see Appendix E.2)
Corollary 2.11 (Asymptotic efficiency). The asymptotic variance Viee is no greater than that of the target-only estimator (t, a). Further, if at least one source site provides consistent estimate of (t, a). then (t, a) is strictly more efficient (smaller asymptotic variance).
Remark 2.12 (Selection consistency). The asymptotic validity of (t, a) relies on selection consistency with respect to the oracle selection set. This guarantees that post-selection inference by our variance estimator Died remains valid, even in the presence of biased sources.
Remark 2.13 (Efficiency gains). Under the mild regularity conditions stated Appendix E.2 namely compact covariate support, bounded density ratios, and finite variance covarriacc matrices of the EIFs across sites, we show that our federated estimator recovers the following oracle-optimal weights:
nta arg min nt,.=0, vkgg
Viled (1r,))
where Si, denotes the oracle selection set, Vi (Tr,a) denotes the asymptotic variance of the federated estimator under weight vector ni.a" The target-only estimator corresponds to the special case
Tr. = (1, 0), so its variance is no larger than that of any federated estimator. If the bias term Xu.t.a remains asymptotically non-zero, then n/.a + O, ensuring exclusion of biased sites. Proofs are adapted from Han et al. 2023 2025).
We note from Corollary 2.11 that strict efficiency gains rely site-k consistency for at least some k. This can be achieved under site-k CCOD. It is also possible that site-k CCOD is violated, but site-k consistency still holds by chance; this case, our method can still yield efficiency gains.
To preserve target-anchored validity while borrowing information, our methodology anchors each source estimator the target-site estimate and incorporates source data only through the augmented term of the EIF in Theorem 2.5 In the federation stage, we solve a global optimization problem that determines each site's contribution to the final estimator. While this resembles one-shot aggregation, the optimization operates over all source summaries and explicitly targets variance reduction while preserving the target estimand.
Although the framework allows both randomized and observational sites, many survival-date applica- tions involve randomized treatment assignment, in which case the propensity score 7 known and the target-site estimator is consistent. For observational studies, it is common to have larger sample sizes that enable flexible nonparametric and machine learning methods estimate nuisance functions with greater robustness to model misspecification.
3 SIMULATION STUDY
We conducted simulations to assess the performance of our federated estimator (FED) to four competing approaches: target-only (TGT), pooling (POOL), inverse variance weighting (IVW) and a meta-analytic IVW (META-IVW) estimators. TGT uses only target-site data (R = 0), POOL aggregates data from all sites without adjustment, IVW combines site-specific estimators using livers-varriade weights, and META-IVW applies additional covariate density correction to IVW. We refer to it imeta-analyee to correct for covariate shift. This comparison allows us to assess the efficiency gains and double robustness properties of FED under varying degrees of site heterogeneity and against several baselines.
as it parallels classical IVW meta-analytic pooling, augmented
7

---

Published as a conference paper at ICLR 2026
3.1 DATA GENERATING PROCESS
Under a given data generating process (DGP), we generate 500 independent datasets, with n ΣK-1 TA observations distributed across K = 5 sites. The target site (k = 0) was fixed at ng 300 observations, while source sample sizes were varied as n E {300, 600, 1000} for k = ,, 4, representing small, moderate, and large external data. Covariates, treatments, and outcomes were generated according to the mechanisms described in Appendix B. The estimand truth was derived by averaging survival outcomes over super-population of size super 10 from the target distribution. Our main DGP reflects nearly randomized studies that are more common in practice, where each site k has the propensity score N **(( {a X) weakly associated with X and good overlap. We also include a scenario with n4 = 300 (A: = 1, 2, 3, 4) where the target-site propensity score T ( X) is more dependent on X ("limited overlap") to highlight a regime with larger efficiency gains.
We modeled survival outcomes over a one-year horizon, with administrative censoring at day 200. Performance was evaluated at days 30, 60, and 90. To investigate consistency and efficiency under distribution shifts, we introduced five cases: (i) Homogeneous: all sites follow identical DGP; (ii) Covariate Shift: covariate distributions vary: (iii) Outcome Shift: conditional outcome distributions vary; (iv) Censoring Shift: censoring mechanisms vary; and (v) All Shifts: simultaneous covariate, outcome, and censoring vary across sites. Figure Appendix
[.. B. depicts survival curves under outcome and covariate shifts, illustrating how site-specific heterogeneity can affect target estimation.
3.2 PERFORMANCE METRICS AND RESULTS
We evaluated methods using three metrics: (i) Bias: assessed via boxplots of estimation bias across 500 replications, (ii) Relative root mean square error (RRMSE): defined as the RMSE of method divided by that of TGT; values below 1 indicate efficiency gains, and (iii) Coverage probability (CP%): ): the proportion of 95% Wald-type confidence intervals containing the truth. Values near 95 indicate better inference. Details of these metrics are provided in Appendix B.2 Simulation results for all scenarios appear in Appendix B.3. here we summarize representative findings in Figure 3
Good averiap
Good overlap
Limited overiap
RRMSE
POOL
0.1 a.s spca oas Sa3
Doo oue was c.s koa
3:39
Mathodi
META-NWW
01e
29112
140
TGT
Limted overtap
MEW.-WW
POOL
OP%
100
NW
95
012 54
55
POOL
META-NW
00
TGT
82
914
01.3
21.4
502
68
902
62.4
91.8
-02
Gosslate manoggetton
Camsnngg
As
Compaite Shift
Smits
Skaft
Hanogemmeson
Chatano cite Shitt
Contaniig Shit
Shitl
Solt
(A) Estimation Bias
(B) RRMSE (relative to TGT) and CP (nominal level: 95%)
Figure 3: Simulation results for the target-site treated- arm (A = 1) survival function at day 30 with Mu 300 and n = 300 (k- = 1, 3, 4) under good and limited target-site propensity score overlaps.
From Figure 3 A). FED exhibits negligible bias across all scenarios. META-IVW shows small biases under Homogeneous, Covariate Shift, and Censoring Shift, but becomes substantially biased under Outcome Shift and All Shifts. In terms of efficiency. FED consistently outperforms TGT: Panel (B) shows up to 59% reductions in RMSE by FED across all settings, especially under the limited overlap scenarios. META-IVW is more efficient when the outcome does not shift, but its efficiency drops sharply under Outcome Shift. These confirm that FED preserves consistency compared to META-IVW and consistently improves efficiency relative to TGT. Across all scenarios in Appendix B.3, FED achieves larger efficiency gains at carlier time points, when site-specific survival curves more closely resemble the target (see Figure 5) and the source-site EIFs align better with the target. Under limited target-site propensity score overlap, FED also attains higher efficiency, likely because the improved overlaps at the source data help stabilize the source-site estimators.
In terms of inferential validity, both FED and TGT maintain CP% closer to 95% across scenarios, validating the fiffeence-functionnnae variance estimator. Further diagnostics, reported in Figures
8

---

Published as a conference paper at ICLR 2026
6 and 7 show that federated weights Mr. decrease systematically site- site-specific bias measures
1,.,taa )² increase. Thus, FED adaptively upweights sites aligned with the target and downweights or excludes biased ones; the target site receives higher weights under covariate or outcome shifts, while contributions vary over time depending on alignment of survival functions.
Although POOL and IVW exhibit lower variability (narrower boxplots), they perform poorly under Covariate, Outcome, or All shifts: bias is substantial such that RRMSE is elevated, and CP% drops far below 95%. The exception is under Censoring Shift, but this arises because censoring treated as nuisance function and estimated separately within each site, reducing sensitivity to between-site heterogeneity in censoring distributions.
REAL DATA ANALYSIS
We illustrate our method with two applications. The first analyzes the coordinated aatibody-mediial
prevention (AMP) trials (Corey et al.) [021] [Ning et al.] (023), which enrolled 4,611 participants assess whether bn reduces HIV-1 acquisition. The second uses the fl chain dataset (7,874 participants across three biomarker-defined groups) to study sex differences in all-cause mortality. For brevity, we focus on the AMP trials and present the fichain results in Appendix C.2
The AMP trials considered HIV diagnosis by week-80 as the primary endpoint, a rare event with only 3.77% incidence. Loss to follow-up was relatively low (less than 10% per treatment arm)
(Corey et al. 202T). The participants were from four regions (sites): (i) SA: South Africa, (ii) OA: other sub-Saharan African countries, (iii) BP: Brazil or Peru, and (iv) US: United States or
Switzerland. Participants in (i) and (ii) were women, while those in (iii) and (iv) were cisgender men or transgender individuals, reflecting population differences. Because of event sparsity, we applied 2-fold cross-fitting. Conditional survival and censoring functions were estimated via an ensemble of Kaplan- Meiier, Cox proportional hazards regression, and survival random forests from the survSuperlearss
package (Westling et al. 2024). Propensity scores and density ratios were estimated using ensembles of logistic regression and LASSO via Supe r Lea rner (van der Laan et al.,, (007). Predictors included baseline age, standardized machine-leaming dechine-leaminn-ddrrieeed HIV risk score, and body weight.
Results with South Africa as target site. We highlight main results in Figure 4 Additional analyses treating OA, BP, or US as the target, as well as comparisons of regional survival curves and baseline covariates, appear in Appendix [ Table [Talle[[shww shows that OA closely resembles SA, while BP and US differ markedly in baseline risk score, weight, and HIV prevalence, consistent with covariate and outcome shifts. This pattern is reflected in the federated weights: Figure (B) shows SA receiving the highest weights on average, followed by OA, US, and BP.
FED
NETA-IW
Trasled
GooD
Commol
Relartive officinncg
Day 146 330
512
312
1.0
76
600
Time Ida))
Cornoi (gres)
191
Region
Fatarivw afficency
(A) Estimoted Tr Treatment-speenee Curvss
(8) Time- Mine--ppoiffo Foderated Weights
Figure 4: AMP results with SA as the target site. (A): Relative efficiency is the ratio of the estimated standard deviation that of the TGT estimator, 148, 330, and 512 days. (B): Federated weights with locally weighted smoothing (only a representation tool; see Cleveland & Devlin (188)) )
Figure (A) A) shows that FED, META-IVW and TGT produce similar survival curves, although META- IVW does not provide point estimates at earlier time points arising from inverses of small site-specific variances. Compared to TGT, FED offers narrower confidence intervals in some cases. In particular,

---

Published as a conference paper at ICLR 2026
TGT fails to yield valid intervals at certain early time points due to unstable or unavailable variance estimates, driven by the insufficient effective sample size of individuals who experience the event at those times, while FED recover intervals by borrowing information from aligned sites. These efficiency gains mirror our simulation findings, highlighting the ability of FED to improve inference without introducing bias.
Although POOL and IVW exhibit lower variance (smaller relative efficiency), they deviate from the trends of TGT and FED, suggesting bias under distribution shifts. Moreover, same META-IVW, IVW fails to return estimates at early times due to extreme weights, underscoring a practical limitation in survival applications.
DISCUSSION
We developed a federated learning framework for estimating treatment-spectne survival functions in a target population. By leveraging external sources with potentially shifted covariate and outcome distributions, while preserving privacy, our method achieves efficiency gains under oracle selection and mild regularity conditions. In the absence of timing and censoring, our estimator reduces to the FACE estimator Han et al. (2025) when the survival outcome is replaced by the binary indicator I(T(() > r). With censoring (i.e., missingness in this binary outcome), FACE would require modification to incorporate inverse-proot -roorabiipbbbbitttyyy weights under
atsssing-t-randdon assumption. Our method also extends to multiple non-mergeable target sites by anchoring each one separately and solving a target-specific federated aggregation problem. When target sites are comparable (e.g.. satisfy CCOD), transfer learning may be leveraged to further improve nuisance estimation.
Limitations and future directions. Several limitations suggest opportunities for future work. First, although Theorem 20.0 and our simulations demonstrate efficiency gains, developing potentially more efficient covariate-adaptive weighting schemes remains crucial. In addition, when data sharing is permitted but the CCOD assumption fails, it is unclear whether any method-including the pooled estimator-ccn outperform the target-only semiparametric efficient estimator (TGT in our simulation) and our federated approach. Future work may pursue more aggressive borrowing without CCOD by allowing non-regule non-regular behavior (trading regularity for efficiency). Furthermore, while time- time-specific weights are flexible, they may yield non-smooth trajectories and increase computation and communication costs continuous-time settings (communication scales with the evaluation grid size ni, at On n1z)). Future work should develop smoothing strategies to capture temporal trends more efficiently and extend the framework to incorporate time-varying covariates.
Additionally, violations of the positivity assumption can render target estimand unidentifiable, e.g.. when the two treatment groups in some sites differ systematically in their covariate distributions, or when certain participants are ineligible for specific treatments. Future work should investigate or leverage techniques to address such violations in our framework (Cheng et al.) (002)) Xue et al.) 2024). Furthermore, future work should consider settings where covariates differ across sites have limited overlap; in such cases, density ratio estimation becomes difficult and requires additional sensitivity analysis. Finally, although our density-ratio weighting effectively addresses covariate shift, investigating alternative weighting strategies, such as extending the collaborative propensity score weighting (Guo al. 2024) to survival data, is left for future work.
Our framework also connects to several other extensions, including incorporating alternative estima- tors such as TMLE and C-TMLE (van der Laan & Rubin 2006, Stitelman van der Laan 2010 van der Laan & Gruber 2010). adapting to external controls settings such as FedECA (Ogier du Ter
rail et al. (022) as well as extending their inverse probability weighted-Cox approach to incorporate
EIF and ensemble learning, surrogate-assisted causal inference (Han et al., 2022; Gao et al. 2024).
dynamic treatment regimes (Zhang et al. 2013), and data-driven selection of external sources (Gao al. 2025).. It also opens opportunities for constructing two-sided conformalized prediction intervals for event times by leveraging the ElF-based conformal scores for survival outcomes developed
(Farina al.] 2025) with federated learning for predicting missing outcomes (Liu et al.) 2024a). Our
approach could be adapted to other estimands such as restricted mean survival time Han 2023).
competing risks (Lok et al. 2018) left-truncation (Han) 2024 Wang al. 2024).
10

---

Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
Research reported in this publication was supported by the National Heart, Lung, And Blood Institute of the National Institutes of Health under Award Number T32HL079896. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
REPRODUCIBILITY STATEMENT
Software. A user-friendly R package FuseSurv (data fusion for causal survival analysis) im- plementing our method is available from the first author's GitHub: https://github/ com/ Saliu1998/Fuse$un
Programming. All simulation studies and real data analyses were performed using the sta- tistical language R (version 4.4.2). The dependent R packages include: CFsurvival. SarvSuperLearnet superLearner (version 2.0.29), glmnet (version 4.1.8), caret (ver- sion 6.0.94) and tidyverse (version 2.0.0). enhance computational efficiency, parallel com- puting packages foreach (version 1.5.2) and doPare aeaarelllll (version 1.0.17) were employed. The replication simulations was carried out using 200 CPU cores by Duke Computing Cluster.
Real data. The two real datasets are publicly available. The AMP trial data can found at https: 20HPTNS view and the "flchain"
data can be found at htttps://rde 1o/ ctt///aan///uu/ http:///ra//ssrraaaaa//aa/////aa html by typing command data (flchain) in after loading the survival package.
ETHICS STATEMENT
This work complies with the ICLR Code of Ethics. We used only publicly available datasets with appropriate licenses and did not involve human subjects sensitive personal information. We acknowledge potential risks of misuse (e.g., unfair application, misinterpretation, or unintended deployment beyond the intended research scope) and discuss limitations and safeguards in the paper. All results are reported transparently, and code will be released to support reproducibility.
REFERENCES
Alejandro Almodóvar, Juan Parras, and Santiago Zazo Propensity weighted federated learning for treatment effect estimation in distributed imbalanced environments. Computers in Biology and Medicine, 178: 108779, 2024.
Alberto Archetti, Eugenio Lomurno, Francesco Lattari, André Martin, and Matteo Matteucci. Hetero- geneous datasets for federated survival analysis simulation. In Companion of the 2023 ACM/SPEC International Conference on Performance Engineering, pp. 173-180, 2023.
Peter C Austin. Generating survival times to simulate cox proportional hazards models with time- varying covariates. Statisties Medicine, 31(29):3946-3958, 2012.
Xiaofei Bai, Anastasios A Tsiatis, and Sean M O'Brien. Doubly-robust estimators of treatment- specific survival distributions in observational studies with stratified sampling. Biometrics, 69(4): 830-839, 2013.
Peter Bickel, Chris AJ Klaassen, Peter J Bickel, Ya' Ya'acov Ritov, Klaassen, Jon Wellner, and YA' Acov Ritov. Efficient and adaptive estimation for semiparametric models, volume 4. Springer, 1993.
Michael Borenstein, Larry V Hedges, Julian PT Higgins, and Hannah R Rothstein. Introduction to meta-analysis. John Wiley & Sons, 2021.
Kate Bull and David Spiegelhalter. Tutorial in biostatistics survival analysis in observational studies. Statistics Medicine, 16(9): 1041-1044, 1997.
11

---

Published as a conference paper at ICLR 2026
Zhiqiang Cao, Youngjoo Cho, and Fan Li. Transporting randomized trial results to estimate counter- factual survival functions in target populations. Pharmaceutical Statistics, 2024.
Chao Cheng, Fan Li, Laine E Thomas, and Fan Li. Addressing extreme propensity scores in estimating counterfactual survival functions via the overlap weights. American Journal of Epidemiology. 191 (6):1140-1155 2022.
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whit- ney Newey, and James Robins, Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 2018.
William S Cleveland and Susan Devlin. Locally weighted regression: an approach to regression analysis by local fitting. Journal of the American Statistical Association, 83(403):596-610, 1988.
Bénédicte Colnet, Imke Mayer, Guanhua Chen, Awa Dieng, Ruohong Li. Gaël Varoquaux, Jean- Philippe Vert, Julie Josse, and Shu Yang. Causal inference methods for combining randomized trials and observational studies: a review. Statistical Science, 39(1): (111)):15-19999 2024.
Lawrence Corey, Peter B Gilbert, Michal Juraska, David C Montefiori, Lynn Morris, Shelly T Karuna, Srilatha Edupuganti, Nyaradzo M Mgodi, Allan C Decamp, Erika Rudnicki, et al. Two randomized trials of neutralizing antibodies to prevent hiv-1 acquisition. New England Journal of Medicine, 384(1 (84(11):1003-111 2021.
David R Cox. Regression models and life-tables. Journal of the Royal Statistical Society Series B: Statistical Methodology, 34(2):187-202, 1972.
Yifan Cui, Michael Kosorok, Erik Sverdrup. Stefan Wager, and Ruoqing Zhu. Estimating heteroge- neous treatment effects with right-censored data via causal survival forests. Journal of the Royal Statistical Society Series Statistical Methodology, 85(2):179 221, 2023.
Rebecca DerSimonian and Nan Laird. Meta-anaalssi in clinical trials. Controlled Clinical Trials, 7 (3):177-188, 1986.
Iván Díaz. survtmlerct: Efficiency guarantees for covariate adjustment RCTs with survival out- comes, 2020.
ctt.abb.coommmiaaasaa.s wttp:/sstsssuuttteeeeeeee. R package version
1.0.0.
Iván Díaz, Elizabeth Colantuoni, Daniel F Hanley, and Michael Rosenblum. Improved precision in the analysis of randomized trials with survival outcomes, without assuming proportional hazards. Lifetime Data Analysis, 25:439-468, 2019.
Angela Dispenzieri. Jerry A Katzmann, Robert A Kyle, Dirk R Larson, Terry M Therneau. Colin L Colby, Raynell Clark, Graham Mead, Shaji Kumar, L Joseph Melton III, et al. Use of nonclonal serum immunoglobulin free light chains to predict overall survival in the general population. Mayo Clinic Proceedings, 8766)511--533, 2012.
Xianqiu Fan, Jun Cheng, Hailing Wang, Bin Zhang, and Zhenzhen Chen. A fast trans-lasso algorithm with penalized weighted score function. Computational Statisties Data Analysis, 192: 192:107899 10789, 2024.
Rebecca Farina, Eric Tehetgen Tchetgen, and Arun Kumar Kuchibhotla. Doubly robust and efficient calibration of prediction sets for censored time-to-event outcomes. arXiv preprint arXiv:2501.04611 2025.
Chenyin Gao, Peter B Gilbert, and Larry Han. On the role of surrogates in conformal inference of individual causal effects. arXiv preprint 4rXiv:2412.123655 2024.
Chenyin Gao, Shu Yang, Mingyang Shan, Wenyu Ye, Ilya Lipkovich, and Douglas Faries. Improving randomized controlled trial analysis via data-adeptive borrowing. Biometrika, 1 12(2))asac069, 2025.
Richard Gill and Soren Johansen. A survey of product-integratin with a view toward application in survival analysis. The Annols of Statistics, 1501):555 1990.
12

---

Published as a conference paper at ICLR 2026
Tianyu Guo, Sai Praneeth Karimireddy, and Michael Jordan. Collaborative heterogeneous causal inference beyond meta-analysis. arXiv preprint arxiv:2404.15746, 2024. 2024.
Larry Han. Breaking free from the hazard ratio: Embracing the restricted mean survival time in clinical trials. NEJM evidence, 210)EEVDDe230042, 2023.
Larry Han. Truncated, not horgotien-hhadllng left truncation in time-to-event studies. NEJM evidence, 3(12)EEVIe2200034, 2024.
Larry Han, Xuan Wang, and Tianxi Cai. Identifying surrogate markers real-world comparative effectiveness research. Statistics in Medicine, 4 41(26):520-5330 2022.
Larry Han, Zhu Shen, and Jose Zubizarreta. Multiply robust federated estimation of targeted average treatment effects. Advances in Neural Information Processing Systems, 36:70453-7046 70482, 2023.
Larry Han, Yige Li, Bijan Niknam, and José Zubizarreta. Privacy-preserving, communication- efficient, and target-flexible hospital quality measurement. The Annals of Applied Statistics, 18(2): 1337-1359, 2024.
Larry Han, Jue Hou, Kelly Cho, Rui Duan, and Tianxi Cai. Federated adaptive causal estimation (face) of target treatment effects. Journal of the American Statistical Association, 120(551):1503-1555 2025.
Miguel A Hernán. The hazards of hazard ratios. Epidemiology, 21(1):13-15, 2010.
Beilin Jia, Donglin Zeng, Jason JZ Liao, Guanghan Liu, Xianming Tan, Guoqing Diao, and Joseph G Ibrahim. Inferring latent heterogeneity using many feature variables supervised by survival outcome. Statistics in Medicine, (0(13)3318133993 2021.
Edward L Kaplan and Paul Meier. Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282):457-485 1958.
Robert A Kyle, Terry M Thernea,, S Vincent Rajkumar, Dirk R Larson, Matthew F Plevak, Janice R Offord, Angela Dispenzieri, Jerry Katzmann, and I. Joseph Melton III. Prevalence of monoclonal gammopathy of undetermined significance. New England Journal of Medicine, 354(13))1362--136 2006.
Dasom Lee, Shu Yang, and Xiaofei Wang. Doubly robust estimators for generalizing treatment effects on survival outcomes from randomized controlled trials to target population. Journal of Causal Inference, 10(1):415-44 440, 2022.
Fan Li and Fan Li. Using propensity scores for racial disparities analysis. Observational Studies, (1):59-68, 2023.
Huiyue Li, Yi Liu, Yunji Zhou, Jiajun Liu, Dezhao Fu, and Roland A Matsouaka. Variance estimation for weighted average treatment effects. Statistics in Biosciences, pp. 1-73, 2025.
Sai Li, Tianxi Cai, and Rui Duan. Targeting underrepresented populations in precision medicine: A federated transfer learning approach. The Annals of Applied Statistics, 17(4):977-2299,, 2023.
Sijia Li and Alex Luedtke. Efficient estimation under data fusion. Biometrika, 110(4):0441-104, 2023.
Xiangru Lian, Ce Zhang, Huan Zhang, Cho-Jui Hsich, Wei Zhang, and Ji Liu. Can decentralized algorithms outperform centralized algorithms? case study for decentralized parallel stochastic gradient descent. Advances in Neural Information Processing Systems, 30, 2017.
Jiajun Liu, Yi Liu, Yunji Zhou, and Roland A Matsouaka. Assessing racial disparities in healthcare expenditure using generalized propensity score weighting. BMC Medical Research Methodology, 25(1):64, 2025.
Yi Liu, Alexander W Levis, Sharon-Lise Normand, and Larry Han. Multi-source conformal inference under distribution shift. Proceedings of Machine Learning Research, 11:311344-31888 2024a.
13

---

Published as a conference paper at ICLR 2026
Yi Liu, Huiyue Li, Yunji Zhou, and Roland A Matsouaka. Average treatment effect on the treated, under lack of positivity. Statistical Methods in Medical Research, 33(10))1689-117)) 1717, 2024b.
Judith J Lok, Shu Yang, Brian Sharkey, and Michael D Hughes. Estimation of the cumulative incidence function under multiple dependent and independent censoring mechanisms. Lifetime Data Analysis, 24:201-223, 2018.
Disha Makhija, Joydeep Ghosh, and Yejin Kim. Federated learning for estimating heterogeneous treatment effects. arXiv preprint arXiv:2402.17705 2024.
Fulgencio Marín-Martinez and Julio Sánchez-Meca. Weighting by inverse variance by sample size in random-effects meta-analysis. Educational and Psychological Measurement, 70(1):56-74 2010.
Roland A Matsouaka, Yi Liu, and Yunji Zhou. Overlap, matching, or entropy weights: what are weighting for? Communications statisties-Simalation and Compulation, 54(7):2672-2691, 2025.
Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera Arcas. Communication-effio efficceett learning of deep networks from decentralized data. In Artificial intelli- gence and statistics, pp. 1273-1282 PMLR, 2017.
Chirag Nagpal, Vedant Sanil, and Artur Dubrawski. Recovering sparse and interpretable subgroups with heterogeneous treatment effects with censored time-to-event outcomes. arXiv preprint 1rXiv:2332.112044 2023.
Xi Ning, Yinghao Pan, Yanqing Sun, and Peter B Gilbert. A semiparametric cox colen transformation model with censored data. Biometrics, (114)3111-33 2023.
Jean Ogier du Terrail, Quentin Klopfenstein, Honghao Li, Imke Mayer, Nicolas Loiseau, Mohammad Hallal, Michael Debouver, Thibault Camalon, Thibault Fouqueray, Jorge Arellano Castro, al. Fedeca: federated external control arms for causal inference with time-to-event data in distributed settings. Nature Communicutions, 16(1) 16(1):7496, 2025.
Khellaf Rémi, Bellet Aurélien, and Josse Julie. Federated causal inference from multi-site observa- tional data via propensity score aggregation. arXiv preprint arXiv:2505. 17961, 2025.
Paul R Rosenbaum and Donald Rubin. The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1):41-55, 1983.
Ori M Stitelman and Mark van der Laan. Collaborative targeted maximum likelihood for time to event data. The International Journal of Biostatistics, 6(1):1-46, 2010.
Mark J van der Laan and Susan Gruber. Collaborative double robust targeted maximum likelihood estimation. The International Journal of Biostatistics, 6(1):1-71, 2010.
Mark J van der Laan and Daniel Rubin. Targeted maximum likelihood learning. The International Journal of Biostatistics, 2(1), 2006.
Mark J van der Laan, Eric C Polley, and Alan E Hubbard. Super learner. Statistical Applications in Genetics and Molecular Biology, 6(1):1-21, 2007.
AW Van der Vaart and JA Wellner. Weak Convergence and Empirical Processes. Springer & Verlag New York, 1996.
Yiming Wang, Yi Liu, and Shu Yang. Rate doubly robust estimation for weighted average treatment effects. Journal of Causal Inference, 13(1):0224003, 2025.
Yuyao Wang, Andrew Ying, and Ronghui Xu. Doubly robust estimation under covariate-induced dependent left truncation. Biometrika, pp. asac005, 2024.
Lan Wen, Jon A Steingrimsson, Sarah Robertson, and Issa J Dahabreh. Multi-source analyses of average treatment effects with failure time outcomes. Lifetime Data Analysis, pp. 1-29, 2025.
14

---

Published as a conference paper at ICLR 2026
Ted Westling, Mark J van der Laan, and Marco Carone. Correcting estimator of a multivariate monotone function with isotonic regression. Electronic Journal of Statistics, 14(2):3032, 2020.
Ted Westling, Alex Luedtke, Peter B Gilbert, and Marco Carone. Inference for treatment-specific survival curves using machine learning. Journal of the American Statistical Association, 119(546): 1541-1553, 2024.
Charles J Wolock, Peter B Gilbert, Noah Simon, and Marco Carone. A framework for leveraging machine learning tools estimate personalized survival curves. Journal of Computational and Graphical Statistics, pp. 1-11, 2024.
Jun Xie and Chaofeng Liu. Adjusted kaplan- meier estimator and log-rank test with inverse probability of treatment weighting for survival data. Statistics in Medicine, 24(20):308-311 10, 2005.
Ruoxuan Xiong. Allison Koenecke, Michael Powell, Zhu Shen, Joshua T Vogelstein, and Susan Athey. Federated causal inference in heterogeneous observational data. Statistics Medicine, 42 (24):4418-4439, 2023.
Wu Xue, Xiaoke Zhang, Kwun Chuen Gary Chan, and Raymond KW Wong. Rkhs-based covariate balancing for survival causal effect estimation. Lifetime Data Analysis, 30(1):34-58, 2024.
Shu Yang and Peng Ding. Combining multiple observational data sources to estimate causal effects. Journal of the American Statistical Association, 2019.
Baqun Zhang, Anastasios A Tsiatis, Eric B Laber, and Marie Davidian. Robust estimation of optimal dynamic treatment regimes for sequential treatment decisions. Biometrika, 100(3)6681-69) 2013.
Hui Zou. The adaptive lasso and its oracle properties. Journal of the American Statistical Association, 101(476);441--142, 2006.
15

---

Published as a conference paper at ICLR 2026
A USE OF LLMS
We acknowledge the use of ChatGPT-5.0 exclusively for language polishing and grammatical correc- tions. No large language models (LLMs) used for any other aspects of this work. The research ideas, conceptualization, methodology development, and all experiments are entirely original contri- butions of the authors.
B SIMULATION DETAILS AND ADDITIONAL RESULTS
B. DETAILS DATA GENERATING PROCESS
Three covariates X, X2, and X3 are sampled as transformations of Beta random variables with site-ppecffie parameters:
- X1 ~ 33 -eta(1.110.55//66
+
X2 52 Beta(1.5 + 2y(k)) +7+ (7)),
X3 (4 + (k)) Beta(1.5 Xx - 50 2,0)) +.11((k)))))))))))))))))
where 7(k) represents some function of site k, specified later. We then generate the treatment assignment probabilities π (X) using the logistic function:
logit(((X)) 1105 + log (1.3 + exp( X1/10) + exp(-2 + X2/3) + exp(-2 + X3
and treatments A are sampled as A Bernoulli(=(X)) For the scenario with limited propensity score overlap the target site, we modify the target-site propensity score model T(X) to such that
logit(m (X) 1.05 log (0.3 exp(-120 + X1) exp(-6 + x2) + exp( + Xz/4))
and generate A Bernoulli (TT (X) accordingly for target-site samples only. This increases the dependence of A on X and induces reduced overlap.
Next, we consider the mechanisms of event and censoring times. The hazard rates for event times and censoring times given by the following exp(hı and exp(he), respectively, where h4
-5.02 + 0.1 25) - 0.1 Xx2 25) + 0.05 X3 - 2) + Dr(k) 0.1 25) + A 8r(k) 0.1(X, + X2 Xg- -50), and h. 4..7
25) 0.02(X² 25) 0.01 (k( (166)-00110 0.1(X, 25) + (k) 0.1 0.1 + X2 + X3 50)-
87 + (x01
Here, (r))). (k(.). (k) and (k) are some site-specific indicators, specified later, for varying the treatment effects and trends of survival curves for different sites. Then, event times and censoring times are sampled as:
T = (----1i log(Ui) (olo
1/P
C c-(- log(U2) exp(ha) (=(-esp(ha)
1/P
eex((h)) (-exp(ha))
with p = 1.2, À = 0.6, and U2 ~ Uniform(O. 1). This technique follows Austin (0122). Thus, the observed times and event indicators are Y min C), = I(T < C),, respectively.
Under this data generating process (DGP), the event time is generated to mimic days in year (365 days), and truncate the censoring time 200 days mimic the end of follow-up in survival analysis. Our DGP allows the following scenarios based on site-specific distributional heterogeneity:
Homogeneous: Homogeneous covariates and hazard rates across sites. We let 7(k) Dr(k) = (k( (k)) dk(k) = ( for h = 1, 1....,4. .4.
Covariate Shift: Covariates X,, X2, and X3 vary across sites. We let 7(k = and Dr(k) Do(k) (k:) (k) 0, for 0, 1.
Outcome Shift: Conditional outcome distribution varies across sites. We assign (k) = 0. for = 0, 1.
Dr(k) = 6 = k. and (c = (k)
Censoring Shift: Censoring mechanism varies across sites. We let 7(k) 0, Dr(k) 6y(k) = 0 and (k) = (kk = k, for k = 0,
All Shift: Covariates and both event and censoring effects vary across sites. We let 7(k)
(r(k) De(k) 5r (k:)) dc k. for k 0, 1,...,4.
16

---

Published as a conference paper at ICLR 2026
Figure[ibeloo plots the true treatment-spent sppecfffttttteiiiieeee survival curves under the Covariate Shift and Outcome Shift scenarios, as defined by our designed DGPs, to illustrate the effect of site differences on survival outcomes.
Source sites have covariate shift only:
Target Site
Site
Source he
Source Site
Source ite
SunAel Probacsityy
0.8-
0.6-
9.6
0.6
0.6
0.4-
0.4
0.4
0.4
50
23
10 Time
75
25
50
75
0
25
30 Time
13
25
30 Time
15
Time
Time
Source sites have outcome shift only:
Target Site
Source Site
Source ftte2
Source
S Source itt
Sunvel Probabiity
1.0
0.8
0.6-
0.6
Treated
0.4
0.4
0.4
0.4
25
50 Time
15
25
50 Time
75
25 50 Time
75
0
25
50 Time
75
25 50
75
Time
Figure 5: Site- and treatment-speciffe survival curves, each based random sample of 10 from the true DGP of each site. The two dashed curves each source site panel are the target site survival functions for reference. Under covariate shift, curves preserve their shapes and trends but differ scale, whereas outcome shift produces marked changes in shape and treatment effects.
B.2 PERFORMANCE CRITERIA DEFINITIONS
The simulation performance criteria considered in Section 3.2 with an additional metric 95% confi- dence interval (CI) width for the complete simulation results are defined follows.
Let denote the true target parameter, and let 0, and o; be the point and standard error estimates, respectively, from the ith Monte Carlo replication of competing method, i = 1,..., 500. Then:
Estimation bias: 0: 10,, i 1, .. 500, summarized via boxplots;
RRMSE: the RMSE of method relative that of the TGT estimator, where RMSE / /500-1 - 500 (0, 0)2 By definition, the TGT estimator has RRMSE 1. Smaller
RRMSE values indicate higher efficiency relative TGT;
CP%: the proportion of replications in which the Wald-type CI contains 0: 100% x 500-1> -i T[O 0 1.96Ø,, 0, + 1.965,]). The closer CP% is to 95, the more reliable the inference based on ã; and
95% CI width: the average CI width across replications, where the CI from the ith replication is 0, + 1.96, o. Thus, CI width 3.92 x 500
500
Σi=1 0,.
B.3 COMPLETE SIMULATION RESULTS
Figures 6 summarize the results comparing the federated and source-site shifts, well as the corresponding discrepancy values (x+ (2h Figures 8 report the full simulation results under good target-site propensity score overlap for varying source sample sizes. Figure [I] shows the corresponding results under limited target-site overlap, where FED consistently achieves higher relative efficiency compared with TGT.
17

---

Published as a conference paper at ICLR 2026
Homageneous
Coveriate Outcome
Censoring
0.5
0.4
Site
0.2
Target
Federated weights
0.0
Source
Source
Source
30
60
90
90
30
60
90
30
60
90
60
Time
Figure 6: Average federated weights of each site different time point by site heterogeneity cases.
This figure uses the case where Tig 300 (k 2 1) as an illustration for weights.
Source
Source
Source 3
Source 4
1.00
0.75
Homoggeneeua
0.50
0.25
0.00
1.00
1.00
Federated weights
0.75
Outcome
Time
0.50
0.25-
0.00
1.00
0.75
Centoring
0.50
0.25
0.25
0.00
0.0
0.3.
0.2
x² value
Figure 7: Scatter plots of site-specific federated weights vs. discrepancy measure (x (X values, under scenarios of site heterogeneity and selected time points (days 30, 60 and 90). Sites 2-4 under Covariate Shift and All Shift have more larger (Kt y² values with clear trends of decreasing weights. The pink dashed lines indicate weight 1/5, i.c., one over five sites. This figure uses the case where n4 300 (k 2 1) illustration for weights.
18

---

Published as a conference paper at ICLR 2026
Homogeneous
Covariate Siift
Outcome Shift
Censoring Shitt
5
0.0
-0.1-
Method
-0.2
FED
NETA-WV
POOL
-0.1
-0.2-
30
60
90
30
60
90
30
60
90
30
60
90
30
60
50
Time (day)
Homogeneous
Covariate 5 hift
Outcome Shift
Censoring Shift
All Shift
POOL
0.25
0.19 0.23
4.35 5.82
6.81
12
3.00
1.31
0.25
0.22
0.23
20.06
10.46
6.47
0.25
0.23
3.27
0.23
11.49
CORIOO
RRMSE
META-IVW
0.38
0.32
0.74
TGT
40
FED
0.88
0.88
0.87
0.86
0.81
0.82
0.78
0.76
0.77
0.87
0.87
0.94
0.83
0.8
0.85
30
POOL
0.18
0.22
0.2
2.21
3.76
4.01
25.34
19.02
8.86
0.16
0.22
0.18
47.39
32.36
27.23
20
IWW
0.25
2.94
19.68
32.
10
META-IVW
0.65
TGT
1
1
FED
0.91
0.88
0.96
0.91
0.95
0.88
0.84
0.89
0.87
0.87
0.59
0.94
0.9
0.93
0.94
30
90
30
90
60
60
60
Time (day)
Homogennous
Covariate Shift
Outcome Shift
Censoring S Shift
All hiff
POOL
93.4
93.4
2.2
54.4
03.4
91.8
92.6
4.
92.6
MET -VW
86
84.2
CP%
TGT
94.2
95.4
94.8
95
94.6
95.2
94.2
95.2
94.0
94
95.6
95
95.2
94.6
95.4
FED
91.7
92.6
95.8
96
97
92.4
92.8
93.4
91.8
91.6
93.8
95.6
96.8
97.2
75
POOL
93.6
94.4
92.6
36.8
16.2
0.2
93
93.6
93.4
50
96.2
92.8
87.2
25
META-IVW
79.6
91.8
0.7
1.2
73
89
78.8
TGT
92.4
94.4
93.6
94.2
96
94
93.4
92.2
93.4
94
FED
90
96.8
88
96.5
95.8
96.2
91
92.8
90.6
90.2
91.6
88.6
96.6
97
96.2
30
90
30
60
90
30
90
30
60
Time (day)
H Homogeneous
Covariste S hift
Outzome Shift
Censoring Shift
All Shift
POOL
0.06
0.08
6.09
0.07
0.08
0.09
0.06
0.07
0.08
0.08
0.09
0.07
0.08
0.08
0.06
0.08
0.07
0.08
0.08
CI width
META-IVW
0.06
0.07
0.08
0.06
0.07
0.08
0.07
0.08
TGT
0.34
0.17
0.2
0.14
0.17
0.2
0.14
0.17
0:14
0.17
0.2
0.14
0.17
0.2
FED
0.31
0.14
0.17
0.14
0.37
0.2
0.12
0.14
0.36
0.12
0.15
0.17
0.14
0.17
0.19
0.15
POOL
0.05
0.07
0.08
0.06
0.07
0.08
0.06
0.07
0.08
0.05
0.07
0.09
0.06
0.07
0.08
0.05
0.07
0.10
ME META-IVW
0.05
0.07
0.07
0.08
TGT
0.31
0.16
0.18
0,11
0.16
0.18
0.11
0.16
0.18
0.11
0.16
0.18
0.11
0.16
0.18
0.05
FED
0.1
0.13
0.15
0.11
0.16
0.18
0.1
0.14
0.15
0.1
0.13
0.16
0.12
0.16
0.18
30
Time (day)
Figure 8: Estimation bias (boxplots), relative root mean square error (RRMSE) compared to TGT, coverage probability (CP%) with 95% nominal coverage level, and width of 95% CI under 300 (k: 1,2,3, 4), with good propensity score overlap in the target site, evaluated days 30, 60 and 90 in simulation.
19

---

Published as a conference paper at ICLR 2026
Homogeneous
Covariate Shift
Outcome Shift
Censoring Shift
5
-0.1
Method
FED
-0.2
NETA-WV
0.1
POOL
0.0
-0.1-
-0.2
30
60
90
30
60
90
30
60
90
30
60
90
30
60
90
Time (day)
Homogeneous
Covariate 5 hift
Outcome Shift
Censoring Shift
All
POOL
0.11
0.12 0.12
6.4
8.35
7.12
11.37
4.76
0.11
0.14
0.17
27.19
14.03
6.34
0.32
6.12
5.48
7.85
10.7
5.06
0.14
CORIOO
RRMSE
META-IVW
0.22
4.69
0.22
0.75
TGT
FED
0.77
0.86
0.9
0.88
0.89
0.88
0.76
0.79
0.81
0.8
0.86
0.89
0.85
0.88
0.87
40
POOL
0.11
0.12
0.12
3.36
4.89
5.5
32.59
21.69
11.36
0.11
0.13
0.13
72.32
43.04
28.05
IWW
0.11
2.39
23.45
0.16
34.73
20
META-IVW
21.94
0.36
2.29
TGT
1
1
1
FED
0.92
0.91
0.91
0.94
0.92
0.88
0.89
0.91
0.86
0.93
0.94
0.94
0.94
0.89
0.91
30
90
30
90
60
60
60
Time (day)
Homogennous
Covariate Shift
Outcome Shift
Censoring S Shift
All Shift
POOL
93
16
92.2
90.4
IWW
92.4
91.4
MET -VW
80.2
91.6
83
61.6
69
CP%
TGT
94.4
96.4
94.8
96.4
96.4
95.0
94.2
96.6
94.6
94.4
96.4
94.0
96.4
95.4
95.6
FED
94
95.2
93.6
97.2
96.6
96.8
94
96.4
93.8
93.4
93.8
97
95.4
96.4
75
POOL
92.8
96,4
7.8
0.2
92.4
93.8
50
92.2
94
25
META-IVW
87.4
89.4
89.2
80.8
62.8
73.8
50.7
45.6
TGT
93.2
92.5
94
92.8
92.8
93.2
92.8
92.6
FED
92.2
91.8
90.2
93.8
94.4
93.6
93.8
91.4
91.2
92
92.2
90.8
94.6
94.5
94.4
30
90
30
60
90
30
90
30
60
Time (day)
H Homogeneous
Covariste S hift
Outzome Shift
Censoring Shift
All Shift
POOL
0.05
0.06
0.06
0.05
0.06
0.05
0.06
0.06
0.06
0.07
0.05
0.06
0.06
0.06
0.05
0.06
0.06
CI width
META-IVW
0.05
0.06
0.06
0.05
0.05
0.06
0.06
0.06
TGT
0.34
0.17
0.2
0.14
0.37
0.2
0.14
0.17
0.14
0.17
0.14
0.17
0.2
FED
0.32
0.15
0.17
033
0.37
0.19
0.12
0.15
0.17
0.12
0.35
0.18
0.13
0.17
0.19
0.16
POOL
0.04
0.05
0.06
0.04
0.06
0.06
0.05
0.05
0.06
0.04
0.06
0.06
0.05
0.06
0.06
0.12
0.04
0.05
0.08
ME META-IVW
0.04
0.06
0.05
0.05
0.06
TGT
0.31
0.16
0.18
0,11
0.16
0.18
0.11
0.16
0.18
0.11
0.16
0.18
0.11
0.16
0.18
0.04
FED
0.1
0.14
0.16
0.11
0.15
0.18
0.11
0.14
0.16
0.1
0.34
0.16
0.11
0.15
0.18
30
Time (day)
Figure 9: Estimation bias (boxplots), relative root mean square error (RRMSE) compared to TGT, coverage probability (CP%) with 95% nominal coverage level, and width of 95% CI under 600 (k: 1,2,3, 4), with good propensity score overlap in the target site, evaluated days 30, 60 and 90 in simulation.
20

---

Published as a conference paper at ICLR 2026
Homogeneous
Covariate Shift
Outcome Shift
Censoring Shift
5
0.1
Method
FED
-0.2
NETA-WV
0.1
0.0
POOL
-0.1-
11
13
30
60
90
30
60
90
30
60
90
30
60
90
30
60
90
Time (day)
Homogeneous
Covariate 5 hift
Outcome Shift
Censoring Shift
All Shift
POOL
0.08
0.07 0.06
6.53
10.08
9.24
13.99
5.9
1.74
0.08
0.08
0.00
27.01
17.5
8.54
0.07
0.05
5.7
9.79
13.74
5.85
0.09
RRMSE
META-IVW
0.17
0.29
13:11
5.76
0.19
1.51
TGT
60
FED
0.89
0.85
0.86
0.93
0.92
0.91
0.87
0.78
0.84
0.88
0.86
0.9
0.92
0.91
0.92
40
POOL
0.08
0.09
0.07
3.2
5.14
6.51
38.36
21.33
12:22
0.09
0.09
0.07
66.01
47.71
32.06
IWW
0.08
2.5
31.75
23.02
0.09
52.
META-IVW
22.54
0.25
3.45
TGT
1
1
FED
0.94
0.92
0.93
0,97
0.95
0.94
0.92
0.92
0.95
0.93
0.92
0.93
0.95 0.94
0.93
30
90
30
90
60
60
60
Time (day)
Homogennous
Covariate Shift
Outcome Shift
Censoring S hhif
All hift
POOL
91.4
7.6
92.6
92.6
90.2
92.2
1.8
92.8
92.4
MET -VW
77.6
76.6
46
CP%
TGT
94.6
94.4
94.4
93.4
94.6
94.4
94
94.6
94.4
94
94.4
93.4
FED
94.8
94.4
35.2
94.5
95.4
94.2
94.8
94.8
95
05.2
94.8
94.8
95.4
95
94.4
75
POOL
95
94.4
95.4
0.5
0
95.6
94.2
96
50
95
95.4
25
META-IVW
90.2
48.4
20
15.6
TGT
92
91.2
94.2
93.8
92
93.8
91.4
FED
91.2
93.2
93.4
92.4
91.2
94.2
92.6
93.4
94.4
91.2
93.8
93.8
92.8
91.2
94.4
30
90
30
60
90
30
90
30
60
Time (day)
H Homogeneous
Covariste S hift
Outcome Shift
Censoring Shift
All Shift
POOL
0.04
0.05
0.05
0.04
0.05
0.05
0.04
0.04
0.05
0.05
0.05
0.04
0.05
0.04
0.04
0.05
0.05
CI width
META-IVW
0.04
0.05
0.04
0.05
0.04
0.04
0.05
0.04
0.05
TGT
0:34
0.17
0.2
0.14
0.37
0.2
0.14
0.17
0.2
0.14
0.17
0.14
0.17
0.2
FED
0:33
0.16
0.18
0.14
0.37
0.2
0.13
0.16
0.18
0.13
0.35
0.18
0.14
0.17
0.19
0.15
POOL
0.03
0.04
0.05
0.03
0.04
0.05
0.04
0.04
0.05
0.03
0.04
0.05
0.04
0.04
0.05
0.10
0.03
paseall
ME -VTA-IVW
0.03
0.04
0.04
0.04
0.05
TGT
0.31
0.16
0.18
0,11
0.16
0.18
0.11
0.16
0.18
0.11
0.16
0.18
0.11
0.16
0.18
FED
0.31
0.15
0.17
0.11
0.16
0.18
0.11
0.15
0.17
0.11
0.15
0.17
0.11
0.16
0.18
30
Time (day)
Figure 10: Estimation bias (boxplots), relative root mean square error (RRMSE) compared to TGT,
coverage probability (CP%) with 95% nominal coverage level, and width of 95% CI under nı = 1000 (k: 1,2,3, 4), with good propensity score overlap in the target site, evaluated days 30, 60 and 90 in simulation.
21

---

Published as a conference paper at ICLR 2026
Homogeneous
Covariate Shift
Outcome Shift
Censoring Shift
0.0
Method
-0.2
FED
-0.4
0.4
NETA-WV
0.2
POOL
0.0
-0.2-
-0.4
30
60
90
30
60
90
30
60
90
30
60
90
30
600
90
Time (day)
Homogeneous
Covariate 5 hift
Outcome Shift
Censoring Shift
Shlft
POOL
0.15
0.12 0.12
4.35
3.9
2.89
7.46
2.95
0.91
0.16
0.11
0.11
20.88
8.17
4.14
0.35
6.12
3.5
3.77
2.73
0.13
17.79
4.12
RRMSE
META-IVW
0.22
0.47
0.36
0.17
0.35
TGT
20
FED
0.76
0.74
0.76
0.79
0.77
0.76
0.72
0.72
0.79
0.78
0.72
0.71
0.81
0.78
0.76
15
POOL
0.06
0.06
0.07
0.62
1.03
1.43
5.04
4.30
2.79
0.06
0.08
0.15
14.64
11.11
8.86
10
IWW
0.07
0.1
0.14
91
10.76
META-IVW
0.16
5.84
TGT
1
FED
0.41
0.52
0.68
0.51
0.62
0.65
0.43
0.52
0.69
0.42
0.51
0.67
0.54 0.6
0.68
30
90
30
90
60
60
60
90
Time (day)
Homogeneous
Covariate Shift
Outcome Shift
Censoring S Shif
All hift
POOL
96.2
5.4
6.2
45.8
96.4
98.6
96
96.2
1.2
95.8
96.4
95.8
MET -VW
90.4
25.8
90.2
CP%
TGT
95.2
94.8
94.8
93.2
94
94
95
94.2
94.2
94,0
94.8
93.4
FED
92.4
89.6
91.6
97
96.7
95.8
87.2
89
02.6
88.6
90
96.8
95.4
95.6
75
POOL
97
97.6
44.6
24.2
17.2
96.6
93.8
90.8
50
91.2
95.4
27.8
0.2
85.2
82.8
0.2
25
META-IVW
83.4
0.7
1.8
74.6
75.6
TGT
92.2
94.2
93.6
95.2
92.4
95.8
95.2
91.8
94.4
96.2
FED
89.8
90
89
93.4
96.4
97.4
92.2
93.2
90.8
90.6
90.8
90.4
94.4
96.8
97.2
30
90
30
60
60
90
30
90
30
60
Time (day)
H Homogeneous
Covariste S hift
Outzome Shift
Censoring Shift
All Shift
POOL
0.06
0.08
6.09
0.07
0.08
0.09
0.07
0.08
0.08
0.08
0.09
0.07
0.08
0.09
0.06
0.08
0.07
0.08
CI width
META-IVW
0.06
0.07
0.08
0.06
0.07
0.08
0.08
0.09
TGT
0:35
0.21
0.25
0.15
0.21
0.25
0.15
0.21
0:25
0:15
0.21
0:25
0.15
0.21
0.25
0.30
FED
0.32
0.16
0.18
0.15
0.2
0.23
0.13
0.15
0.18
0.12
0.15
0.19
0.15
0.19
0.23
0.25
0.20
POOL
0.05
0.07
0.09
0.06
0.08
0.09
0.06
0.07
0.08
0.06
0.08
0.09
0.07
0.08
0.09
0.15
0.05
0.08
paseall
ME -VTA-IVW
0.05
0.08
0.07
0.08
0.09
0.10
TGT
0.2
0.28
0.3
0.2
0.28
0.31
0.2
0.28
0.2
0.28
0.3
0.2
0.28
0.31
FED
0.32
0.17
0.19
0.16
0.23
0.26
0.14
0.18
0.19
0.12
0.17
0.19
0.17
0.23
0.26
30
60
60
Time (day)
Figure 11: Estimation bias (boxplots), relative root mean square error (RRMSE) compared to TGT, coverage probability (CP%) with 95% nominal coverage level, and width of 95% CI under 300 (k: 1,2,3, 4), with limited propensity score overlap in the target site, evaluated at days 30, 60 and 90 in simulation.
22

---

Published as a conference paper at ICLR 2026
C ADDITIONAL RESULTS FOR REAL DATA ANALYSIS
C.I AMP TRIAL DATA
Table presents summary statistics for bascline covariates and outcomes in the AMP trial data, stratified by region and treatment group. Comparing the treatment groups- both overall and within each region-wee we observe that the treated group consistently shows a lower average event proportion. Additionally, some covariates appear to shift across regions; for example, among treated participants, the standardized risk scores exhibit notably different means when comparing SA to BP and US.
Treated (bnAb) group
Total
SA
OA
BP
US
3, 076)
(n 679)
(n = 608)
(n 846)
(n 943)
Age (year) at baseline
25.9 (4.60)
27.0 (5.19)
25.4 (4.59)
25.1 (3.70)
26.2 (4.68)
Standardized risk score
0.0 (1.00)
+0.01 (1.00)
0.02 (1.00)
0.76 (0.67)
-0.68 (0.71)
Weight baseline (kg)
72.8 (15.64)
68.8 (14.24)
65.2 (12.63)
70.9 (12.42)
82.3 (16.43)
HTV diagnosis by week-80
107 (3.48%)
27 (3.98%)
20 (3.29%)
46 (5.44%)
14 (1.49%)
Control (placebo) group
Total
SA
OA
(n = 1, 535)
(n = 3410)
(n = 297)
(n = 428)
(n = 470)
Age (ycar) at baseline
25.9 (4.72)
26.6 (5.28)
25.4 (4.78)
25.2 (3.94)
26.1 (3.79)
Standardized risk score
0.0 (1.00)
0.02 (0.92)
-0,02 (0.98)
0.75 (0.67)
-0.68 (0.73)
Weight at baseline (kg)
72.5 (16.35)
67.6 (14.77)
65.1 (13.64)
71.1 (12.84)
81.8 (17.5)
HIV diagnosis by week-80 67 (4.36%)
16 (4.71%)
13 (4.38%)
29 (6.78%) (1.91%)
Table I: Summary statistics of AMP trial data by treatment group and region. The standardized risk score is baseline score built by machine learning models (Corey et al. (02TT) that is predictive to the time-to-event outcome. Age, standardized risk score and weight are summarized by mean (standard deviation), while the HIV diagnosis by week-80 is summarized by count (percentage).
In Figure 12
we plot the region-specific survival curves of all the 4 regions we considered (SA, OA, BP and US) for direct comparison on region heterogeneity, using their target-site-only (TGT) estimators, to showcase the heterogeneous effects of the bnAb antibody treatment on different target populations.
SA 15 nuth Africa)
OA (Other Sub-Saharan African Countries)
Survia Prbbbaity
2.50
Group
Group
2.40
Conkel Inxed
Conaal
Troktd
2.54
2.22
0.32
(day)
Iday!
BP (Brazil or Perul
US (The United States or Switzerfand)
1:50
Sunkel Prdobiiyy
Sunvall Preboblity
3.48
0.40
Group
Group
Contrl
Contal
Traaad
Trasted
Time (day)
Time Iday)
Figure 12: Estimated region-specific survival curves of the HVTN 704/HPTN 085 and HVTN 703/HPTN 081 trials. SA (our target region in the main text) and OA exhibit relatively similar curves, indicating less heterogeneity of these two regions. In contrast, both BP and US regions show significant differences to SA, which also confirms why they often have small or zero federated weights in Panel (B) of Figure in the main text. The BP and US also show a substantial difference on their curves.
23

---

Published as a conference paper at ICLR 2026
OA (Other Sub-Saharan African Countries)
META-IVW
Control proup
1.00
Suvvall Survhal Pubaesby
Sunial Pobabalty
Group
Group
0.88
Twore
Texkd
0.22
200
402
800
430
Time (day)
Time
Troated group
T0T
IWW
POOL
Tagic
Thes (day)
Estimated survival curves
Federated weights
BP (Brazil or Peru)
FED
META-IVW
Gontrol group
1.50
SUrVA Picbatiity
Survhel Provabiity
2.00
Group
8.00
Group
2.08
Contrll
8.00
Control
Teotee
0.04
0:02
Thme May
Time
200
Time (cay)
600
Therled group
IWM
POOI
Region
Thredeed
Timee 1Ay
Federated weights
Estimated survival curves
US (The United States or Switzerland)
FED
META-IVW
Control
1.00
Regior
Sarvted Probebiity
0.0
Group
Group
666
Cantel
Trated
Teaadd
0.34
0.80
0.39
0
200
400
500
100
400
600
Time (day)
Tine (dag)
Thested group
PWW
POOL
1:20
Creup
Thr (hre
Estimated survival curves
Federated weights
Figure 13: Additional data analysis results when treating the other three regions (OA, BP and US) as the target site. Time-specific federated weights with locally weighted smoothing (only a representation tool; Cleveland & Devlin (1888).
24

---

Published as a conference paper at ICLR 2026
Furthermore, in Figure 13 weights- using three regions other than SA as the target population. For the federated weights, similar to Figure in the main text, we applied locally weighted regression (Cleveland & Devlin 1988) to smooth the observed weights over the study period, providing a clearer visualization of temporal trends in this specific example.
present the souults-includingg survival curve estimations and federated
From Figure [3 we observe that for each region, the FED method yields results similar to the TGT estimator, while also recovering some interval estimations at earlier time points. This finding is consistent with the observations made in Figure 4 In contrast, the IVW and POOL methods deviate noticeably from the TGT and FED results especially for the BP and US regions indicating potential biases introduced by site heterogeneity.
Finally, regarding federated weights, the results for the OA region resemble those of SA in Figure However, for the BP and US regions, the federated weights are nearly for the target site and 0 for all other sites. This pattern suggests that when targeting the survival curves of BP or US, other sites contribute substantial biases-ad observation that corroborates our findings in Figure T2
C.2 "FLCHAIN" DATASET FROM PACKAGE SURVIVAL
The "flchain" dataset, obtained from the Mayo Clinic Study of Serum Free Light Chain (FLC) and Mortality, comprises data on 7,874 individuals followed between 1995 and 2009 to investigate the prognostic value of serum free light chains for survival (Dispenzieri al. 2012 Kyle et al. 2006). This dataset is freely available
package survival.
This dataset does not contain natural treatment variable, but to illustrate and extend the use of our framework, we investigate the sex difference in mortality. Since sex (female vs. male) is assigned birth, it be viewed as "treatment" variable for methodological purposes, as precedes the occurrence of any outcomes. While not manipulable in the conventional sense, causal inference methods allow us frame sex as an exposure to quantify disparities in survival outcomes, rather than as an intervention subject to policy or clinical decision-making Similar approaches have been employed to assess disparities associated with mon-manipulable variables such as race (Li & 2023
Liu et al. 2025 2024b, Matsouaka et al. 2025 Li et al. 2025).
Male
Total
Group A (n 972)
Group
Group C (n 1, 123)
(n 3, 524)
(n 1, 429)
Age (year) at baseline
63.1 (9.62)
60.1 (7.80)
62.6 (9.25)
66.4 (10.5)
MGUS
0.01 (0.11)
0.04 (0.20)
0.00 (0.05)
0.00 (0.00)
Sample year
1996.9 (1.84)
1996.7 (1.72)
1996.9 (1.87)
1996.9 (1.90)
Concentration of x light chain
1.5 (1.01)
0.9 (0.34)
1.4 (0.45)
2,2 (1.44)
Concentration of light chain
1.8 (1.19)
1.1 (0.35)
1.6 (0.47)
2.5 (1.77)
Mortality
1,004 (28.5%)
159 (16.4%)
372 (26.0%)
473 (42.1%)
Female
Total
Group A (n = 1999
Group B (n= 1,17)
Group C (n = 1, 180)
(n. = 4, 350)
Age (year) at baseline
65.2 (11.01)
62.2 (9.57)
65.0 (10.8)
69.1 (11.8)
MGUS
0.02 (0.
0.05 (0.21)
0.00 (0.05)
0.0 (0.00)
Sample year
1996.7 (1.70)
1996.6 (1.55)
1996.7 (1.68)
1996.9 (1.87)
Concentration of light chain
1.4 (0.78)
0.9 (0.34)
1.3 (0.43)
2.1 (1.03)
Concentration of light chain
1.6 (0.88)
1.1 (0.35)
1.6 (0.46)
2,4 (1.22)
Mortality 1,165 (26.8%)
231 (16.5%)
455 (25.7%)
479 (40.6%)
Table 2: Summary statisties of "fichain" data by sex group and the site variable we defined. All bascline covariates are summarized by mean (standard deviation), while the mortality is summarized by count (percentage).
25

---

Published as a conference paper at ICLR 2026
We include age, the presence of monoclonal gammopathy of undetermined significance (MGUS) and sample year as baseline covariates for nuisance models. The primary outcome consists of follow-up time in days and an event indicator for all-causes death (mortality).
categorical variable (fic. gzp, taking values 1, ..,.... 10) related to x and concentration levels is available in the data. We construct the "site" variable (R in our notation) based fic.grp grp as follows: (i) Group for flc.grp E (1; (ii) Group B for flc. grp E {3, 4, 6, 9): and (iii) Group C for gic E {7,8, 10). Several categories were merged in this way ensure sufficient sample size within each group, allowing 5-fold cross-fitting to train different nuisance functions reliably.
addition, we allow the groups share nearby values of fic. grp (e.g.. 5 in Group A, 6 in Group B, and 7 in Group C) so that each site retains comparuble information, enabling borrowing across groups. We emphasize that this grouping method is adopted solely for illustrative purposes in demonstrating our framework.
Table 2alle2preseee presents the summary statisties of baseline covariates and mortality for the "flchain" data. Across Groups A, B, and C, we observe clear covariate shifts, accompanied by differences in the marginal death rates. In contrast, when comparing the two "treatment" (sex) groups, the distributions of baseline covariates and mortality appear overall similar.
We analyzed the sex-specific survival curves over the first years for the three groups Figure 14 We used 5-fold cross-fittingg and estimated conditional survival for both event and censoring processes by an ensemble of Kaplan-Meics Cox regression and survival random forest models via the survSuperLearne (used in federated method) models were fitted by the ensemble logistic regression and LASSO using the Super Learner (van der Laan et al. 2007).
package (Westling et al. 2024). The propensity
and density ratio
Overall, the FED method yields point estimates that closely track those of the TGT estimator, while producing slightly narrower confidence bands. By calculations, the efficiency gain (by estimated standard стrоr of FED to that of TGT) can achieve 3%-10% consistent with the findings from both our simulation studies and the AMP trial data. The IVW and POOL estimators exhibit noticeably different survival curve patterns relative to TGT and FED when Groups A and C are regarded as targets, suggesting potential biases. The META-IVW method yields similar but slightly different curves compared to TGT and FED.
26

---

Published as a conference paper at ICLR 2026
Group A
FED
META-IVW
Famria
1.0
Buvvel Probabiity
Sex
0.8
Pemalo
0//
Mrste
0.7
0.5
0.6
1000
680
1.000
Time (day)
2000
3000
1000
2000 Time (dba)
(009
Mide
FWW
POOL
Tiao (k4g)
Traks
Time pkyy
Estimated survveatcuvves
Federated weights
Group B
FED
META-IVW
Female
1.0
Group
Sumiee Protessity
0.0
Sex
Sex
0.8
Fumull
09
Famull
0.7
0.
0.6
1000
500
2000
1000
2000 Time (day)
5000
2000 (day))
Mali
TGT
NWV
POOL
Time (1:00
Time 30mm
Time 1MV
(day)
Federated weights
Estimated survival curves
Group C
FED
META-IVW
Famale
Goop
Procaberty
Barveel Probabity
Sox
Sec
3.6
Femulo
ea
Famale
0.7
1000
2000
3000
Time HMM
Tima (pwy)
Time (daa))
Male
Goo
TGT
IVM
PODK
1503
1.50
2000
300B
100
300
Tine 1891))
Estimated survival curves
Federated weights
Figure 14: "flchain" data analysis results. Estimated sex-specific survival curves and federated weights for sites (Groups A, B and C defined by fl grp variable) Time-specific federated weights with locally weighted smoothing (only a representation tool; CCeveland & Devlin (1988).
27

---

Published as a conference paper at ICLR 2026
D IMPLEMENTATION DETAILS
In the following Algorithm we detail the double machine learning procedure (Wang et al. 2025 Chernozhukov et al. 2018) for fitting and predicting nuisance functions in Algorithm
Algorithm Double/debiased machine learning algorithm for nuisance function estimations and influence function calculations in Algorithm 1 at given time point and treatment.
I: Input: Observed
(X; A,, Y, A,, R.), right-censored data O {0: K. K_ - -13 and O* represents 1,...,mi ,,,) - O U O² .... OK-1 where R; E {0,1,...,
the data for site R k; Given treatment group A and specific time point I; The num-
ber of disjoint folds into which the data are split, M, where M E (2.3.... nn/ with min 11...... ,nK-1)-
2: Output: Estimated influence functions for each individual.
3: Partition O into M approximately equal-sized, disjoint validation folds vp.... Vs allowing size difference of at most + between folds.
4: for m , M do
5: Define the training set T = OU pu
mi
6: Fit nuisance functions 5 Go. TO on T using some methods ensemble from suzvSuperLearner and LuperLearner;
7: Predict nuisance functions Vo S Go and Am
8: end for
9: Train model of 5 by the entire data of the target site Oº, denoted 50,full using chosen methods ensemble from survSuperLearrer.
for k l..... K do
11: Partition O into M approximately equal-sized, disjoint validation folds Vt,... Vk allowing size difference of most +1 between folds.
12:
for m = ,..... M do
Define the training set T = Ok) :
14:
Fit the density ratio w using only covariate data of T U.K or by just passing through some coarsening level summary statistics; Fit nuisance functions G n on T using chosen methods ensembles from sur survSuperLearne and SuperLearner;
15:
Predict above nuisance functions on y* as Gk @4.0 and ** Predict nuisance function S DE using the pre-trained S0,full model, and denote the predicted value by Sk
17:
18: end for
Aggregate all predicted nuisance functions over M folds as St Gh. @1.0 and # end for
20:
Return: The estimated EIFs, by plugging-in their predicted nuisance function values,
(O:P) = k.kk0 SO 5o 2 a k)),) for all & E
Remark D.1. To ensure the monotonicity of the estimated survival curves, we invoke isotonic regression techniques (Westling al. 2020), which enforce con-increasing constraint on the site- specific survival and censoring estimates S and G for E 10, I,.., ....K K 1}. thereby maintaining their logical consistency over time.
E TECHNICAL PROOFS
We begin by recalling notation for probability, expectation, and variance. Throughout, denotes the true probability under the data geaneeeenrtingg distribution, In the empirical average, and fo the evaluation with estimated nuisance functions introduced in the main text). addition, E denotes the population expectation, V the population variance, and Cov the population covariance.
We further adopt the following notation throughout this appendix: (i) Po denotes general probability limit, and the nuisance functions under Po are denoted with subscript co, e.g., So for the limit of
So (ii) P means the corresponding nuisance functions replaced by their estimates, and ie may
28

---

Published as a conference paper at ICLR 2026
converge general limit Poc: (iii) Pw [[()) 1i 1iml-IEeeza average on the m-th validation set Vm by cross-
EiEI,, f(O,) to denote the empirical cross-fittings fitting, m =1,. ,,...,
Furthermore, we distinguish notation P(f) and Ep(s): P(f) = P f(O)dP denotes an integral over new observation O ~ IP. treating f. which possibly depends on training data (c.g., some estimated parameters for nuisance functions), as fixed. In contrast, Ef((f is the usual mathematical expectation of random variable/element f under distribution P. fixed value without randomness.
E.1
THEORY OF THE LOCAL ESTIMATOR
E.1.1 PROOF OF THEOREM 2.5
Recall that a mean zero, finite variance function sP (O; P) called an function of the target estimand (a functional) (t, a) = (t,, a; PP) at P if, for any one-dimensional regular parametric
submodel (P, :c E [0,1)] through P = Po.
a
0 (t. P. de
= E (1;)(())))))
c=0
where ((0) is the score function of the submodel at e = 0 (i.e., typically, i(O) 0 log (p. (O))
(O))/a2 / l)=),, where (-) denotes the probability density (likelihood) function under submodel F. (Bickel et al. 1993).
Recall the site-k CCOD assumption made in Theorem 2.5
(t a, (t a, X) almost surely.
To find the EIF, we begin by writing the following equation:
② 00 a)
0 = de
ESS (1 a. X) R -2 0}
de
k=0
0
Bt(S (t a. X)
(t) R=o
R = 0} + E
5 a(,))
u(dx)
R= 0
de
c=0
[t( a. X) (t, (t.,a)(x
R 03 E
/
2
0
st(()) x.
p(dx)
R= 0 J
de
c=0
where pd-) denotes the distribution of induced by and, for any sets of variables V and W, Mli denotes the conditional score function of V given W. v..... typically å (pe(V) WW)/0e le=o-note note that such scores always satisfy (fv w W) = 0 (Bickel et 1993)
For the derivative of Sh with respect to e, by the chain rule, we decompose it as (aS /8A") x (de)/ For the first part as* / we leverage Theorem 8 in Gill & Johansen 1990). Specifically. the mapping H + 5 (t; H) == No. (1 + H(du)} is Hadamard differentiable at relative to the supremum norm with derivative
a+ S (t:HH) H) S'
(u-: a(du). St(u; H)
Thus, by letting (t) = AE I x) and the chain rule, the integrand in the second term becomes
0 [ (1 - Ad a, x)}
=-s a S/' st a,,)) i N (du a, x)
de
St x) de
(0.4)
l==0
c=0
Furthermore, recall that
( a, X) = 1-¹ [du a. X)
Duu a, X)
where (/ a, X) P(Y X, R = k). Hence,
1 a, X, k) and 1* a, X) = PPY 21 A=
2 Nk Nt. (du
a, x) Ie=0
2
Du x, le=o Nt. a, x)
A (duu a, dc
de
3e
( x,
Dd l
1mm0
29

---

Published as a conference paper at ICLR 2026
In addition,
① (du (du a,x)
No. de
P. (Y S - k)
de
c=0
k=0
/| TTy S u, = 1)P, (dy, d8 a. x, k:)
Ōc
1c=0
/] I[ u, d 1)ē(u.8 P dõ a, k))
J 1(6 - 1),(u, 8 x,,)) a, a,
and
8 PY u A a, X = x, R R:)
(u a,x)
de
Ōe
le=0
l==0
0
/| Ily 2 u)l (dy, d8 a, x, k:)
Ōe
c==
// 1(y S u)(ly, 5 a, x)) d5 a, x)).
We can then express the integrand of 3 as
0
// [ (1
(d a, x))
dc.
(0.1)
10=0
-/ll -I(y 11,0 (t((t x) a, x)
8 a, d5 |aa x, k))(dx)
(h(( a, x) xx
(t a, x) (u- x.
I(u (tustusy)
5h a. xA (u x)
x Eyy, 0' a, (, dō a, x, N (du a, xx))
-/ʃ T(y $1,0 1) (t a, x. (y
a, x.x) Ö(y. 5 x, k)) d5 a, x, (dx
S* D (y) x)
+ +]]l Sh((1 (xx) 10 AAy (u xx
N2 Nf (des xx
5t a, D) x)2
x x(ly.0 81 a, (x))))ddy... d6 a, a. (()
E s*((t) a, x) T(A a)
(H, T(Y (1,= (Y- (xX))
(a
( (Y x.
x :((Y A a. X, R X, 1b)
where
(t,,,,)) = (-' st( a, N/ ax)))
sk(u (+ (u x,x)²
Now, we note that
IIY <1,-= - 1) AAX)
{"
** a. x) (dy |
E
A= x, R k
5*(Y | (Y (,X)
Sh a. x. [x
30

---

Published as a conference paper at ICLR 2026
and
E(H A,X) A= x x, R= k3
(u xx
I(u S u) 5h(u1 a, xx (u a, x)2
Pidy a. k,)
/ IP u A= = x, R k) 5H [, (f a.
P(dy a,
(h a, (u a,
(* Nf x, (h (x xx
Therefore,
E ( 1(Y St
A,X, R k -0
S* A, (Y A,
almost surely. By properties of score functions and the tower property, the above implies that
0
π (1
(du a, {()x))
0e
(0.g)
c=0
I(R k) I(A = a)
E
5t a, X)
P(R= k X) n X)
{ T(Y S 1, A (Y 1,X)
5* A. D A,
Combining these results with the facts that Nf (du a. //x (u a, A a, x) and
(* x,x) = (u (M a, x), we can rewrite
3 at the beginning as follows:
①
p a)
de
c=0
I(R = 0)
I(R - 0)
I(R = k:)
=
[S (1 a, X) - (l)))))((()
F. s* a, X)
P(R 0)
P(R = 0)
P(R = k. X)
I(A = a)
1(Y <t,
tty
Adu x)
i((0) x]]
7a X)
(K(()) X)G (v) x)
J
Sh X)G* a, X)
I(R = 0)
I(R = k) P(R = X)
=
(1*(1 x) - 0 (1, (1))))((
E
sh(L a. X)
P(R = 0)
P(R 0) P(R = X)
T(A- a)
{ I(Y 1t,4=1)
A a x,
[((0)]
(aaa X)
(s(yy X)O*(V x)
St X) a, X)
Therefore, an EIF of 60 (t(t a) at P is found as
Pt.a .
(O:P) P)
I(R = 0)
(5 a, X)- (t, k) 0 X)
S*t x)
P(R- = 0)
= 0)P(R = X)
a)
I(Y St,a=1) 1)
MAY
A (du a, x
(a X)
5" a. XY a. X)
5t a, X)G (u a. X
Observe that, by Bayes's rule,
P(R 0 X) P(R l X) P R= P(R k))
P(X
R = 0)
P(R 1R=00 0)
(x)
where (X)) is covariates density ratio function. We then find that the EIF form in Theorem 2.5
+4.0 Pr. (O:P) = I(R = 0)
II = k)
P(R 0) (5 a. X) (t.
P(R = k) w 4.0 (X)s (t a, X)
1(A= a) x
I(Y St, 1)
AAY
A a, X)
*a x)
S a. XY a, x)
5t x* (u X
31

---

Published as a conference paper at ICLR 2026
E.1.2 REGULARITY CONDITIONS FOR THEOREM 2.6
We now state regularity conditions for Theorem 2.6 For site R = k, we denote x², G*, who A and 5th the truths of nuisance functions. We use "0 who G A and S to denote some general probability limits for nuisance function estimators.
Condition E.1. There exist The t G A and S such that
1
(a) max IP
tp 0;
A (a
X)
(a X)
(b) max 17
k.0 (X)
k.0 (X)]
-pp 0;
(c) inax
sup mE(0.r)
tp 0,
m
G (u a, X)
Gh o, X)
S (t
a, x.
S a, X)
(d) max IP
sup vE
-pp O.
Sk (u
0, X)
S a, X)
Condition E.2. There exists an r E (0, 00) such that for Pallmost all x, * (a x) 2 1/1, * (a
x) 2 1/n, 1/n S (x) S. n. 1/1 S w S n, Gh (1 a, 2 1/n, and G (1 a, x) 2 1/n
with probability tending to
Condition E.3. Define
"m (a X) (a x)H a, a,X) S* a, x)1
n,,,0,1 max
m
[60 (X) - w. :x ( n n,t,a,2 = Iax m
a, X) - St a. x)}
and
(t a, x) 1/' G a X)
}
-1) (du | a, X)
T n,,,,,33
max
3k m
a, X)
Sk
m
Then, it holds that rk n,t.a,1 Op (1122), r n,,,, (121) and r
n,t,,,33 (1(221
Next, to prove Theorem 2.6 we first introduce some useful results and lemmata in the next section.
E.1. USEFUL LEMMATA FOR THE LOCAL ESTIMATOR
We start by examining the difference (1, a) (t, Recall Im is the empirical distribution corresponding to the m-th validation set V. from the entire data O, and denote Gm the corresponding empirical process. A result exactly following WWstling al. (2024) is that
A.D a. (t, (t, = PrIpo
+
Σ M
1
M
1/2
Gm p
m=1
Mnm
00 (t,aa))
P
Praa
(4)
M
n
We then establish the (IP) norm distance (bound) between the estimated EIF and its underlying limit for the local estimator by the following lemma.
Lemma E.1. Under Condition m, n, t and a,
E.2 there exists a universal constant C = C(()) such that for each k.
6
Pi
R.O poc.taa
2 < C(n) Σ
A m.n.mt
j=1
32

---

Published as a conference paper at ICLR 2026
where
A M,,,m, P
pm (R 0)
P(R = 0)
A 2,n,m,L,a
Pm ( R = k))
IP(R = k))
4 m,n,m.t.a
[ (al X)
w x)]
1
A M,,,a
Fm (a X
T (a x)
A A.n.m,r.a
P
sup ME(0.0]
G (u a, X)
G Gh (u 0, x)
5 (t a, X)
S (t a,
A A.a.m.L
P
sup [E|00,]
Sk (u
a, a,X)
8 (u
X
Proof. We first denote
B*(mm) I(A a) **(a X)
(t((t) x,
1(Y S A 1)
Lo
MAY
A | a, X)
S*(Y a, XY a, x)
S a, X) (u a, X)
C* C*(Ym) B* (Vmn ) Then, first have the following decomposition:
Σ
Uk
j..m,t
5=1
where
T(R 0)
T(R 0)
U L.n.m..a
30 a,,),
P 0)
P(R 0)
I(R 0)
Uk
[30 (1 (4 a,,))
So (1 (a,x)
2,n.m.t.a
P(R = 0)
U. U,n.m.tta
I(R k)
I(R kb)
O+ (Vm),
Im R kt)
P(R=k) k)
Uk U.n.m...a
I(R k)
[Ok (Vm. C+ (Vir))
P(R = k:)
Now, for Uk
t,a. we further decompose as
I(R k) PPR = k)
U U.m.m.
Vk
j=1
where
Vk
L.m.tt.. = B2k (Vm)
[1) (X) - w 4.0 (X)
Vi
2,n.m,taa = (X)
Db (Vm) - B& (Vm)
The expression of D (Vm) - B& (Vm) is exactly the same as the Lemma 3 in Westling et al. (2024), while we only need replace the corresponding nuisance functions by the site-k ver-
k. 12 S sion here, so the detail omitted. By the triangle inequality, we have Plpta po.
{E ( /2 Therefore, under Assumption 2.3 and Condition E.2 there ex- ists universal constant C C(r) such that the result in the statement holds.
33

---

Published as a conference paper at ICLR 2026
Furthermore, we need to bound the empirical process term G This is formally shown below in Lemma E.2
P Pnmmtta
by op(nn 1/22).
poo,f,a
Lemma E.2. If Conditions E.
F.2 hold, M- 2m=1 n
Mri
1/2 2mm
Pn.m.t
2k.0
f00,tra0)
k.n
Proof.
We follow notation Lemma E.1 First, we note that
Mm
1/2
M(um nn/M + n/M)2// M Mln n/MP4/2 + Mn/MMu/2
M
M
n
n
for since Mmm n/Ml S by assumption on fmm Then, we have that M
M 1/2
sup
Gm
-k.0
Pn.m.ca
8.0 Poc,ta
n
aE[0.4]
M
S
O(n 1/2
sup
G
2ko Pn,mea Poo.t,
M
m
since = (())
Therefore, we turn to show M- Σ M argument, we write
m==
Gm
k,0 Pn.m.t,
k.0 Poe,rra
=p((1. Using conditional
G
Pn,m.t,a
4.0 Po0,1,
E
G
Pn,mata
Poo.t.a
Tm
where Tm = vm is the m-th training set. Note that the randomness in the inner expectation of the nigh--hand-side above, by conditioning the training set, is only induced from G by averaging over the observations on the validation set. Therefore.
E
E
- Pn,m,ta
0.O Po0.1,
Tm =P
GT (Fr,m,
K.O (ec,,t,))
Defining FI. t,m,t,a as the singleton class of functions P
Poo,ta we further have
k.0
P G (ptt (Pn,m,t
(oo,tta))
P sup fE m.m.t.
G (f]
n.m.t.a t.a is for all E, k.0 so the uniform entropy integral (1, J amm is relative the natural envelope Pn,m,,, Po,, Therefore, there is universal constant C' such that
By Theorem 2.1.14 in Van der Vaart & Wellner 1996). the covering number of Fk.0
k
1/2 )2)
sup TeRao
G (0)
< C²
K.D
S
C"
Σ A Ajn.m,,ta
following M-
definition of Aj,mm,t terms in Lemma Poc,ria is bounded up to C" E 5= IE [maax, ( (Aj.m,m...))
E.
So that
EM for some constant zero, M-
1m=1
m.m.t. a
C" is straightforward that by Conditions E. and B.2 this upper bound tends to M
k.0 Pn.m.t.
4.0 Poo,t. (1).
m=
Finally, the only difference we have not characterized in 4 is P Pl show it below.
(O;P()) - (1, a), which we
Lemma E.3. Consider some general nuisance functions under Pos, denoted by 5o,, Sk Gh a [O: (t, a) equals
and k.0 (equals / if k: = 0). Then, PP k.0
(X)
S*
(y
a, x)
P(R = sk
X)
5k (y
a, X)
h.0
(y/
a. x T (a X)
} (A AF) a, x,
x
(1)
X
aa
X
34

---

Published as a conference paper at ICLR 2026
Proof. By direct P(et
(t, a) equals
I(R = 0)
(x) b. (X) (1 a, X) x X)
E
P (S (1 a. X) - ( a a, x))
P(R k)) w
T (a X)
Sh (y- 0 (x a. X.X)
(A A (dy a, x,x)
Sk a, X a,
qº (X)
(X) w
n" (a X) xx (a X)
E
PPR = (S a. X) (t a, x,
s (t a, X)
P(R = 0) (x)
x 1/ 5 (v 0, x G* (yl a.
(A A (dy a, x)
Sh (v G (y/
a. x)
In the second "E" after "=", we used the following relationship:
P(R 0)
P(R = k)
by Bayes's rule. Furthermore, by Duhamel equation in Gill & Johansen (990)).
- (t)
= (X)
P(R 0) sk (t X. L/ 5h a, X)
8 (1 a, x)
w (*) (y/ a, x)m (al X)
who w (y a, X n (a x) 1 (A
-dy a, xx
(5)
E.1.4 PROOF OF THEOREM 2.6
By with + = nh, w = wk., G = Gk, and Sk = Skk
M
1/2
Oz.0 (1,0) - (1)), = Pnilettaa xk.0
M Mr
G
Pn,m,t.
k.0 92.0
M
122
M Mnm
P
a.00 Pro
(t.taa))
M
By Conditions
E.
and
E.2
the second summand on the right-hand-side is op (1/22) by Lemma E.2
By Lemma
E.3
P
0 (1, a) equals
(X)
//" 8 Sh a, a, X)
E
5 (1 a, X)
(RR = 0)
Sk (y X)
(X)G*(v (v) a, X) X)
- 1 ( ( Ah / a, xx]
w.o (X) Gh (v
a m (a X)
By Duhamel equation in Gill & Johansen 1990) and Condition E.3 we find that the above bias term can be bounded by ( (m..a.11 have
+ +rrk
r ) over m. Since M- MM 2m=1= n M S 2, we
M Σ
Mn
10.0 PtR
S 2n² frih + buk n.aa2 + rh, r,, = (n-nn-112)
M
m==1
by Condition
E.3 This established the pointwise RAL property: 0: (t, a) = 00 a) + (P((, =k.0
(n- Since
Pr.a uniformly bounded, ((((ttt))) oo and since IP tk.0 0, then
2/2p, ( - (1(p ())))))))))))))))))))))))
35

---

Published as a conference paper at ICLR 2026
Remark E.4 (Technical version of Theorem 2.8 If we only need the consistency of 6t (t, a), then
condition Th = n², whe0 wh: G G", and S SS can be replaced by the following
statement: For P-almost all X, there exist measurable sets St. G C [0,t] such that St U G [0. t
and (f a, X) = A (u a, X) for all u E S and G(u | a, X) G (u a X) for all u E g
In addition, if 8 is strict subset of [0, f). then (a X)
n (a X) and (X)
& (X)
as well. Then, 0% (1, a) is consistent if Conditions E.1 and E.2
hold.
To prove Remark E.4 we decompose the integral Su as Jst + +sst. where She is the complement
of set S and S C 9 by definition. Then, it is straightforward to verify that when the statement in Remark E.4 holds, the following integral
!'" Sh a, X) 5 (y
G* (y
a, x) & (a X) 1) (A Ady aa X)
a, xxxx(((e (a)
X)
a, x)
Sh (y-
X)
G (v a, x) (a X) -1] (1 -A a. x) 0,
G* (y) a, xt (a X)
+ J0z)
st
5% (y)
a, X)
which further implies Pl = 0.
E.2 THEORY FOR THE FEDERATED ESTIMATOR
In this section, we present the properties of the federated estimator. Given that our proposed weights, Tr,a are both time- and treatment- -seenfff,,,,,iiic, we focus on the pointwise convergence properties.
Let the set of all source site indices be S = fl....,k. K 1}. We then define the oracle selection space for ni,a and the corresponding weight space as:
Sr (k E S (1, (t, a) = (t), and Ri = (Mra E RR-11 the = =0 & Sia),,
respectively.
The space Sy,, is both time- and treatment- varying, indicating that source site may not consistently be useful or unhelpful across different time points or treatments. However, offers the advantage of increased flexibility and adaptivity, allowing for more effective borrowing of information at different points along the survival functions. Based on the theory presented Section E. for k E Si the
site-specific estimator 0t.0(1,
(1, a) is consistent for (L,,, a) for any given E [0, T] and {0, 1).
We begin by assuming fixed nr.a Th. v/.a "))))))) We invoke Lemmata 4 and in Han et al. (002)), which state that the proposed adaptive estimation for v/. as shown in allows for (i) the recovery of the optimal vi.a by the estimator it. and (ii) the uncertainty induced by ok
Tha is negligible when estimating 8 a). We require regularity Conditions E. E.2 and E.3 for
the pointwise convergence result in Theorem 2.6 hold. Let us denote the federated estimator by plugging-in the fixed Mr,
11 tha (ttta))+ + E ni,b 15.0%.00 (t, (t) KES
(t, a: (,,a)))
RES
Recall that notation He, defined in T
I(Y =t,o=
MAY
A(du a, X)
He (O:S,) 5(Y a, X)G(Y a, X)
S(u a, X) a, x)
Let us then write
I[A = a)
(0) = St a, X)
Mr (O.S".A.G)))
Ta X)
I(A = a)
(0000(0)) = w 4 (X) S* a, x)
He, (O: A G*),
*ta x)
(0))(()) = (t a. X) - 00 (t,
36

---

Published as a conference paper at ICLR 2026
and nk E for k 0, 0.......A K
Then.
Tnadl (1, u; M1,0) ( (4)
- Σ Thsa
{ (t, - ( -0((ta))
+ Σ
Nh.a { { (p: - (t)
KES
KEB
Σ [RR 0) (0.)
no
KES
I(R, O)TE, (O.)
Σ
I(R, S*..0.00
140
KES Tkkk
KE
n
(0,) -0,0 P(R; 0)
Σ Tha
I(R =0)
12
KES
(0.) P(R;=0)
(0.) P(R, = k)
TR = 0)
Σ le
1 Σ n
IR, = k)mE
(6)
KES
RES t=
The asymptotic variance of Oleed (t, t, denote it Vi yeod mr.a) We highlight its dependence to the federated weights vector nh here because in the below (8) we consider an optimization program for deriving the weights based on
Mi.a) equals the variance of the influence function of 6 Let us
minimizing the (estimated) asymptotic variance.
Under the assumption of i.i.d. participants within each site, we have
((gu.12) (O.) R. 0)
yfedd
1
P(R; =
kES
((C0,22) R4 = 0}
Σ
Tha
P(R: = 0)
WES
R = 0
2
1 Σ Thaa
Σ Tha
KES
WES
R = k}
Σ (nita) keS
P(R: k)
With appropriate boundedness conditions on conditional variance and covariance terms above, ffd < 00 (see Lemma E.6 Consequently, the asymptotic distribution of glod (t, a: ni,a) given by
Vn ( (tod a; n..)) 8t a)} - (O, Vie))
Remark E,5. Based on the derivations in (6) and 77. an infiuence-funnciio--aesd asymptotic variance estimator of (f,d(t, a) (ffe in Theorem 2.10 is obtained by replacing the population proportions, variances, and covariances in ? with their sample (empirical) counterparts and plugging in the estimated weight vector Me...
We further define the optimal adaptive weights tho as follows:
Mia arg min Vood (T..a)
(8)
n vkggs
We adapt two lemmata from Han et al. 2025 for recovering the optimal weights th. with negligible uncertainty for estimating (t, a) we estimate 77, using (2), akin to adaptive Lasso (Zou 2006
Fan et al. 2024).
37

---

Published as a conference paper at ICLR 2026
E.3 along with the following mild conditions on covariates support and covariances: (i) The covariates X and density ratio wx.((()) (X) are in compact sets X [- B\" and (x)) E [-B, B) for all l -.....K KK 1 with probability 1; and (it) The variance of (g [e, M, and the variance-covariance matrix
Lemma E.6 (adapted from Lemma 4 in Han al. (025)). )). Under Conditions E.1
y ((((0.)) <0.22) ((())))) | R = 0]
has eigenvalues in (e, for some positive constants and B. Then,
holds that
lim PM E <R-ia)= =1, 7--20
17, Tr.ll = Op(nn1)/2),
for all (1, E [0, x 10.
Lemma E.7 (adapted from Lemma in Han et al. 2025)). ). Under conditions in Lemma E.6.
vii (Thd a; (i, - 0 (1,a)) -a (1, (i (Ar,a))
for all at, E [0, {0, 13.
The consistency of Diad = (rr,) follows when we can effectively approximate ()r,)) with Fied Thus,
1,a'
(idd /n/Vod ta
n/
Dio
- (t, -a 10,
We now analyze the efficiency gain resulting from the federation process. The estimator relies only on the target data is denoted as 6%, (t, a) = (t, a; mf.a), where v/f. assigns all weights to the target and none the source. In contrast, the estimator that leverages the proposed adaptive ensemble approach denoted as grad (t, a; a) Here Mr, can recover the optimal weights th that are associated with the minimum asymptotic variance. Consequently, the variance of 0lr a; Mr.a) is no larger than that of the estimator relying solely the target data since n/.a generally not the variance minimizer.
To establish that the asymptotic variance of Ōf:ed (1, Tt.a) is strictly smaller than that of the estimator
based solely on the target data 6% (t, a), we adopt Proposition in Han et al. (0255) with a modified informative source condition (modified Assumption 3(b) in Han et al. (2025)
Specifically, for each source site Si we define (te a; mf...) federated estimator where vi.. is the optimal ensemble weight of site s if we only consider target site and this source site for the federation. Then, the modified informative source condition given as
Cov [Vnêh (t. a), /n (aic (t, a: nf.))
Chit. a) 2E,
for some > O. where @rod (t, ai.a) (t, a) can be expressed
0.rl ai (i.a) 82
= (lod (t, 0; niaa) - (t, a))
- -
I(R 0) ni,a) (O,) - (O,)
1 " I(R, 0)ni (O.)
P(R: 0)
n
(R: = 0)
TT = s)1i.a (o))
TI (R: = 0) (0.(2) (O.) - (O.)
P(R = =)
P(R: 0)
i=1
(O,) Σ
n
(O,)
I(R; = 0)Ti. Pi:R: = 0)
I(R, = s)Mi.o
TP R, = 8)
Therefore, straightforward to see that the modified condition can be achieved if n/,, > 0.
38


Reviews:
[{"review_id": "rErAnz9Z68", "reviewer": "Reviewer_YJNV", "summary": "A federated learning framework for survival analysis is a missing topic. This paper presents a federated learning framework for causal survival analysis that aims to estimate treatment-specific survival functions across multiple heterogeneous datasets while respecting privacy constraints. The formulation mostly follows previous papers where we have to adjust the distribution shift between the source and the target dataset.", "questions": "None", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper does a good job transitioning the federated causal inference problems to this setting. They provide a complete and clean framework. I list the strengths below.\n\n1. Timely and important setting\n2. Clear formulation \n3. Doubly robust estimator\n4. Accounting for biased source states\n5. Complete simulation and real-world studies", "weaknesses": "I believe that there are some important questions that the paper should answer:\n\n1. Comparison with standard setting. Consider a degenerated case in survival analysis, where we do not have censoring and only have time step 1. This becomes the non-survival setting. In this case, does the estimator proposed degenerated to estimators the same with Han et al.? Answering this question can help the reader understand any tricky or nontrivial part in extending the estimator to survival analysis.\n\n2. Estimating the distribution shift is the core problem in causal inference. It would be interesting to see discussions on different strategies on density ratio analysis and their effect on the final result.\n\n3. It would be interesting to see the extension to better estimators, for example, Guo et al. (2024) proposes an estimator that is more efficient than meta-analysis styled estimators.\n\nGuo, Tianyu, Sai Praneeth Karimireddy, and Michael I. Jordan. \"Collaborative heterogeneous causal inference beyond meta-analysis.\" arXiv preprint arXiv:2404.15746 (2024).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "G8TEnnjVQN", "reviewer": "Reviewer_shKn", "summary": "The paper proposes a federated learning strategy to estimate population-level causal effects in survival outcomes, combining the data of multiple sources, under the violation of the _common conditional outcome distribution_ (CCOD) assumption.\nThe strategy is composed by several steps:\n1) Each different site estimate the nuisance functions for its own data: _survival function ($S$), cumulative hazard function ($\\Lambda$), propensity score ($\\pi$), the conditional survival function of censoring ($G$), etc._.\n2) Also, global parameters have to be estimated from coarse statistics: $w^{k,0} = \\frac{P(X|R=0)}{P(X|R=k)}$, $P(R=k)$, etc.\n3) In each site, the target parameter, i.e., the treatment-specific survival function has a closed-form solution, given the nuisance functions and the coarse statistics.\n4) Weights for collaborative learning are computed for each time-point and each site, aligning the EIF with the target distribution, and introducing an $\\ell_1$ loss to remove the sites that do not contribute positively to the estimation of the target parameter in the target site. This problem is solved by optimization.\n5) A federated estimator that combines the sites is the final estimation of the targed parameters.\nThe method has been validated over 1) synthetic data in which the distribution shifts are varied  on demand, and 2) real AMP and _flchain_ data (although _flchain_  is defered to the appendix). \n\nResults show better RMSE and coverage percentage compared with Pooling and IVW, especially when there exist covariate and outcome shift.", "questions": "> Minor concerns\n\n- In the introduction, it seems that no previous work on _federated causal survival analysis_ have been done (especially in `line 054`: 'these methods remain focused on single-study...'). I miss some references to Van der Laan collaborative learning [1, 2]. So I would add some information about this work and its (likely) follows ups. (disclaimer: I have nothing to do with those works.)\n\n- Can we get a explicit definition of what nuisance functions are? From the context, it can be understood that _survival function, hazard function, propensity score, etc._, are nuisance functions, but is not clear in the introduction.\n\n- Can authors provide more intuition about  Equation of line 128? E.g. what the role of $\\frac{I(R=0)}{P(R=0)}$ is? what the role of $\\frac{I(A=a)}{\\pi(a|X)}$ is? It would help to understand the equation. \n\n- What does the RAL property in Appendix E.1.2 mean? It does  not seem to be defined.\n\n- In `line 430-431`, authors say thet TGT yields wider confidence intervals than FED, but I cannot see that in the Figure 4(A). In fact, what I observe is that FED, between aproximately day 80 and day 180, has wider intervals than TGT. Am I missing something?\n\n- Figure 4 (B) represents the weights given to each site in each timepoint. We can observe two curves, one very noisy, and another _filtered curve_. The _smoothing_ component is part of the approach or is it only a representation tool in the figure?\n\n- In general, I would recommend to add more intuitions about theoretical implications and, especially, under which assumptions the oracle-optimal weights are recovered. The pointers to the appendix are overwhelming, and more plain intuitions would be very helpful\n\n> Summary\n\nIn general, I would say that this is a good paper. It is based on other similar approaches that leverages _Efficient influence function theory_ and both the theoretical approach and the results are sound. However, I have several concerns, reflected in _Weaknesses_ and _Questions_ section that prevents me to give a higher score. I would be happy to raise my score if those concerns are solved.\n\n> References\n\n[1] van der Laan, M. J., & Gruber, S. (2010). Collaborative double robust targeted maximum likelihood estimation. The international journal of biostatistics, 6(1), 17.\n[2] Stitelman, O. M., & van der Laan, M. J. (2010). Collaborative targeted maximum likelihood for time to event data.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The motivation of the paper is clear and strong, the authors aim to give an unbiased estimators of population-level causal effects in survival outcome, given that the CCOD (common conditional outcome distribution) assumptions does not hold.\n\n- The theoretic contribution is based on _Efficient influence function_ theory, and, although I have not studiend in detail the proofs of Appendix E, they seem to be consistent and valid.\n\n- The results present clearly the improvements in comparison with pooling the data and using Inverse Variance Weighting (IVW). Both RMSE and CP show that TGT and FED are consistently better in synthetic data. The results in real data are not that conclusive, but still points in the right direction.", "weaknesses": "- My main concern relies on the data-adaptive weighting criterion. The parameters $\\eta$ are the coefficients of a linear regression (omitting $\\ell_1$) to fit the _local_ EIF ( $\\hat{\\varphi}_{t,a}^{*0}$ ), with the site specific EIFs. Therefore, if the local EIF is not well specified, the other sites may contribute negatively to the estimation. That is, the federated learning strategy can be harmful if the local-estimator is not well specified. Is not that exactly what federated learning tries to fight? The same thing happens with the $\\ell_1$ regularization: the optimization process will remove the $\\theta$ that are far from the local estimation, thus benefiting the errors in the local estimation to persist. Is there any explanation for this fact, or any way to fight this positive feedback?\n\n- I have marked the contribution as _fair_ because this is an adaptation of (Han, 2025) to time-to-event data. Although that does not prevent me for accepting the paper, I do not see the contribution as disruptive.\n\n- Results in real data are not conclusive. All the charts present similar results between TGT and FED. Coverage percentage is the most of the time equal, and metrics about RMSE statistical difference are not provided. This limits the interpretation and the strength of the results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "K0chUws0xV", "reviewer": "Reviewer_n3Hz", "summary": "This paper aims at introducing a new procedure to perform generalization of treatment-specific survival functions to a centralized target population, where the effect is learned of multiple decentralized sources, distinct from the target, and without exchanging raw individual data as in a federated constraint. The authors tackle the challenging setting where the source and target are heterogeneous on their joint distributions. The method proposed is 1. each site fits local nuisance models and EIF for pseudo outcomes at each treatment arm and time point reweighted by estimated density ratios to account for covariate shift relative to the target, 2. the EIFs and other summary statistics are sent to the server and aggregated with weights $\\eta^k$ which penalize sites according to their estimated discrepancy from the target's nuisance functions.", "questions": "**1. Conceptual positioning and relation to prior work**  \nIn what concrete ways does your approach differ from a meta-analysis of EIF-based local estimators, possibly preceded by a density-ratio reweighting step?  \nCould the proposed method be viewed as a *meta-analysis with adaptive weighting*, and if so, what theoretical or practical advantages does it offer?  \nMore broadly, how does the framework go beyond FACE [5]? Since the EIF-based structure and penalized weighting are similar, what are the novel theoretical or computational insights specific to survival analysis?  \nWhat new theoretical guarantees are provided beyond those of FACE and Westling et al. [1]?\n\n---\n\n**2. Methodology, penalization, and communication**  \nHow is the regularization parameter $\\lambda$ in the penalized objective cross-validated in practice?  \nIs cross-validation performed locally, centrally, or through additional communication between sites?  \nPlease also specify what quantities are exchanged across sites (EIF evaluations, sample moments, covariance matrices, or model parameters), especially for moment matching and density-ratio estimation.  \nThis clarification would help determine whether the method is truly *federated* or rather *semi-centralized*.  \nAdditionally, what is the communication complexity of the procedure in terms of the number of sites $K$, covariate dimension $d$, and time grid $\\tau$?  \nCan the method accommodate time-varying covariates, and is it computationally scalable for large $d$ or fine-grained time grids?\n\n---\n\n**3. Assumptions and robustness**  \nAre there assumptions ensuring that the density ratios $\\omega_{k,0}(X) = P(X|R=0)/P(X|R=k)$ are well-defined, particularly when $P(X|R=k)$ can approach zero under strong covariate shift?  \nThe framework also assumes overlap (Assumption 2.3) across all sites—how robust is the method to *partial or local violations* of this assumption?  \nIn such cases, meta-analysis often remains a strong competitor. Can you demonstrate scenarios where the proposed method outperforms simpler aggregated-data baselines?  \nFurthermore, could the approach extend to situations with multiple non-mergeable target sites or a target population defined as a subpopulation of the sources, as is common in observational studies?\n\n---\n\n**4. Empirical evaluation and interpretation**  \nThe variance gain from the federated estimator relative to the local-target estimator appears small in the experiments.  \nAre there particular regimes (e.g., limited target size, moderate source bias, partial overlap) where the federated approach provides substantial efficiency improvements?  \nAdditionally, why is the pooled estimator biased under shift scenarios in the simulations?  \nWould adjusting for the site indicator as a covariate in the pooled model yield comparable results?\n\n---\n\n**5. Privacy and model specification**  \nCould you specify which privacy risks the method actually mitigates and how this differs from standard aggregated-data sharing?  \nSince no formal privacy guarantees (e.g., differential privacy) are implemented, should the term *privacy-protected* remain in the title?  \nFinally, the density ratios are estimated via exponential-tilt models—could you justify calling these models “flexible,” and discuss how misspecification of the density-ratio model affects the bias or variance of the estimated survival functions?\n\n---\nFinally, I am willing to increase my grade if the authors properly address the above questions, especially my concerns on the framing of the article around privacy, explaining clearly what objects are communicated and the challenges in small sample sizes and high(er) dimensions, and the true contribution of the paper beyond the mere extension of the FACE article to the survival analysis setting.\n\n---\n\n### References\n\n[1] Westling, Ted, et al. “Inference for treatment-specific survival curves using machine learning.”  \n*Journal of the American Statistical Association* 119.546 (2024): 1541–1553.  \n\n[5] Han, Larry, et al. “Federated adaptive causal estimation (FACE) of target treatment effects.”  \n*Journal of the American Statistical Association* (2025): 1–14.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The work presented here is a nice foundation for survival curve estimation in a federated setting, which is crucial and where privacy is clearly a big stake.\n\nThe theory seems solid with asymptotic unbiasedness and accounts for heterogeneous designs. I believe most of the strength of the paper is on theorem E.1, which should be the central theorem of the (main) paper instead of the appendix. Assessing how original this theorem is in comparison to [1] is not properly addressed though.\n\n[1] Westling, Ted, et al. \"Inference for treatment-specific survival curves using machine learning.\" Journal of the American Statistical Association 119.546 (2024): 1541-1553.", "weaknesses": "## Main Concerns\n\n- **Related work and positioning:**  \n  The discussion of related work is incomplete. Several recent contributions have explored federated causal inference, such as [1] and [2], which could serve as meaningful baselines even if they are not specific to survival outcomes, or at least be referred to. In addition, key competitors have been omitted, including [2] and [3], who propose federated external control arm approaches that relax the positivity assumption across sites. The paper should more clearly situate itself within this literature and justify how it advances beyond these frameworks.\n\n- **Conceptual novelty:**  \n  The proposed method appears to be primarily a survival adaptation of FACE [4]. The paper would benefit from a clearer statement of what new theoretical or methodological insights are introduced beyond this earlier work. Without this clarification, the contribution could read as incremental.\n\n- **Motivation and scope:**  \n  The motivation for privacy preservation in survival analysis should be better articulated. The authors should explain why privacy is especially relevant for survival data and why access to large multi-institutional datasets is critical for estimating treatment-specific survival functions. Currently, the privacy aspect appears overstated relative to its actual methodological treatment.\n\n- **Estimator behavior and information sharing:**  \n  The estimator defined in Theorem 2.5 does not seem to borrow information from other source sites, effectively acting as a generalization of local survival estimators to the target population. The moment-matching aggregation step does not clearly improve efficiency. This raises questions about how much “federation” is actually achieved beyond one-shot aggregation of summaries.\n\n- **Density-ratio estimation and implementation details:**  \n  The section on density-ratio estimation is too brief (line 194) and should be expanded. Estimation under federated constraints is non-trivial, and the so-called “flexible” exponential tilt models are better described as parametric. The authors should discuss the impact of misspecification and the propagation of density-ratio estimation error to the final survival estimates. Moreover, the requirement to share empirical covariance matrices across sites raises both dimensionality and privacy issues that require explicit discussion.\n\n---\n\n## Minor Comments\n\n- The section titled “Limitations of existing work” should be reformulated into a proper **Related Work** section, emphasizing connections to federated causal inference and data fusion.  \n- Clarify in the text which quantities are exchanged across sites (EIFs, covariance matrices, model parameters). This will help readers understand whether the method is truly federated or semi-centralized.  \n- Some claims about “flexible” models or “privacy-protected” estimation should be moderated or substantiated.  \n- Improve notation consistency and specify whether expectations and variances in key equations are empirical or population-level.  \n- The role of overlap (Assumption 2.3) should be highlighted in the discussion of limitations, as it may fail in realistic multi-center observational data.\n\n---\n\n## Recommendations\n\n1. **Reorganize and expand the related work section** to include recent developments in federated causal inference, external control arm methods, and meta-analytic baselines.  \n2. **Clarify the contribution relative to FACE**, identifying specific theoretical or computational advances introduced in this survival extension.  \n3. **Provide additional methodological details** on the estimation of density ratios, including how model misspecification on exponential tilt model and limited covariate overlap affect performance.  \n4. **Discuss communication and privacy aspects** more explicitly, particularly the trade-offs between sharing covariance matrices and preserving data confidentiality.  \n5. **Include stronger baselines** (e.g., meta-analysis and FedECA) in empirical evaluations to demonstrate under which regimes the proposed method achieves measurable improvements.\n\n---\n\n### References\n\n[1] Xiong, Ruoxuan, et al. “Federated causal inference in heterogeneous observational data.” *Statistics in Medicine* 42.24 (2023): 4418–4439.  \n\n[2] Archetti, Alberto, et al. “Heterogeneous datasets for federated survival analysis simulation.” *Companion of the 2023 ACM/SPEC International Conference on Performance Engineering.* 2023.  \n\n[3] Terrail, Jean Ogier du, et al. “FedECA: A federated external control arm method for causal inference with time-to-event data in distributed settings.” *arXiv preprint* arXiv:2311.16984 (2023).  \n\n[4] Han, Larry, et al. “Federated adaptive causal estimation (FACE) of target treatment effects.” *Journal of the American Statistical Association* (2025): 1–14.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)