Flexible Doubly Robust Proximal Causal Inference for Censored Outcomes

1 Introduction

• Unmeasured confounding undermines observational causal claims, and right censoring means event times are only partially observed.Standard causal methods rely on conditional exchangeability, which assumes all variables affecting treatment and outcome have been measured.Source: §1 · p. 1
• Existing proximal survival work uses restrictive parametric bridges, while flexible proximal learners generally target scalar rather than censored survival-curve outcomes.Censored-outcome work also targets mean survival time rather than the full curve.Source: §1 · p. 2
• PSB combines survival-specific proximal identification with flexible kernel and neural outcome-bridge learners and a targeted treatment-bridge stage.The framework yields outcome-bridge, treatment-bridge, and doubly robust estimators; PSB-K suits smoother regimes, while PSB-N is designed for harder misspecification.Source: §1 · p. 2
• The evaluation covers SEM, a proximal semi-synthetic ACTG dataset, and real-world Framingham, plus modern baselines and proxy and censoring stress tests.A separate randomized clinical trial provides an external treatment-effect benchmark for Framingham.Source: §1 · p. 2

2 Related Work

• Flexible kernel and neural proximal learners use maximum-moment-restriction objectives but target scalar outcomes, not censoring or survival-curve targets.For right-censored outcomes, [46] identify treatment-specific survival curves using IPCW and low-dimensional parametric bridges.Source: §2 · p. 2
• Prior censored-outcome work proposes OB, TB, and DR estimators, including nonparametric minimax bridges, for differences in mean survival times.Source: §2 · p. 3
• This work combines existing survival identification with flexible proximal moment-restriction learning to estimate time-indexed outcome bridges and counterfactual survival curves.The causal graph in Figure 1 depicts the proximal survival setting, with latent U and observed T = min(Te, Tc).Source: §2 · p. 3, Figure 1 · p. 3

3 Proximal Assumptions, OB, TB and DR Estimators

• Identification assumes latent ignorability, proxy and censoring conditional independence, treatment and censoring positivity, informative proxies, and completeness.Source: §3 · p. 3

Treatment Bridge (TB)

• A treatment bridge q(Z, X, a) based on observed quantities identifies the counterfactual survival curve through a weighted, IPCW-adjusted estimator.Its balancing property reweights the treatment arm so functions of (W, X) behave as if treatment were independent of latent confounding; the result requires bridge existence, the stated assumptions, and completeness.Source: Treatment Bridge (TB) · pp. 4–5

Outcome Bridge (OB)

• An outcome bridge h(t, W, X, a) satisfies a conditional integral equation and identifies the counterfactual survival curve without observing U.The bridge can be optimized at one time horizon or over t > 0, subject to its existence, the stated assumptions, and completeness.Source: Outcome Bridge (OB) · p. 5

Targeted Learning of TB using OB

• The targeted q-learning condition aligns weighted observations with a fitted outcome bridge at each time, replacing the full balancing family with bridge-specific moments.Source: Targeted Learning of TB using OB · p. 5

Doubly-Robust (DR) Estimator

• The DR estimator is consistent at each fixed time when censoring survival is consistently estimated and either the treatment or outcome bridge converges to a solution.The implementation learns q through alignment with fitted h; if h is inconsistent, q need not be a valid treatment bridge, so the implementation need not retain the theorem's full double robustness.Source: Doubly-Robust (DR) Estimator · pp. 5–6

4 Model Specification

• The bridge is learned from an IPCW pseudo-outcome by enforcing a conditional moment restriction across conditioning variables and time horizons.A conditioning kernel probes (A, Z, X), while a time kernel couples residual moments across horizons.Source: §4 · p. 6
• Self-normalized Hájek variants are also reported because self-normalization can mitigate instability from extreme weights and generally reduces error in the experiments.Monotone rearrangement did not affect the reported results at displayed precision.Source: §4 · pp. 6–7

PSB-K: Kernel Bridge Representation

• PSB-K represents the outcome bridge in a tensor-product time and bridge-input RKHS, using regularization to control bridge complexity.Its representer expansion is reduced with Nyström sample and time landmarks to a smaller Kronecker-product linear system.Source: §4 · p. 7

PSB-N: Neural Bridge Representation

• PSB-N replaces the bridge RKHS with a neural network while retaining the same kernel-based test functions for the conditional moment restriction.It minimizes the common projected risk by minibatch gradient descent and uses the learned bridge to form OB, TB, and DR estimators.Source: §4 · pp. 7–8

Targeted Treatment-Bridge Learning

• Both learners share a treatment-bridge stage that fits q with the outcome bridge held fixed, aligns moments over the time grid, and imposes normalization.The implementation uses a neural q with positive output.Source: §4 · p. 8

Comparing PSB-K and PSB-N

• Held-out risk selects PSB-N on Framingham and PSB-K on ACTG; on SEM, the DR estimators are nearly indistinguishable (MAE 0.022 vs. 0.021 at N = 2000).The authors present this as practical guidance rather than a theoretical guarantee.Source: §4 · p. 9

SEM: matched structure favors the parametric and kernel bridges

• In SEM, DR-PSB-K offers the most balanced accuracy–coverage tradeoff among flexible methods, with PDR and DR-PSB-K coverage near nominal at 0.94–0.97.The benchmark is generated from the proximal survival model, and PSB-K remains the strongest flexible family at every sample size.Source: §5 · p. 9, Figure 2 · p. 10

Proximal semi-synthetic ACTG: flexible bridges repair the parametric outcome bridge

• On ACTG, PSB-K attains the best average bias and MAE, while both flexible learners sharply reduce POR average absolute bias from 0.156 to 0.010 and 0.024.PSB-N remains close; flexible bridge learning helps where the reference outcome bridge is poorly matched to the nonlinear, censored data-generating mechanism.Source: §5 · pp. 9–11

Framingham: neural bridges are most stable on the observational-to-RCT transfer task

• On Framingham, OB-PSB-N is strongest overall (average absolute bias 0.138, MAE 0.151); PSB-N is the only flexible bridge family stable throughout.The external benchmark is a randomized trial in a different population, and the 30-run coverage reflects computational variability rather than sampling uncertainty.Source: Figure 2 · p. 10, §5 · p. 11

Modern baselines and stress tests

• When SEM proxy–confounder correlation falls to 10%, DR-PSB-K bias is 0.0471 and coverage is 0.57; with up to 60% ACTG censoring, DR bias stays between 0.002 and 0.007.An invalid treatment proxy raises SEM bias to 0.0128, while modern no-unmeasured-confounding baselines have higher reported bias than DR-PSB-K on SEM and ACTG.Source: §5 · p. 11

Hájek normalization

• Hájek normalization lowers ACTG average absolute bias for all four weighted estimators to 0.004–0.008, but does not make them the best overall option.On Framingham, PIPW RMSE falls from 22.0 to 0.634 while its bias remains 0.633; learned-weight ESS is 51.8% there.Source: §5 · p. 11

6 Conclusion

• PSB-K performs well in smoother SEM and remains competitive on ACTG, while PSB-N is most stable on the more challenging Framingham task.The conclusion attributes the preferred bridge family to the complexity of the underlying confounding structure.Source: §6 · p. 12
• The framework identifies treatment-specific survival curves through time-indexed outcome and treatment bridges, yielding OB, TB, and DR estimators.The estimators build on the doubly robust identification in [46].Source: §6 · p. 12

AI use statement

• Generative AI supported methodology feedback, code development, manuscript drafting and editing, brainstorming, literature search, and presentation; the authors say they reviewed and verified AI-assisted content.The authors take responsibility for the final work.Source: AI use statement · p. 12

A Limitations

• Completeness is not directly testable from observed data, proxy availability and quality are application-dependent, and Framingham evaluates hazard ratios rather than full survival-curve contrasts.The theory is population-level and establishes no rates, uniform-in-time consistency, or asymptotic distributions for learned bridges.Source: A Limitations · p. 17

B Extended Related Work

• The discussion of prior work notes that [46] lacked complete proofs for both bridge identifications, used largely favorable synthetic settings, and had no external real-world benchmark.This work states that it addresses these gaps.Source: B Extended Related Work · p. 17
• Prior survival bridges use low-dimensional parametric forms, while flexible proximal learners for scalar outcomes do not directly address censoring or survival-curve targets.PSB combines survival identification from [46] with flexible proximal moment-restriction learning, without new identification results.Source: B Extended Related Work · p. 17

C Proof of Proposition 1; D Proof of Proposition 2

• The outcome- and treatment-bridge proof steps invoke conditional-independence conditions and completeness; the weighting argument also invokes censoring positivity.The passages identify completeness as the condition used to obtain bridge equalities and positivity as ensuring a ratio is well-defined.Source: C Proof of Proposition 1 · pp. 18–19, D Proof of Proposition 2 · p. 20

E Proof of Lemma 2

• The IPCW pseudo-outcome has conditional expectation equal to the event-time survival probability, which the outcome bridge represents conditional on proxies and covariates.The displayed proof uses the tower property to connect the weighted observed survival indicator to the bridge.Source: E Proof of Lemma 2 · p. 20

F Comparing the two bridge learners

• PSB-K uses an RKHS bridge with a closed-form solution and favors smoother bridges, whereas PSB-N uses gradient descent and can adapt to sharper nonlinearities but needs more data and careful tuning.The comparison describes PSB-K as excelling on SEM and PSB-N as more robust under harder Framingham misspecification.Source: F Comparing the two bridge learners · p. 21

G.1 Setup and notation

• PSB-K represents its bridge in the tensor-product RKHS using a bridge kernel on (W, A, X) and a time kernel; its data include an IPCW pseudo-outcome matrix over time horizons.The conditioning kernel instead defines test functions enforcing the moment restriction.Source: §G.1 · pp. 21–22

G.2 Representer theorem and Nyström reduction

• The Nyström approximation restricts the bridge expansion to landmarks, reducing the unknown coefficients from nM to rsrt.The landmark selections include sample landmarks stratified across treatment arms and uniformly spaced time landmarks.Source: §G.1 · pp. 21–22, §G.2 · p. 22

G.3 Derivation of the Kronecker system

• The regularized risk yields a Kronecker linear system whose terms contract conditioning and time kernels and penalize the landmark-represented RKHS norm.With rs = 128 and rt = 48, the system is 6144 × 6144; a 10^-6 jitter is added for numerical stability.Source: §G.3 · p. 22

G.4 Prediction

• At a new point and time, PSB-K predicts the bridge using bridge-kernel and time-kernel evaluations against their respective landmarks.Predictions over observations and the full time grid are computed as Φb α C⊤.Source: §G.4 · p. 23

G.5 Computational complexity

• The full n × n conditioning kernel is the main large-sample computational bottleneck, while the Nyström system avoids iterative optimization and gradient computation.The conditioning kernel is computed only once.Source: §G.5 · p. 23

H Details about Hájek estimator variants

• The Hájek DR estimator is consistent under Theorem 1's conditions when its normalizer stays bounded away from zero; the Hájek TB estimator inherits consistency from the standard version.Asymptotic normality carries over by the delta method where it holds jointly for the standard estimators and the normalizer.Source: H Details about Hájek estimator variants · p. 23

I Compared approaches and metrics

• The evaluation compares the parametric proximal bridge family of [46] and weighted Cox baselines that do not solve proximal bridge equations.Reported metrics are average absolute bias, MAE, RMSE, and empirical coverage; coverage applies only to SEM proximal bridge estimators.Source: I Compared approaches and metrics · p. 24

J.1 SEM simulation design

• The SEM simulation follows Ying et al. [46], with censoring generated independently and approximately 22% of observations censored.The parametric comparator is especially informative in SEM because the data-generating mechanism favors its functional form.Source: §J.1 · p. 24, I Compared approaches and metrics · p. 24

J.2 Framingham cohort study

• Framingham uses 32 Offspring Exam 6 covariates allocated between proxy blocks, and treats the external trial's hazard ratio of 0.75 as a reference rather than the cohort's true effect.The benchmark evaluates survival curves at normalized horizons 0.3, 0.5, and 0.7; the trial enrolled a different population.Source: §J.2 · p. 25

• The generator keeps the original ACTG175 baseline covariates fixed and regenerates the latent, proxy, treatment, event-time, and censoring variables.Source: §J.3 · p. 25
• Using standardized clinical features, the generator forms three nonlinear observed summaries.Source: §J.3 · p. 25
• Potential event times are generated on a 240-point grid.Source: §J.3 · p. 26

K Evaluation metrics

• Survival-curve experiments target ψ_r(t) = S_1,r(t) − S_0,r(t), while Framingham targets an external benchmark hazard ratio.Coverage intervals differ by dataset; Framingham runs share one cohort, so across-run spread reflects computational variability.Source: K Evaluation metrics · p. 28

L Weighted Cox baselines

• The weighted Cox baselines use logistic-regression propensity scores and treatment-only weighted partial likelihood, unlike proximal methods that solve bridge equations.Source: L Weighted Cox baselines · p. 29

M Detailed Results

• Results report bias, MAE, RMSE, and coverage across displayed horizons; bias is signed in per-horizon columns, while Table 5 averages absolute bias.Source: M Detailed Results · p. 29

Modern baselines

• Modern causal-survival baselines assuming no unmeasured confounding remain biased by construction and have poor SEM coverage of 0.06–0.66.Source: N Additional robustness analyses · p. 29

Weak and invalid proxies

• Weakening SEM proxies leaves the unadjusted Kaplan–Meier contrast at average absolute bias 0.0309; proxy noise is varied while structural coefficients stay fixed.Source: N Additional robustness analyses · p. 30

Censoring intensity and model

• With ACTG censoring increased from about 25% to 60%, DR-estimator bias stays in the same range, while DR-PSB-K MAE rises from 0.0136 to 0.0177.The authors limit these conclusions to the censoring mechanisms studied.Source: N Additional robustness analyses · pp. 31–32

Uncensored ACTG

• Without censoring, POR average absolute bias falls from 0.156 to 0.010, while the two flexible outcome bridges change by less than 0.01.Source: N Additional robustness analyses · p. 32, Table 10 · p. 33

Weight and proxy diagnostics

• Learned-weight effective sample size is 96.8% of n on ACTG and 95.6–99.1% on SEM, but 51.8% on Framingham; SEM proxy relevance falls from 0.45 to 0.03 as proxies weaken.The proxy-relevance statistic is unchanged by the invalid-proxy perturbation.Source: N Additional robustness analyses · pp. 32–33

Monotonicity

• Monotone rearrangement changes 47–83% of weighted estimates but leaves every reported metric unchanged at printed precision.It changes 0.1–0.4% of outcome-bridge estimates, by essentially zero.Source: N Additional robustness analyses · p. 33

Code and computational setup

• Supplementary materials provide anonymized PSB-K and PSB-N code, SEM and ACTG generators, and the Framingham pipeline without its data.Each neural and kernel bridge run used one NVIDIA H200 GPU.Source: O Additional implementation details · p. 33

Architecture dimensions

• The architecture table defines dW, dZ, and dX as the dimensions of the outcome proxy, treatment proxy, and observed covariates.Source: Table 11 · p. 34

Framingham implementation

• For Framingham, PSB-N uses 48 time centers and 200 AdamW epochs, while PSB-K fits its outcome bridge by a regularized kernel solve with 128 sample and 48 time support points.Source: O Additional implementation details · p. 34

ACTG implementation

• For ACTG, PSB-N uses 1,000 AdamW epochs and PSB-K uses a regularized kernel solve with 128 sample and 48 time support points; results aggregate 30 datasets.Source: O Additional implementation details · p. 34

SEM implementation

• For SEM, each reported sample size uses 500 simulations; PSB-N uses 1,000 AdamW epochs and PSB-K again uses 128 sample and 48 time support points.Source: O Additional implementation details · p. 34