Flexible Doubly Robust Proximal Causal Inference for Censored Outcomes

Dhanajit Brahma; Ruolin Meng; Lawrence Carin; Ricardo Henao

stat.MLOct 14, 2026

Flexible Doubly Robust Proximal Causal Inference for Censored Outcomes

Authors: Dhanajit Brahma, Ruolin Meng, Lawrence Carin, Ricardo Henao

Organizations: Duke University

Abstract

Estimating causal effects on right-censored time-to-event outcomes is challenging under unmeasured confounding. Proximal causal inference addresses unmeasured confounding through proxy variables, but existing proximal survival methods rely largely on restrictive parametric bridge models. We introduce Proximal Survival Bridge learning (PSB), a flexible doubly robust proximal framework for censored outcomes. Building on existing proximal identification of treatment-specific survival curves, the framework yields outcome-bridge (OB), treatment-bridge (TB), and doubly robust (DR) estimators, with two flexible bridge learners: PSB-K, a kernel estimator with a low-rank time decomposition, and PSB-N, a neural estimator trained by a projected maximum-moment-restriction loss. Unlike existing flexible proximal learners for scalar outcomes, PSB directly handles censoring and survival-curve targets. On a standard SEM benchmark, a proximal semi-synthetic ACTG study, and the real-world Framingham study, PSB-K performs strongest on SEM and ACTG, whereas PSB-N is the most stable on Framingham. Additional results with modern causal survival baselines, weakened or invalid proxies, and heavier censoring assess robustness across challenging settings.

Figures & tables

Panel of Figure 1 — Figure 1: Causal DAG for proximal survival setting. $U$ is latent; $T=\min(T_{e},T_{c})$ is observed. Missing edges encode proxy independence $(W,T_{e}(a))\perp\!\!\!\!\perp(A,Z)\mid U,X$ and censoring independence $T_{e}\perp\!\!\!\!\perp T_{c}\mid A,X,W,Z$ .

Weighted Cox baselines
	SEM			ACTG
Method	$N=500$	$N=1000$	$N=2000$	$N=2139$
Uniform	0.017 / 0.028 / 1.00	0.016 / 0.022 / 1.00	0.018 / 0.020 / 1.00	0.072 / 0.072 / 0.34
IPW	0.030 / 0.035 / 1.00	0.029 / 0.031 / 1.00	0.030 / 0.031 / 0.67	0.068 / 0.071 / 0.43
OW	0.030 / 0.035 / 1.00	0.029 / 0.031 / 1.00	0.030 / 0.031 / 0.67	0.068 / 0.071 / 0.43
Parametric proximal bridge [ 46 ]

Table 1: SEM and semi-synthetic ACTG results. Entries report average absolute bias / MAE / Coverage. SEM results vary by sample size; ACTG results use the full ACTG175 cohort (

N{=}2139

) and are averaged over horizons

\{201,401,602,802\}

days. Appendix Tables 5 and 3 show the corresponding RMSE values and per-horizon breakdowns. SEM coverage is that of pointwise 95% Wald intervals with influence-function standard errors over 500 replicates (Cox baselines: replicate-dispersion reconstruction); ACTG coverage is for replicate-dispersion coverage over 30 datasets.

Figure 2: Top: SEM at

N=2000

, where PSB-K performs comparably to the parametric bridge in [ 46 ] . Bottom: Framingham hazard ratios over 30 runs vs . the external benchmark

0.75

(dashed). PIPW, PDR, and TB/DR- PSB-N are shown in their Hájek variants (Appendix H ); OB- PSB-N has the smallest average absolute bias.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

	PSB-K	PSB-N
Bridge representation	RKHS (kernel)	Neural network
Conditioning kernel $K_{c}$	Exact ( $n\times n$ )	Exact ( $B\times B$ per batch)
Time structure	Nyström projection $C_{t}$	Nyström projection $C_{t}$
Optimization	Closed-form (linear algebra)	SGD (AdamW)
Tuning	ridge $\lambda$ ; median-heuristic bandwidths	standard neural tuning
Suited to	smooth bridges, moderate $n$	sharper nonlinearity, larger $n$

Appendix

Table 2: Comparison of the two outcome bridge learners. Both methods use the same conditioning kernel

k_{c}

and Nyström time projection

C_{t}

; they differ in bridge representation and optimization.

Weighted Cox baselines
	201 d				401 d				602 d				802 d				Avg.
Method	Bias	MAE	RMSE	Cov	Bias	MAE	RMSE	Cov	Bias	MAE	RMSE	Cov	Bias	MAE	RMSE	Cov	Bias	MAE	RMSE	Cov
Uniform	0.081	0.081	0.084	0.133	0.120	0.120	0.126	0.267	-0.028	0.029	0.040	0.667	-0.058	0.058	0.060	0.300	0.029	0.072	0.078	0.342
IPW	0.084	0.084	0.086	0.067	0.128	0.128	0.134	0.267	-0.014	0.026	0.033	0.767	-0.047	0.047	0.049	0.600	0.038	0.071	0.076	0.425
OW	0.083	0.083	0.085	0.067	0.127	0.127	0.132	0.267	-0.014	0.026	0.033	0.767	-0.047	0.047	0.049	0.600	0.037	0.071	0.075	0.425
Parametric proximal bridge [ 46 ]

Appendix

Table 3: Detailed ACTG results for Table 1 . The displayed ACTG horizons are

\{201,401,602,802\}

days. Bias is signed, and the final average columns are simple averages over these horizons. Cov is replicate-dispersion coverage over 30 datasets (Appendix K ).

Weighted Cox baselines
	2184 d				3640 d				5095 d				Avg.
Method	Bias	MAE	RMSE	Cov	Bias	MAE	RMSE	Cov	Bias	MAE	RMSE	Cov	Bias	MAE	RMSE	Cov
Uniform	1.026	1.026	1.045	0.000	1.026	1.026	1.045	0.000	1.026	1.026	1.045	0.000	1.026	1.026	1.045	0.000
IPW	0.515	0.515	0.552	0.300	0.515	0.515	0.552	0.300	0.515	0.515	0.552	0.300	0.515	0.515	0.552	0.300
OW	0.184	0.184	0.208	0.567	0.184	0.184	0.208	0.567	0.184	0.184	0.208	0.567	0.184	0.184	0.208	0.567
Parametric proximal bridge [ 46 ]

Appendix

Table 4: Detailed Framingham results for the standard (non-Hájek) estimators. The displayed horizons are

\{2184,3640,5095\}

days. Bias is signed, and the final average columns are simple averages over these horizons; Table 6 reports the Hájek variants of the weighted estimators. Cov is the rate at which the external benchmark lies within 1.96 across-run standard deviations of each run’s estimate, over 30 runs on a fixed cohort (Appendix K ); it is not the coverage of a confidence interval. Runs whose hazard ratio is not finite (an estimated survival curve projected to one) are excluded: 7, 7, and 3 of the 30 runs for PIPW and 1, 1, and 0 for TB- PSB-K .

$N=500$
	$t=0.25$				$t=0.50$				$t=0.75$				Avg.
Method	Bias	MAE	RMSE	Cov	Bias	MAE	RMSE	Cov	Bias	MAE	RMSE	Cov	Bias	MAE	RMSE	Cov
Uniform	0.010	0.020	0.025	1.000	0.018	0.030	0.037	1.000	0.023	0.035	0.043	1.000	0.017	0.028	0.035	1.000
IPW	0.019	0.024	0.030	1.000	0.032	0.037	0.045	1.000	0.038	0.044	0.053	1.000	0.030	0.035	0.043	1.000
OW	0.020	0.024	0.030	1.000	0.032	0.037	0.045	1.000	0.039	0.044	0.053	1.000	0.030	0.035	0.043	1.000
POR [ 46 ]	-0.000	0.021	0.026	0.034	-0.001	0.031	0.039	0.068	-0.001	0.035	0.045	0.128	0.001	0.029	0.037	0.077

Appendix

Table 5: Detailed SEM results for all three sample sizes. Each panel reports Bias, MAE, RMSE, and Coverage at

t\in\{0.25,0.50,0.75\}

together with their simple averages. Coverage for the proximal bridge estimators is that of pointwise 95% Wald intervals with influence-function standard errors over 500 replicates, while the SEM Cox baseline coverage values use the replicate-dispersion reconstruction described in Appendix K ; the average Bias column reports the average absolute bias.

Dataset	Bridge	Estimator	Variant	Avg. abs. bias	MAE	RMSE	Coverage
ACTG	Ying et al. [46]	PIPW	Standard	0.025	0.033	0.038	0.98
ACTG	Ying et al. [46]	PIPW	Hájek	0.008	0.017	0.020	1.00
ACTG	Ying et al. [46]	PDR	Standard	0.015	0.022	0.025	0.97
ACTG	Ying et al. [46]	PDR	Hájek	0.008	0.017	0.020	1.00
ACTG	PSB-N	TB	Standard	0.013	0.027	0.036	0.98
ACTG	PSB-N	TB	Hájek	0.005	0.014	0.018	1.00

Appendix

Table 6: Standard versus Hájek-normalized weighted estimators. The rows labeled Ying et al. [46] correspond to the parametric proximal bridge. ACTG averages are over

\{201,401,602,802\}

days and Framingham averages are over

\{2184,3640,5095\}

days. Coverage is replicate-dispersion coverage; for Framingham, the target is the external benchmark; see Appendix K . Framingham PIPW (Standard) excludes the runs whose hazard ratio is not finite (Table 4 ).

Method	SEM	ACTG
Survival TMLE	0.0191	0.0137
IPCW-AIPW	0.0256	0.0058
Causal survival forest	0.0256	0.0057
DeepSurv + IPW	0.0312	0.0858
DR- PSB-K	0.0017	0.0021
DR- PSB-N	0.0124	0.0058

Appendix

Table 7: Average absolute bias of modern causal survival baselines and our DR estimators (SEM with

N=2000

; ACTG with

N=2139

$\mathrm{corr}(U,\text{proxy})$ as a fraction of baseline	1.00	0.75	0.50	0.25	0.10
Noise SD $s_{Z}/s_{W}$	0.50/0.50	0.80/0.82	1.33/1.37	2.79/2.89	7.08/7.32
Proxy-relevance statistic	0.45	0.23	0.10	0.03	0.03
Avg. abs. bias	0.0017	0.0021	0.0073	0.0329	0.0471
Coverage	0.94	0.93	0.87	0.70	0.57
Invalidity $\delta$	0	0.10	0.25	0.50
Proxy-relevance statistic	0.45	0.45	0.45	0.46

Appendix

Table 8: DR- PSB-K under weakened (top) and invalid (bottom) proxies on SEM (

N=2000

Censoring rate	25%	41%	60%
DR- PSB-K	0.0021 / 1.00	0.0036 / 1.00	0.0026 / 0.99
DR- PSB-N	0.0058 / 0.99	0.0065 / 0.99	0.0071 / 1.00

Appendix

Table 9: DR estimators on ACTG under increasing censoring (average absolute bias / replicate-dispersion coverage).

Method	25% censored	Uncensored
OB- PSB-K	0.0096	0.0044
OB- PSB-N	0.0235	0.0137
POR [ 46 ]	0.1562	0.0104
DR- PSB-K	0.0021	0.0006
DR- PSB-N	0.0058	0.0003

Appendix

Table 10: Average absolute bias on ACTG with and without censoring.

Component	Architecture / parameterization
PSB-N outcome bridge (ACTG, Framingham)	Input $(t,a,w,x)\in\mathbb{R}^{2+d_{W}+d_{X}}$ ; FC $(2+d_{W}+d_{X},32)$ , ReLU; FC $(32,16)$ , ReLU; FC $(16,1)$ ; Softplus output.
PSB-N outcome bridge (SEM)	Input $(t,a,w,x)\in\mathbb{R}^{2+d_{W}+d_{X}}$ ; FC $(2+d_{W}+d_{X},16)$ , ReLU; FC $(16,16)$ , ReLU; FC $(16,1)$ ; Softplus output.
Targeted $q_{\phi}$ for ACTG	Signed input $\tilde{d}(a,z,x)=(-1)^{1-a}(1,a,z^{\top},x^{\top})^{\top}$ ; FC $(2+d_{Z}+d_{X},32)$ , ReLU; FC $(32,16)$ , ReLU; FC $(16,1)$ ; positive output $1+\exp(\cdot)$ .
Targeted $q_{\phi}$ for Framingham	Signed input $\tilde{d}(a,z,x)$ ; FC $(2+d_{Z}+d_{X},32)$ , ReLU; FC $(32,32)$ , ReLU; FC $(32,1)$ ; positive output $1+\exp(\cdot)$ .
Targeted $q_{\phi}$ for SEM	Signed input $\tilde{d}(a,z,x)$ ; FC $(2+d_{Z}+d_{X},8)$ , ReLU; FC $(8,1)$ ; positive output $1+\exp(\cdot)$ .
PSB-K outcome bridge	Treatment-matched kernel on $A$ , RBF kernels on standardized $(W,X)$ and $(A,Z,X)$ , $128$ sample support points, $48$ time support points, and ridge regularization $\lambda=10^{-2}$ .

Appendix

Table 11: Neural and kernel architectures used by the proposed survival bridge estimators. Here

d_{W}

d_{Z}

, and

d_{X}

denote the dimensions of the outcome proxy, treatment proxy, and observed covariates.

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