Doubly-Robust Functional Average Treatment Effect Estimation

TL;DR

DR-FoS tackles estimating the Functional Average Treatment Effect (FATE) in observational studies with functional outcomes by extending double-robust causal inference to function-on-scalar settings. It defines a DR-FoS estimator based on nuisance functions for both the treatment propensity and the outcome regression, and proves that the estimator converges to a Gaussian process, enabling simultaneous confidence bands via functional inference techniques and cross-fitting. Through simulations, DR-FoS demonstrates robustness to misspecification and superior performance relative to outcome regression and IPW, while ensuring valid simultaneous coverage. An empirical application to the SHARE data shows that chronic conditions exert persistent adverse effects on functional quality-of-life trajectories, with effects growing over time. The work provides a practical, theoretically grounded tool for causal analysis of high-dimensional functional data and lays groundwork for extensions to broader causal structures involving functional outcomes.

Abstract

Understanding causal relationships in the presence of complex, structured data remains a central challenge in modern statistics and science in general. While traditional causal inference methods are well-suited for scalar outcomes, many scientific applications demand tools capable of handling functional data -- outcomes observed as functions over continuous domains such as time or space. Motivated by this need, we propose DR-FoS, a novel method for estimating the Functional Average Treatment Effect (FATE) in observational studies with functional outcomes. DR-FoS exhibits double robustness properties, ensuring consistent estimation of FATE even if either the outcome or the treatment assignment model is misspecified. By leveraging recent advances in functional data analysis and causal inference, we establish the asymptotic properties of the estimator, proving its convergence to a Gaussian process. This guarantees valid inference with simultaneous confidence bands across the entire functional domain. Through extensive simulations, we show that DR-FoS achieves robust performance under a wide range of model specifications. Finally, we illustrate the utility of DR-FoS in a real-world application, analyzing functional outcomes to uncover meaningful causal insights in the SHARE (Survey of Health, Aging and Retirement in Europe) dataset.

Figures (5)

• Figure 1: Simulation results based on the data generating process described in Section \ref{['sec:simulate']}. The first panel (leftmost) shows an example of true FATE $\beta$ (purple) as generated by our simulation scheme, and the estimates provided by IPW (orange), OR (green) and DR-FoS (blue) -- together with simultaneous confidence bands around the latter. In this example we set $\alpha_\mu=0.25$ and $\alpha_\pi=0.75$. The second panel shows the average (across simulation replicates) estimation error as captured by the $\text{MSE}$ when the estimated propensity score is increasingly corrupted by random noise, and the regression functions are well-specified ($\alpha_\mu=0$, $\alpha_\pi\neq0$). While DR-FoS (blue) maintains excellent performance, IPW (orange) performs progressively worse, as expected. The third panel displays the opposite situation, where the estimated regression functions are increasingly corrupted by random noise and the propensity score is well-specified ($\alpha_\mu\neq0$, $\alpha_\pi=0$). Again, the performance of DR-FoS (blue) remains excellent, while the performance of OR (green) progressively deteriorates. Finally, the last panel (rightmost) shows the average coverage levels achieved under misspecifications of the regression functions (light blue, $\alpha_\pi=0$) and of the propensity score (brown, $\alpha_\mu=0$). The grey dashed line represents the nominal $95\%$ coverage level.
• Figure 2: SHARE application results. Each panel displays a different estimated causal effect. Blue continuous lines correspond to DR-FoS estimates; blue bands are 95% asymptotic simultaneous confidence bands obtained by repeatedly sampling from the Gaussian process; grey dotted horizontal lines correspond to 0.
• Figure D.1: Results of simulation study run as described in Section \ref{['sec:simulate']}. Each panel displays estimation performance of DR-FoS, IPW and OR under different $\alpha_\mu$ and $\alpha_\pi$. For each box, the center line represents the median; the lower and upper hinges correspond to the first and third quartiles; the upper and lower whiskers span 1.5 times the interquartile range. DR-FoS consistently matches or outperforms the other estimators, demonstrating the strength of double robustness.
• Figure D.2: Results of the simulation study described in Supplementary Material Section \ref{['suppsec:expl_sim']}. Each panel shows the estimation performance of DR-FoS, IPW, and OR under both well-specified and misspecified settings. Misspecification is introduced by fitting nuisance functions using transformed covariates $\Tilde{X}_i$, $i = 1, \dots, n$. Misspecified outcome regressions are denoted by $\mu^{(a)}_m$, and misspecified propensity scores by $\pi^{(1)}_m$. For example, OR$\mu^{(a)}_m$ refers to the outcome regression estimator fitted with a misspecified regression function, DR $\pi^{(1)}_m$ denotes the DR-FoS estimator fitted with a misspecified propensity score but well-specified regression functions, while DR $\pi^{(1)}_m,\mu^{(a)}_m$ denotes the DR-FoS estimator constructed using both a misspecified propensity score and misspecified regression functions. For each box, the center line represents the median; the lower and upper hinges correspond to the first and third quartiles; the upper and lower whiskers span 1.5 times the interquartile range. DR-FoS consistently matches or outperforms the other estimators, demonstrating the strength of double robustness.
• Figure E.3: SHARE application results using FunGCN as model for the regression function. Each panel displays a different estimated causal effect. Blue continuous lines correspond to DR-FoS estimates; blue bands are 95% asymptotic simultaneous confidence bands obtained by repeatedly sampling from the Gaussian process; grey dotted horizontal lines correspond to 0.

Theorems & Definitions (16)

• Lemma 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.5
• Remark 3.6
• Example 3.7: Assumption e can be superfluous
• Lemma 3.8: Asymptotic Normality of finite dimensional projections
• Corollary 3.9: Pointwise Asymptotic Normality
• Theorem 3.10: Convergence to Gaussian Process
• Remark 3.11