Doubly Robust Causal Estimands

• Doubly robust causal estimands are statistical techniques that yield consistent effect estimates when either the propensity score model or outcome regression model is correctly specified.
• They leverage semiparametric efficiency and influence functions to maintain robustness, even with high-dimensional covariates and complex data structures.
• Extensions include methods for non-Euclidean outcomes, regularized high-dimensional settings, and multiple imputation, enhancing their applicability across diverse empirical studies.

Doubly robust causal estimands represent a key advancement in semiparametric statistics, providing estimators that remain consistent for target causal parameters—such as the average treatment effect—even if only one of two working models (typically the propensity score model or the outcome regression model) is correctly specified. This double layer of protection against model misspecification has led to widespread adoption of doubly robust methodology across observational studies, high-dimensional settings, missing data problems, non-Euclidean outcomes, data integration, and robust inference under contamination.

1. Fundamental Principles of Doubly Robust Causal Estimands

The defining property of a doubly robust estimator is that it is consistent if either the propensity score model (describing treatment assignment given confounders) or the outcome regression model (describing the conditional mean outcome given covariates and treatment) is correctly specified. For a typical binary treatment $T \in \{0,1\}$ and observed covariates $X$, the augmented inverse probability weighting (AIPW) estimator provides a canonical example: $$\hat{\mu}_1^{\text{AIPW}} = \frac{1}{n} \sum_{i=1}^n \left[ \frac{T_i}{\hat{\pi}(X_i)} (Y_i - \hat{m}_1(X_i)) + \hat{m}_1(X_i) \right]$$

$$\hat{\mu}_0^{\text{AIPW}} = \frac{1}{n} \sum_{i=1}^n \left[ \frac{1-T_i}{1-\hat{\pi}(X_i)} (Y_i - \hat{m}_0(X_i)) + \hat{m}_0(X_i) \right]$$

$$\widehat{\text{ACE}} = \hat{\mu}_1^{\text{AIPW}} - \hat{\mu}_0^{\text{AIPW}}$$

with $\hat{m}_t(X)$ estimating $\mathbb{E}[Y| X, T=t]$ and $\hat{\pi}(X)$ estimating $\mathbb{P}(T=1|X)$. If either $m_t(\cdot)$ or $\pi(\cdot)$ is correctly modeled, the estimator is unbiased (Guo et al., 2015).

This robustness is preserved even when high-dimensional feature selection or machine learning is employed for nuisance estimation (Benkeser et al., 2019), provided suitable cross-fitting or sample splitting is used to mitigate overfitting.

2. Model Construction, Influence Functions, and Efficiency

Doubly robust estimators may be derived from efficient influence functions (EIFs) in a semiparametric framework (Baba et al., 2021). The EIF for an average treatment effect typically takes the form: $$\phi(O; \eta, \theta) = \frac{T}{\pi(X)} \left[ Y - m_1(X) \right] + m_1(X) - \frac{(1-T)}{1-\pi(X)} [Y - m_0(X)] - m_0(X) - \theta$$ and the resultant estimator is obtained by solving the sample analogue

$$\frac{1}{n} \sum_{i=1}^n \phi(O_i; \eta, \theta) = 0$$

where $\eta$ collects all nuisance parameters. This construction guarantees that the estimator is efficient—achieving the semiparametric lower bound—if both nuisance models are correct, and remains consistent if at least one is.

When positivity is violated (i.e., treatment assignment probabilities are close to zero or one), specialized variants such as the $e$-score adjustment further reduce estimator variance and bias (Díaz, 2018). These approaches dimension-reduce the propensity model to target only the bias-correcting part.

3. Extensions to Complex Settings

High Dimensionality and Regularization

Modern applications often involve high-dimensional confounding or selection variables. Penalized empirical likelihood, one-step plug-in methods, and penalized estimating equations (e.g., SCAD, adaptive LASSO) have been successfully integrated to retain double robustness while enabling variable selection (Du et al., 26 Mar 2024, Kang, 23 Jul 2025). These approaches yield doubly robust estimators that can simultaneously perform estimation and feature selection, even under moderate or large $p$.

Non-Euclidean and Structured Outcomes

A significant extension is doubly robust inference for outcomes in non-Euclidean spaces (“random objects”). In this setting, outcomes may reside in spaces of probability distributions, shapes, graphs, or SPD matrices. The estimands are typically (generalized) Fréchet means: $$\beta_t = \operatorname{argmin}_{y \in \mathcal{Y}} \mathbb{E}[d_Y^2(Y_t, y)]$$ where $d_Y$ is a metric on a Polish space $\mathcal{Y}$. The estimator uses an isometric Hilbert space embedding $\rho$ so that standard arithmetic can be performed and influence functions can be constructed in the embedding (Bhattacharjee et al., 28 Jun 2025). The doubly robust, “doubly debiased” estimator thus accommodates continuous treatments and infinite-dimensional nuisance functionals.

Proxies, Unobserved Confounding, and Proximal Identification

Recent advances allow for consistent causal effect estimation under unmeasured confounding by exploiting negative control proxies and bridge functions. Here, doubly robust estimators combine outcome bridge and treatment bridge identification, achieving consistency when either bridge function is valid (Li et al., 2022, Bozkurt et al., 26 May 2025). Kernel mean embedding techniques provide closed-form, nonparametric solutions even in continuous and high-dimensional treatment settings.

Handling Outliers and Data Contamination

To safeguard against influential data, robust outcome regressions (using estimating equations with bounded influence functions) and CBPS for treatment assignment modeling have been incorporated into the doubly robust framework. Penalized empirical likelihood is used to avoid overfitting and to enable valid inference in small samples or under contamination (Kang, 23 Jul 2025).

4. Multiple Imputation, Congeniality, and Missing Data

Combining multiple imputation (MI) with doubly robust estimators requires that imputation models include all variables and functional forms present in either the outcome or propensity model (“congeniality”). Failure to supply the correct variables or use the proper functional forms in the imputation model leads to bias, even if both analysis models are correct (McGowan, 13 Oct 2025). If treatment effect heterogeneity is to be modeled (e.g., via interaction terms), imputation should be performed separately within exposure strata.

The interplay between augmentation and imputation is central to the validity of MI-based doubly robust estimators, particularly when confounders, exposures, or outcomes are partially observed (Mayer et al., 2019).

5. Advanced Inference: Calibration, Confidence Intervals, and Theoretical Guarantees

Standard doubly robust estimators guarantee consistency but—without further regularity—may not ensure correct coverage or asymptotic linearity if only one nuisance model achieves a slow convergence rate. Calibration of the nuisance estimators via isotonic regression (as in calibrated DML frameworks) fixes this mismatch and achieves doubly robust asymptotic normality: valid $n^{–1/2}$-rate inference is possible if either the regression or the Riesz representer (propensity-type function) converges sufficiently fast (Laan et al., 5 Nov 2024).

For finite samples or non-classical outcome structures, the use of conformal inference techniques and cumulant generating function-based confidence intervals yields coverage that is robust to contamination and small sample size (Bhattacharjee et al., 28 Jun 2025, Kang, 23 Jul 2025). Permutation-based tests on kernel mean discrepancies offer powerful nonparametric methods for detecting distributional causal effects (Fawkes et al., 2022).

6. Empirical Validation and Practical Examples

Empirical studies consistently indicate that doubly robust estimators outperform singly robust ones, especially in the presence of nuisance model misspecification, high-dimensional covariates, missing data, or outlier contamination. Simulation studies underline variance reduction (sometimes by factors exceeding 50%) and lower bias relative to standard IPW or outcome regression (Díaz, 2018, Benkeser et al., 2019, Kang, 23 Jul 2025).

Applications span the estimation of average treatment effects in high-dimensional health data (Du et al., 26 Mar 2024), distributional shifts in environmental exposures affecting mortality as random object outcomes (Bhattacharjee et al., 28 Jun 2025), causal survival analysis in multi-arm clinical trials with truncation by death (Tong et al., 9 Oct 2024), and robust causal feature selection in complex, nonlinear systems (Quinzan et al., 2023).

7. Implications and Ongoing Developments

Doubly robust estimands represent a unifying principle for robust causal inference, offering a safeguard against model misspecification and thereby facilitating more credible causal effect estimation in challenging scenarios. Ongoing research directions include: further optimization of variance through collaborative/targeted machine learning, extending robustness to settings of partial identifiability (e.g., instrumental variable and proxy settings), refinement of model selection criteria (such as doubly robust AIC-type methods (Baba et al., 2021)), and the generalization to complex outcomes and treatment mechanisms.

In sum, doubly robust estimators—by integrating model-based and design-based adjustment—form a cornerstone of contemporary causal inference, extending the reach of rigorous methodology to a wide range of modern, data-rich, and challenging empirical domains.