Causal Sensitivity Analysis & Exposure Mappings

• The paper introduces partial identification and sharp bounding techniques that yield robust causal effect estimates even when exposure mappings are misspecified.
• It employs cross-fitted, orthogonal pseudo-outcome estimation to mitigate bias from machine-learned nuisance parameters, achieving quasi-oracle convergence rates.
• The research demonstrates practical sensitivity analysis frameworks for network, shift–share, and continuous exposure designs to quantify mapping errors and inform inference.

Causal sensitivity analysis to exposure mappings examines the robustness of causal effect estimands and inferences when the mapping from underlying treatments or neighborhood configurations to observed "exposures" is either misspecified or subject to omitted variable bias. This area addresses scenarios with interference (where unit-level outcomes may depend on multiple units’ treatments), network-structured treatments, endogenous exposures, and environments in which researchers must specify a reduced exposure mapping out of necessity—but may be uncertain about its fidelity. Recent advances employ partial identification, sharp bounding, and influence-function-based estimation to provide valid, informative inferences under quantified misspecification or unmeasured confounding.

1. Foundations of Exposure Mappings and Sensitivity in Causal Inference

In experimental and observational studies with interference or complex treatment structures, exposure mappings are user-specified functions compressing the (potentially high-dimensional) assignment vector (or environmental context) into lower-dimensional "exposure" variables. For example, in network studies, an exposure mapping $g$ may summarize the treatment status of a unit's neighborhood by a weighted mean, a threshold, or through higher-order neighborhoods. Formally, for unit $i$, $Z_i = g(T_{\mathcal{N}_i})$, where $T_{\mathcal{N}_i}$ indicates the treatments of $i$'s neighbors (Schröder et al., 3 Feb 2026).

The traditional approach in such designs is to assume the chosen mapping fully determines potential outcomes—i.e., for any two treatment allocations $z, z'$ such that $g(z) = g(z')$, $Y_i(z) = Y_i(z')$. However, practical and substantive constraints often preclude this strong assumption ("correct specification"), leading to specification error when the true mapping $g^*$ differs from $g$ (Sävje, 2021). Sensitivity analysis in this context quantifies the impact of such misspecification on the identification, estimation, and interpretation of causal estimands.

2. Partial Identification Under Misspecified Exposure Mappings

Recent work establishes a formal partial identification framework when the exposure mapping may be misspecified (Schröder et al., 3 Feb 2026). Instead of point identification of estimands such as the average direct effect (ADE) or average spillover effect (ASE), one derives sharp upper and lower bounds for these quantities by constraining the possible deviation between the assumed and true exposure-propensity distributions.

For observed exposure $Z$ and true exposure $Z^*$, introduce sensitivity parameters $b^-,b^+$ that bound the ratio $P(Z^*=z|X)/P(Z=z|X)$. The parameterization

$$b^-(z,x)\;\leq\; \frac{P(Z^*=z|x)}{P(Z=z|x)}\;\leq\;b^+(z,x)$$

encodes plausible degrees of misspecification. Causal effects (e.g., direct effect $\psi(1,z)-\psi(0,z)$) are partially identified by optimizing over all data-generating distributions compatible with (i) the observed data and (ii) these propensity ratio bounds. The resulting bounds are closed-form, sharp, and recover point identification if $b^- = b^+ = 1$ (Schröder et al., 3 Feb 2026).

This framework generalizes to canonical exposure mapping classes:

• Weighted mean: Quantifies sensitivity when weights may deviate up to $\epsilon$ from a baseline specification.
• Threshold mapping: Handles uncertainty in threshold value, e.g., true threshold $c^*$ lying in $[c-\epsilon, c+\epsilon]$.
• Truncated/higher-order mappings: Models allowance for neglected higher-order interference.

These bounds are computable for both the conditional (CAPO) and average (APO) potential outcomes.

3. Estimation and Statistical Inference for Sensitivity Analysis

Orthogonal (Neyman-doubly robust) pseudo-outcome estimation is essential for valid inference under sensitivity analysis with machine-learned nuisance parameters (Schröder et al., 3 Feb 2026). Rather than direct plug-in, which suffers bias under flexible regression, a cross-fitted two-stage procedure is employed:

1. Estimate exposure and treatment propensities, conditional quantiles, and bound parameters.
2. Use these to construct an orthogonal pseudo-outcome that targets the upper or lower bound of the APO, achieving second-order bias in the presence of estimation error.
3. Regress the pseudo-outcome on covariates (cross-fitted) to estimate bounds.

A key result is that these estimators achieve quasi-oracle rates: for parametric rates of convergence of nuisance functions, the estimated bounds converge at rate $n^{-1/2}$ and are asymptotically normal. Confidence intervals follow via standard errors from the influence function logic.

In empirical analysis, intervals widen as $b^-, b^+$ are set more conservatively (i.e., as exposure mapping plausibility decreases). Conversely, for mapping errors supported by substantive knowledge, informative and non-degenerate intervals are often attainable (Schröder et al., 3 Feb 2026).

4. Bias Bounding, Misspecification, and Variance Inflation from Exposure Mapping Errors

Exposure mapping misspecification leads to specification error in estimators targeting exposure-based effects (e.g., Horvitz–Thompson estimators). Under general misspecification, the point estimator remains unbiased for a marginalized exposure effect, but its variance is inflated depending on the covariance of the specification errors across units: $\Var(\widehat{\tau}_{a,b}^{HT}) \leq \text{sampling noise} + \frac{20k^2}{p_{min}^2}(\phi_a+\phi_b) + 4(\phi_a+\phi_b) + O(\Delta_{des})$ where $\phi_e$ quantifies the average covariance of specification error for exposure $e$ and $p_{min}$ is the minimal exposure probability (Sävje, 2021). By imposing an upper bound $\Gamma$ for $\phi_e$, one charts a maximum mean squared error (MSE) curve, enabling graphical or tabular assessment of sensitivity.

Variance estimators themselves may be sensitive to misspecification, and partial network or spillover structure can be encoded to produce more conservative variance estimates. This approach enables reporting of effect estimates with sensitivity curves showing under what levels of mapping error the empirical conclusions would change (Sävje, 2021).

5. Sensitivity Analysis for Causal Inference in Networked and Shift-Share Environments

Network interference, shift-share regressions, and price exposure designs involve particularly convoluted exposure mappings. In shift–share designs, for example, exposure mappings aggregate or weight sectoral shocks by region-specific exposure; identification hinges on exogeneity and independence conditions across sectoral shocks. When these are questionable, a simple sensitivity analysis decomposes the 2SLS estimand into a convex average of target effects plus a "contamination bias" term encoding the mapping error. By credibly bounding this term, researchers obtain intervals for target effects and can identify breakdown points for sign changes (Moreno-Louzada et al., 10 Dec 2025).

In matching-based observational studies with continuous exposures, sensitivity to unmeasured confounding is modeled using an exponential tilting approach or, for binary outcomes, via probabilistic lattices. Design sensitivity can be computed, and mixed-integer programming enables sensitivity analysis in heterogeneous effect scenarios, quantifying the identification price of mapping/matching inexactitude (Zhang et al., 2024).

6. Extensions: Continuous Exposures, Partial Identification, and Sensitivity Functions

Sensitivity models have been generalized to continuous treatments with techniques including odds-ratio bounding of the generalized propensity score, Rosenbaum and Marginal sensitivity functions, and pseudo-outcome regression frameworks (Zhang, 9 Nov 2025, Dalal et al., 19 Aug 2025). These allow for:

• Rich, function-valued sensitivity modeling that can vary across exposure levels.
• Nonparametric, machine learning-compatible estimation for bounds of dose–response functions or average derivative effects (ADE).
• Sharp, closed-form identification of bounds via Lagrange or duality arguments.
• Pointwise and uniform confidence intervals achieved via efficient influence function estimation.

In practical terms, these frameworks allow researchers to diagnose under what strength or form of unmeasured confounding, or exposure mapping misspecification, an observed causal effect would be explained away, and to plot or tabulate the worst-case bounds as a function of sensitivity parameters (Zhang, 9 Nov 2025, Dalal et al., 19 Aug 2025).

7. Calibrating Sensitivity Parameters and Practical Implementation

The interpretable calibration of sensitivity parameters (e.g., $b^-, b^+, \Gamma$, contamination bounds) is central to the applied use of these methods. In network studies, the parameter can be related to the plausibility of deviation in exposure weighting or to evidence from sub-sampled validation. In shift–share and continuous exposure studies, parameters can be grounded in sectoral price shock correlations or domain-driven assessment of unmeasured confounding structures. Implementation is facilitated by available R/Python code and packages specifically designed for network exposure sensitivity, shift–share designs, and continuous-treatment settings (Moreno-Louzada et al., 10 Dec 2025, Zhang, 9 Nov 2025, Schröder et al., 3 Feb 2026, Zhang et al., 2024).

These developments render it feasible for analysts to present rigorous, nonparametrically valid, and interpretable intervals for causal parameters under explicit models of misspecification and unmeasured confounding in exposure mappings.