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Sensitivity analysis for causal mediation: bridge score, sharp sensitivity bounds, and calibration

This paper introduces the "bridge score" to derive sharp sensitivity bounds for unobserved mediator-outcome confounding in causal mediation analysis and provides calibration methods and a Bayesian algorithm to operationalize these bounds for robust inference.

Original authors: Yuki Ohnishi, Fan Li

Published 2026-05-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yuki Ohnishi, Fan Li

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out why a specific treatment (like a new medicine or a political message) works. You suspect it works through a specific "middleman" (a mediator). For example, a negative news story might make people feel angry (the mediator), which then makes them support stricter immigration laws (the outcome).

To prove this, scientists usually assume that once they account for the anger, there are no other hidden factors secretly linking the news story to the policy support. But what if there are hidden factors? Maybe the people who got the angry news also happened to be naturally more anxious, and that anxiety drives both the anger and the policy support. If you don't account for this, your results are wrong.

This paper introduces a new, more robust way to test how much those hidden factors could be messing up your results. Here is the breakdown using simple analogies:

1. The Problem: The "Hidden Puppeteer"

In standard analysis, researchers pretend the "middleman" (the mediator) is innocent. But in reality, there might be a "hidden puppeteer" (an unmeasured confounder) pulling strings on both the mediator and the outcome.

The old way: Researchers used to guess a single number (like a correlation coefficient) to represent this hidden puppeteer. The problem is that this guess was often too simple or too rigid, like trying to describe a complex storm with a single wind speed number.
The new way: This paper says, "Let's not guess a single number. Let's build a better map to see exactly how strong that puppeteer could be."

2. The Solution: The "Bridge Score"

The authors invent a new tool called the Bridge Score.

The Analogy: Imagine you are trying to cross a river. On one side is the "Control Group" (people who didn't get the treatment), and on the other is the "Treatment Group" (people who did).
Usually, you look at the people on both sides separately. But the Bridge Score looks at a specific person and asks: "If this person had been in the Control group, how likely were they to have this specific mediator value? And if they were in the Treatment group, how likely were they to have it?"
It creates a two-dimensional "ID card" for every person based on these two probabilities. This ID card acts as a "bridge" that balances the two groups perfectly.
Why it helps: Instead of trying to account for every single detail about a person (age, income, location, etc.) to find the hidden puppeteer, the Bridge Score condenses all that information into this single, powerful ID card. It's like using a high-resolution fingerprint instead of a blurry photo to identify someone.

3. The "Sharp Bound": The Safety Net

Once they have this Bridge Score, the authors calculate a "Sharp Bound."

The Analogy: Think of this as a safety net with a very precise ceiling. If you assume the hidden puppeteer is this strong, the error in your results can't possibly be higher than this ceiling.
Why "Sharp"? Many old methods gave a safety net that was so wide (like a net covering the whole sky) that it wasn't useful. This new net is tight. It tells you the exact maximum amount of error possible given your assumptions. It's the difference between saying "The error could be anywhere between 0 and 100%" versus "The error is definitely between 0 and 12%."

4. Calibration: "The Ruler Test"

The biggest challenge is: "How do we know how strong the hidden puppeteer actually is?" We can't measure it directly.
The paper offers two ways to calibrate (measure) this strength using things we can see:

Method A: The "Benchmark" (The Known Ruler)
- Imagine you have a known variable, like "Income." You know how much Income affects the outcome.
- You ask: "Is our hidden puppeteer stronger than Income? Is it twice as strong?"
- The Innovation: The authors suggest using the Rank of the income (e.g., "Is this person in the top 10%?") rather than the raw dollar amount. This is like measuring height in "inches" vs. "centimeters." Using the rank makes the test fair and stable, regardless of whether you measure money in dollars or euros. It prevents the results from changing just because you changed the units.
Method B: The "Residual Budget" (The Leftover Energy)
- Imagine you have a bucket of "unexplained energy" (variation in the outcome that your model couldn't explain).
- You ask: "How much of this leftover energy could the hidden puppeteer be stealing?"
- You set a limit: "The puppeteer can't steal more than 20% of the leftover energy." This gives you a concrete, data-driven limit on how much error is possible.

5. The Result: A Clearer Picture

By combining the Bridge Score (the ID card), the Sharp Bound (the tight safety net), and the Calibration (the ruler test), the authors provide a way to say:

"Even if there is a hidden puppeteer, and even if it is as strong as [Income] or [20% of the leftover energy], our conclusion about the treatment effect is still likely to hold true. Here is the exact range of how much it could change."

Summary

This paper doesn't just say "be careful about hidden factors." It builds a precision instrument to measure exactly how much those hidden factors could hurt your study.

It uses a Bridge Score to simplify complex data.
It creates a Sharp Bound to define the exact limits of error.
It uses Calibration to let researchers use real-world data (like income or leftover variation) to set realistic limits on their assumptions.

This allows scientists to be much more confident in their conclusions about how and why treatments work, even when they can't measure every single hidden factor in the world.

Technical Summary: Sensitivity Analysis for Causal Mediation via Bridge Score, Sharp Bounds, and Calibration

1. Problem Statement

Causal mediation analysis decomposes total treatment effects into natural direct effects (NDE) and natural indirect effects (NIE). Identification of these effects typically relies on the sequential ignorability assumption, which requires that the mediator is ignorable given treatment and baseline covariates. This assumption is untestable; any unmeasured factor jointly driving the mediator and the outcome violates it.

While sensitivity analysis for mediator ignorability exists, the authors identify three specific statistical challenges in the literature regarding nonparametric sensitivity analysis on the additive scale (mean difference), which is the most common scale in scientific research:

Interpretability vs. Localization: Existing scalar parameters (e.g., residual correlations) collapse violations that vary across treatment arms and covariate profiles. Conversely, fully localized sensitivity functions are difficult to elicit and calibrate in high-dimensional covariate settings. There is a lack of a general low-dimensional reduction that preserves mediator information without requiring full covariate specification.
Lack of Sharp Additive Bounds: Sharp nonparametric bounds exist for the risk ratio scale (suitable for binary outcomes), but no analogous sharp additive (mean difference) envelope is available. Existing additive approaches rely on restrictive parametric bias formulas or uninformative Manski-style bounds.
Fragile Calibration: Operational calibration for continuous benchmarks often depends on arbitrary units or support widths. Furthermore, scale-coherent summaries (like E-values) collapse the two-parameter structure of the bound into a single scalar, losing interpretability.

2. Methodology

The Bridge Score
The authors introduce the bridge score, a two-dimensional vector defined at a common mediator value $m$ and covariate $x$ :
$B(m, x) = (f_0(m | x), f_1(m | x))$
where $f_a(m|x)$ is the density of the mediator under treatment arm $a$ .

Balancing Property: Conditional on the bridge score $B(m)$ , the observed covariates $X$ are balanced with respect to the mediator event $M=m$ . This allows the bridge score to serve as a balancing score for the mediator stage, analogous to the propensity score for treatment assignment.
Dimension Reduction: The bridge score reduces the sensitivity object from a function of the full covariate vector $X$ to a function of the low-dimensional score $B(m)$ , while retaining the arm-specific mediator laws necessary for cross-world identification.

Sharp Sensitivity Bounds
The authors define the bridge score sensitivity function $\Delta_a(m, b)$ as the residual confounding induced by conditioning on the observed mediator within arm $a$ , after conditioning on the bridge score.
Under a latent scalar confounder assumption ( $U$ ), they derive a sharp pointwise additive envelope for $\Delta_a(m, b)$ :
$|\Delta_a(m, b)| \leq \Xi_a(m, b) = \eta_a(m, b) \frac{\gamma_a(m, b) - 1}{\gamma_a(m, b)}$
where:

$\gamma_a(m, b)$ is the residual selection ratio, measuring the tilt in the latent confounder distribution induced by conditioning on $M=m$ .
$\eta_a(m, b)$ is the outcome range parameter, measuring the spread of the outcome mean conditional on the latent confounder within the bridge score stratum.

The authors prove that these parameters at the bridge score level are no larger (tighter) than their counterparts at the full covariate level, provided certain decoupling conditions hold.

Calibration Approaches
To make the bounds operational, two calibration strategies are proposed:

Benchmark Calibration: The latent confounder is calibrated against an observed covariate $W$ . The authors introduce a rank-based version of this calibration. By transforming $W$ to its fractional rank, the resulting envelope becomes invariant to monotone re-expressions of the benchmark and robust to heavy-tailed distributions, while preserving the two-parameter structure ( $\eta, \gamma$ ).
Residual Budget Calibration: This approach scales the outcome range parameter $\eta_a$ based on the residual variation of the outcome after adjusting for the mediator and bridge score. A user-specified parameter $k_a \in [0,1]$ controls the share of residual variance attributed to latent confounding, while a grid value $g_a$ controls the selection ratio.

Inference Algorithm
The framework is operationalized via a Bayesian g-computation algorithm. This approach:

Propagates uncertainty from the observed data model (mediator densities and outcome regressions).
Incorporates uncertainty from the calibration quantities (benchmark or residual budget).
Uses a scalar functional reduction to collapse the infinite-dimensional sensitivity function into arm-specific scalars ( $\bar{\Delta}_0, \bar{\Delta}_1$ ), allowing for finite-dimensional priors on the sensitivity corrections.
Generates posterior draws for the NDE and NIE that fully account for all sources of uncertainty.

3. Key Contributions

Bridge Score Concept: A novel low-dimensional vector that balances covariates with respect to the mediator and serves as the localization scale for sensitivity functions, avoiding the need for full covariate specification.
Sharp Additive Bound: The derivation of a sharp, nonparametric additive envelope for mediator-outcome confounding, parametrized by interpretable residual selection and outcome range parameters. This fills the gap left by risk-ratio bounds which are less natural for continuous outcomes.
Scale-Coherent Calibration: The introduction of rank-based benchmark calibration that is invariant to monotone transformations and preserves the two-parameter structure of the bound, alongside a residual budget approach.
Operational Inference: A Bayesian g-computation implementation that integrates sensitivity analysis directly into the estimation of natural effects, propagating uncertainty from nuisance parameters and sensitivity corrections.

4. Results and Illustration

The paper illustrates the framework using the Brader et al. (2008) immigration framing experiment.

Setup: Treatment ( $A$ ) is a negative tone/Latino framing; Mediator ( $M$ ) is negative emotional response; Outcome ( $Y$ ) is perceived harm from immigration.
Findings:
- The rank-based benchmark calibration produced tighter envelopes than the raw-scale version, demonstrating the stability of the rank transformation against empirical support width.
- The residual budget calibration showed that the envelope widens monotonically with increased selection strength ( $g$ ) and outcome range budget ( $k$ ).
- The analysis demonstrated that robustness conclusions are driven by interpretable amplification factors (e.g., "latent confounder is 1.5x stronger than income") rather than opaque model parameters.
- The Bayesian g-computation successfully propagated uncertainty from the bridge score estimation and calibration into the posterior distributions of the NDE and NIE.

5. Significance and Claims

The authors position this work as a complementary tool rather than a replacement for existing methods like the parametric $\rho$ -based analysis (Imai et al., 2010) or risk-ratio bounds (Ding and VanderWeele, 2016).

The significance of the framework lies in its ability to:

Provide a nonparametric sensitivity analysis on the additive scale without imposing functional form restrictions on the outcome or mediator.
Offer interpretable sensitivity parameters that are localized to the bridge score, allowing for tighter bounds than full-covariate specifications.
Enable scale-coherent calibration that avoids the pitfalls of unit dependence and scalar collapse found in other methods.
Facilitate uncertainty propagation in a unified Bayesian framework, moving beyond point estimates to full posterior inference under sensitivity assumptions.

The paper concludes that the bridge score framework provides a principled, low-dimensional reduction for sensitivity analysis that is particularly practical when baseline covariates are moderately high-dimensional, while maintaining the structural asymmetry required for cross-world natural effect estimation.