Sensitivity Analysis for Unmeasured Confounding in Triadic Structures

In our previous lesson, we established powerful adjustment strategies—backdoor, frontdoor, and double negative controls—to estimate causal effects in the face of confounding. However, each of these methods hinges on a critical, untestable assumption: that we have correctly identified and accounted for all relevant confounders. In your triadic research, where confounders can be latent psychological traits or complex dyadic dynamics, this assumption is particularly fragile.

This brings us to a fundamental question of scientific humility: "What if I'm wrong?" Today, we address this question by learning how to conduct a sensitivity analysis. Instead of assuming away unmeasured confounding, we will quantify its potential impact. Our goal is to determine how strong an unmeasured confounder would need to be to alter our conclusions, thereby assessing the robustness of our causal claims. This moves us from a world of binary "confounded/unconfounded" thinking to a more nuanced understanding of evidential strength.

The Core Idea: Quantifying Robustness

At its heart, a sensitivity analysis is a "what-if" exercise. We've estimated a causal effect, but we suspect an unmeasured confounder, $U$ , might be biasing our result. The analysis proceeds by positing a hypothetical U and parameterizing its relationship with the treatment ( $T$ ) and the outcome ( $Y$ ). We then calculate how strong these relationships must be to change our conclusion—for example, to reduce our estimated effect to zero or make it statistically insignificant.

This video provides an excellent visual and conceptual introduction to this process.

7 - Unobserved Confounding, Bounds, and Sensitivity Analysis

This video from Brady Neal's Causal Inference course offers a clear introduction to sensitivity analysis, starting from the problem of unobserved confounding and using a simple linear model to build intuition.

Please watch the following segments: Introduction: Watch from the beginning to the start of the main section to understand how sensitivity analysis fits in as a response to unobserved confounders. Linear Setting & Contour Plots: Watch from the beginning of the simple linear setting until the start of the proof. Pay close attention to how the bias is parameterized and visualized using contour plots. These plots are a cornerstone of sensitivity analysis.

The key takeaway from the video is the contour plot. It provides a map of how the "true" causal effect ( $δ$ ) changes for different values of the confounding parameters (e.g., beta_u and 1/alpha_u). Your task as a researcher is to argue that the plausible values for these confounding parameters lie within a region of this map that supports your conclusion.

Let's now explore two of the most widely used frameworks for conducting such an analysis.

Approach 1: Sensitivity Analysis in Mediation

Many of your research questions will involve mediation pathways that span multiple perspectives. For example: Does a specific wife-bull activity (T) reduce the husband's jealousy (Y) by first increasing the wife's sense of reassurance (M)?

The problem is that an unmeasured confounder, $U$ —say, the couple's baseline communication quality—could affect both the wife's reassurance ( $U \to M$ ) and the husband's jealousy ( $U \to Y$ ). This violates the "no unmeasured confounder" assumption required for mediation analysis.

A common sensitivity analysis for mediation, developed by Imai and colleagues, quantifies this potential confounding using a single parameter, $ρ$ (rho).

$ρ$ : This parameter represents the correlation between the error terms of the regression models for the mediator and the outcome. If there are no unmeasured confounders that affect both $M$ and $Y$ , then $ρ = 0$ . A non-zero $ρ$ indicates the presence of such confounding.

The analysis then recalculates the Average Causal Mediation Effect (ACME) for various values of $ρ$ . The crucial output is the value of $ρ$ that would be required to make the ACME equal to zero. A larger value of $ρ$ indicates that your result is more robust to unmeasured confounding.

This practical guide explains the methodology and its implementation in R.

A Practical Guide to Causal Mediation Analysis - Frontiers

This article provides a step-by-step walkthrough of causal mediation analysis, with a clear focus on conducting and interpreting a sensitivity analysis.

Please read two key sections: Start with the section titled Sensitivity Analysis. This introduces the parameter \rho and explains its role in quantifying confounding. Next, read Step 3, which shows how to implement this using the medsens() function in R and how to interpret the results. Notice the advice to compare \rho values across studies to benchmark what constitutes a "large" or "small" degree of confounding.

This approach provides a tangible number. If your analysis shows that a $ρ$ of 0.4 is needed to nullify your effect, you can then argue whether it's plausible for an unmeasured factor to induce a 0.4 correlation between the residuals of your mediator and outcome models, after accounting for all observed covariates.

Approach 2: The E-Value for General Causal Effects

While the $ρ$ -based method is excellent for mediation, a more general and increasingly standard metric is the E-value. It can be applied to any estimated causal association, not just indirect effects.

The E-value is defined as:

The minimum strength of association, on the risk ratio scale, that an unmeasured confounder would need to have with both the treatment and the outcome to explain away the observed association.

For example, if you find an activity has a risk ratio (RR) of 2.0 for increasing marital satisfaction, the E-value would tell you the minimum RR an unmeasured confounder U would need to have with both the activity and satisfaction to make the true causal effect an RR of 1.0 (i.e., no effect).

The paper below provides the formal foundation for the E-value. Given your statistical background, the derivation will be quite accessible.

Technical Considerations in the Use of the E-Value - De Gruyter Brill

This paper by VanderWeele and Ding, who developed the E-value, addresses key technical questions about its calculation and interpretation.

Read the Abstract and Introduction to get the core definition. Then, carefully study Section 2, Calculation of E-values. This section formally defines the key parameters RR_UD (confounder-outcome association) and RR_EU (exposure-confounder association) and provides the formula for the E-value: E\text{-value} = RR_{obs} + \sqrt{RR_{obs}(RR_{obs}-1)}. Finally, read Section 3, The E-value as a transformation. This section makes a compelling case for why the E-value, while a simple transformation, is a more intuitive scale for judging robustness than the confidence interval alone.

Applying the E-Value in the Triad

Let's apply this to your context. Suppose you are testing whether a specific communication protocol (T: e.g., "post-date debriefing") reduces the husband's jealousy (Y). You run an analysis and find an odds ratio of 0.5, with a 95% confidence interval of [0.35, 0.71].

Convert to Risk Ratio: The E-value is on the risk ratio scale. Since jealousy is likely a common outcome, we can't assume the OR ≈ RR. We would use a standard conversion formula (or the approximations mentioned in the E-value literature). Let's say the estimated RR is 0.7 and the CI limit closest to the null is 0.8.
Calculate E-values:
- For the point estimate (RR=0.7): $E -value = 1/0.7 + (1/0.7) (1/0.7 - 1) \approx 2.14$ . (We use 1/RR for protective effects).
- For the CI limit (RR=0.8): $E -value = 1/0.8 + (1/0.8) (1/0.8 - 1) \approx 1.81$ .
Interpret: The E-value for the confidence interval is 1.81. This means that an unmeasured confounder U (e.g., the husband's pre-existing attachment style) that was associated with both the use of the debriefing protocol and the husband's jealousy by a risk ratio of 1.81-fold each, above and beyond the covariates already in the model, could potentially explain away the effect. To be robust, you would need to argue that it is implausible for such a strong confounder to exist unmeasured.

From Numbers to Judgment: Interpreting Sensitivity Plots

Both approaches ultimately lead to a judgment call. Is a $ρ$ of 0.3 large? Is an E-value of 1.8 plausible? This is where visual plots and benchmarking become essential.

Let's return to the type of contour plot you saw in the video.

This plot visualizes a sensitivity analysis. The axes represent the strength of an unmeasured confounder's correlation with the mediator (`rCSm`) and outcome (`rCly`). The dashed line shows combinations that would reduce the estimated causal effect (the 'b path' of a mediation) to zero. The solid line shows combinations that would render the effect statistically non-significant. A robust finding requires this "null region" to be far from the origin, demanding very strong confounding to invalidate the result.

This plot provides a map of the danger zone. To defend your finding, you must argue that the true confounding parameters lie somewhere inside the contours, away from the lines that would nullify your result.

How do you make that argument? One powerful technique is benchmarking. You can measure the association of observed covariates with your treatment and outcome and plot them on the same graph. If your most powerful known confounder falls well short of the "null contour," it strengthens the case that an unmeasured confounder is also unlikely to be strong enough to erase your effect.

The following video discusses this idea of benchmarking and introduces the modern, flexible methods that produce plots like the one above.

7.6 - More Flexible Sensitivity Analysis

This clip, a continuation of the previous video, discusses more advanced and flexible approaches to sensitivity analysis, which are highly relevant for complex, non-linear relationships common in psychological research.

Please watch these segments: First, watch the discussion of a paper by Cinelli and Hazlett. This work allows for multiple confounders and non-parametric relationships, which is crucial for your triadic model. Next, watch the segment on informal benchmarking. It explains the logic of using observed confounders as a yardstick but also provides a critical caution about how this can be misleading—a nuance that is important for rigorous analysis. Finally, review the mention of methods that can incorporate arbitrary machine learning models. This points toward the frontier of sensitivity analysis, where you are no longer constrained by simple linear assumptions.

Conclusion

In this lesson, we have moved beyond simply adjusting for confounders to rigorously questioning the limits of those adjustments. A causal claim in observational research is never definitive, but a sensitivity analysis allows you to quantify its resilience.

Key Takeaways:

Sensitivity analysis answers the question: "How strong would an unmeasured confounder need to be to change my conclusion?"
For mediation models, you can use the correlation parameter $ρ$ to quantify confounding. You find the value of $ρ$ that nullifies the indirect effect.
For general causal effects, the E-value provides a standardized metric. It is the minimum association an unmeasured confounder must have with both treatment and outcome to explain away the result.
Visual plots and benchmarking against observed confounders are crucial for interpreting sensitivity parameters and making a substantive argument about the robustness of your findings.

In our last lesson, we used mediation as a hypothetical case for confounding. Now that you have the tools to assess the robustness of such claims, our next lesson will officially begin Module 9 by diving deep into the specifics of building and testing these complex pathways: we will learn to specify a mediation model for a causal pathway that spans multiple perspectives in the triad.