Counterfactual Contrasts for Attributing Satisfaction Changes to Mediating Pathways

1. The Challenge of Multiple Mediating Pathways

Imagine we want to understand the effect of a new living arrangement (let's call it treatment A) on Partner B's satisfaction (Y). We hypothesize that the treatment doesn't just have a direct effect; it also operates through two psychological mechanisms: increasing Partner B's sense of autonomy (Mediator M1) and bolstering his sense of security (Mediator M2).

This situation is represented by the causal graph below. The treatment A influences both mediators M1 and M2, which in turn affect the outcome Y.

A key challenge is that these mediators are rarely independent. A greater sense of autonomy (M1) might itself contribute to a greater sense of security (M2), or both might be driven by a common underlying personality trait (an unobserved confounder like U1). Simply analyzing the A -> M1 -> Y and A -> M2 -> Y pathways separately would yield biased estimates because it fails to account for the influence of the other mediator. We need a method that can parse these intertwined effects.

2. Defining Path-Specific Indirect Effects

To isolate the contribution of a single mediator, we must extend the counterfactual logic from our last lesson. Recall the Natural Direct Effect (NDE) and Natural Indirect Effect (NIE), which gave us a two-part decomposition of the total effect. With multiple mediators, we need to define Path-Specific Indirect Effects (PS-IEs).

A recent paper comparing various methods for multiple mediator analysis provides the formal definitions we need.

Let's translate the definitions from the paper to an additive scale, which is more intuitive for our satisfaction outcome. Let A be the treatment (0 for control, 1 for treated), and M1 and M2 be our two mediators (autonomy and security).

The path-specific indirect effect through M1 (autonomy) can be defined as:

In words, this contrast represents the change in satisfaction under the treatment condition (A=1) that is caused by M1 changing from the value it would take under control (M1(0)) to the value it would take under treatment (M1(1)), all while M2 is held fixed at the value it would have had under control (M2(0)).

Symmetrically, the path-specific indirect effect through M2 (security) is:

This represents the change in satisfaction caused by M2 changing from its control to its treatment value, while A is held at the treatment level and M1 is also at its treatment level.

It's important to recognize that these definitions imply an ordering (first we assess the change through M1, then the change through M2). The sum of these PS-IEs and the NDE may not equal the total effect if there is interaction between the mediators. For a more complete picture, we can turn to the four-way decomposition.

3. A Deeper Look: The Four-Way Decomposition

When mediators can interact with the exposure, the simple NDE + NIE decomposition is insufficient. Tyler VanderWeele proposed a four-way decomposition that splits the total effect into four components, giving us a more precise language for attribution.

As the slides explain, this framework allows us to isolate the portion of the total effect due entirely to mediation (the Pure Indirect Effect, PIE) and the portion due to an interaction between the treatment and the mediator (the Mediated Interaction, INT_MED). The total effect attributable to the mediating pathway is the sum PIE + INT_MED.

When you have multiple mediators, as in our scenario, this "mediated portion" (PIE + INT_MED) is the quantity that you would then seek to attribute to the individual pathways (M1 vs. M2) using methods designed for estimating path-specific effects.

4. Methods for Estimating Path-Specific Effects

Defining these effects is one thing; estimating them from data is another. Given your interest in applied methodology, it's crucial to know what tools are available for this task. The arXiv paper you read earlier compares several modern methods. Two notable approaches capable of estimating path-specific indirect effects for correlated, contemporaneous (i.e., not causally ordered) mediators are Wang's method and Jérolon's method.

From this reading, several key points emerge for your analytical work:

• Capability: Not all mediation analysis methods can estimate path-specific effects. The regression-based approaches you might be familiar with often only estimate the joint indirect effect of all mediators combined. To attribute effects to a single pathway, you need specialized methods like those of Wang or Jérolon.
• Assumptions: These methods rely on strong, untestable assumptions. Beyond the standard assumptions for mediation (no unmeasured confounding of exposure-outcome, exposure-mediator, and mediator-outcome paths), identifying path-specific effects requires an additional assumption: essentially, that there is no unmeasured confounder of the mediator-mediator relationship itself. In our DAG, U1 affecting both M1 and M2 could be problematic if not correctly modeled.
• Performance: There are trade-offs. The paper suggests Wang's method provides better estimates (low bias, good coverage) for path-specific effects but is computationally intensive and harder to implement (SAS macro). Jérolon's method is more accessible (R package) but may have poorer performance for path-specific effects, especially with binary outcomes.
• Data Types Matter: The type of mediator variable (continuous, binary) has a significant influence on which method performs best. This is a critical consideration during the experimental design phase.

5. Application to the Vee Triad

Let's synthesize this for your project. Suppose you want to test whether a new living arrangement (A) increases satisfaction (Y) for Partner B, and you believe it works by increasing his autonomy (M1) and security (M2).

Your analysis plan to attribute the satisfaction change would be:

1. Model Specification: Formulate a structural causal model based on your hypotheses, represented by a DAG like the one shown earlier.
2. Define Estimands: Formally write down the counterfactual contrasts for the path-specific indirect effects PS-IE_autonomy and PS-IE_security.
3. Data Collection: In your experiment, you would need to measure not only the treatment A and outcome Y but also the mediating variables M1 and M2 (e.g., using validated psychological scales for autonomy and security).
4. Method Selection: Based on your mediator data types (likely continuous scores) and computational setup, you would choose an estimation method. For example, you might opt for Wang's method for its superior performance with path-specific effects if you are comfortable with the implementation.
5. Estimation & Interpretation: After running the analysis, you might find that PS-IE_autonomy = +0.8 and PS-IE_security = +0.1 (on a 1-10 satisfaction scale). This would support the conclusion: "The new living arrangement significantly boosts Partner B's satisfaction. Our analysis attributes most of this gain (0.8 points) to the pathway through enhanced autonomy, with a negligible contribution from the security pathway." This level of detail provides a powerful, actionable explanation.

Conclusion

In this lesson, you have learned how to move beyond simple effect decomposition to perform causal attribution. By defining and estimating path-specific indirect effects, you can dissect a total effect and quantify the contribution of individual mediating mechanisms, even when those mechanisms are correlated.

Key Takeaways:

• Attributing an outcome change to a single mediating pathway requires estimating Path-Specific Indirect Effects (PS-IEs).
• PS-IEs are defined using nested counterfactual contrasts that isolate the change in one mediator while controlling for the others.
• Estimating PS-IEs in a multiple-mediator setting requires specialized methods (e.g., Wang, Jérolon) that can model the joint distribution of the mediators.
• The choice of method involves practical trade-offs regarding assumptions, performance, computational cost, and the nature of the data.

This ability to attribute effects is a cornerstone of sophisticated causal explanation. In our next lesson, we will leverage this deep understanding of causal pathways to explore concepts of necessity and sufficiency, asking whether a given intervention is the only way or a guaranteed way to achieve a desired outcome.