Connecting Living Arrangements to Satisfaction: The Role of Psychological Mechanisms

Introduction

In our previous lesson, we established a robust causal model for the vee-triad, incorporating not only direct interference between partners but also the crucial role of shared-environment confounders. By adding nodes for factors like Financial_Situation, we acknowledged that external pressures can create spurious correlations, and we identified conditioning as the primary tool to block these non-causal "backdoor" paths.

This lesson advances our model from asking if a living arrangement affects satisfaction to exploring how and why. We will now focus on the learning outcome: Represent mediating pathways from living arrangement through psychological mechanisms (e.g., security) to satisfaction. A mediator is an intermediate variable that lies on a causal path from a treatment to an outcome. By modeling these pathways, we can start to unpack the psychological machinery of the triad, testing specific hypotheses about which mechanisms (like autonomy, security, or jealousy) are the actual levers of change.

1. The Core Idea: What is Mediation?

At its heart, mediation analysis is about opening the "black box" of causality. Instead of just observing that a treatment $A$ causes an outcome $Y$ , we hypothesize that $A$ causes some intermediate variable $M$ (the mediator), which in turn causes $Y$ .

A fantastic conceptual introduction to this idea is provided in the following video, which uses a clear biomedical example: the effect of sugar consumption on diabetes is mediated by insulin resistance.

What is Mediation Analysis?

This video from the QuantFish channel provides a concise and clear conceptual overview of mediation analysis.

First, please watch the initial segment that defines the core concepts, from the introduction. This will establish the roles of the predictor (X), mediator (M), and outcome (Y). Next, watch the explanation of how the effects are quantified, from direct and indirect effects. This section demonstrates how a total effect is decomposed and how the indirect effect can be calculated as a product of path coefficients (a \times b). Given your background in statistics, you'll recognize this as a straightforward application of substituting one regression equation into another.

As the video illustrates, when a mediator is present, the total effect of the treatment on the outcome can be decomposed into two components:

The Direct Effect (DE): The portion of the effect that does not pass through the mediator.
The Indirect Effect (IE): The portion of the effect that is transmitted through the mediator.

This decomposition is the primary goal of mediation analysis. It allows us to quantify the importance of a specific causal pathway.

2. Formalizing Mediating Pathways in a DAG

We can represent these concepts formally using the language of Directed Acyclic Graphs (DAGs) and counterfactuals. As we've discussed, a mediator is simply a node that causally sits between the treatment and the outcome.

Let's quickly review the graphical definition of a mediator from the UK Reproducibility Network video we examined in a previous lesson.

Introduction To Causal Inference And Directed Acyclic Graphs

This video, which we've used before, gives a very clear rule for identifying a mediator in a DAG.

Please re-watch the brief segment that provides the definition of a mediator. The key insight is its temporal and causal position: it occurs after the exposure and before the outcome, channeling part of the effect.

The standard mediation model is represented by a simple DAG where the treatment A affects both the mediator Z and the outcome Y, and the mediator Z also affects the outcome Y. This structure explicitly defines the direct path ( $A \to Y$ ) and the indirect path ( $A \to Z \to Y$ ). The following resource provides the formal definitions based on Structural Causal Models (SCMs) and counterfactuals.

Chapter 7 Causal Mediation Analysis | [ACIC 2022 ... - tlverse

This guide from the tlverse project offers a concise but formal introduction to mediation, linking the concepts to counterfactual outcomes.

First, read the introduction to grasp the core distinction between direct and indirect effects. Then, review the section "Data Structure and Notation" to see the formal SCM equations and the corresponding standard mediation DAG. In our project, A is the Living Arrangement, Z is a psychological mechanism like 'Security', and Y is 'Satisfaction'. Finally, in the section "Defining the Natural Direct and Indirect Effects", observe how the Average Treatment Effect (ATE) is decomposed into the Natural Direct Effect (NDE) and Natural Indirect Effect (NIE). This decomposition provides the formal counterfactual foundation for mediation analysis.

3. The Critical Assumption: Sequential Ignorability

Estimating the direct and indirect effects is not merely a descriptive exercise; it is a causal claim. As such, it rests on a crucial and very strong assumption. For our randomized experiment, we can be confident that there are no confounders for the $A \to Y$ and $A \to Z$ relationships. However, the $Z \to Y$ relationship is observational. The mediator is not randomly assigned.

This requires an assumption known as Sequential Ignorability.

[PDF] A General Approach to Causal Mediation Analysis - Kosuke Imai

This foundational paper by Imai, Keele, and Tingley clearly articulates the key assumption underpinning causal mediation analysis.

Please read the section titled "Sequential Ignorability Assumption." This assumption has two parts. The first part, ignorability of the treatment, is satisfied by our experimental design. The second part is the critical one for us: we must assume that the mediator is ignorable given the treatment and pre-treatment confounders. This means we are assuming there are no unobserved confounders of the mediator-outcome relationship. For example, in our triad, if we are modeling the path LA -> Security_A -> Satisfaction_A, we must assume there isn't some unobserved personality trait (e.g., U_Anxiety) that both reduces a person's baseline sense of security and their baseline satisfaction, independent of the living arrangement. This is an untestable assumption that relies entirely on your domain expertise.

4. Mediation in Interdependent Systems: Actor and Partner Effects

Now, let's connect this to the complex, interdependent system of the vee-triad. The simple $A \to Z \to Y$ model is a building block, but in our case, one person's treatment can affect another person's outcome. Mediation can happen along these spillover pathways as well.

The Actor-Partner Interdependence Mediation Model (APIMeM) is a framework designed for precisely this kind of situation. The diagram below shows an APIMeM for a patient-caregiver dyad.

An Actor-Partner Interdependence Mediation Model (APIMeM). This shows how mediation can operate in an interdependent system. Notice the pathways for both "actor" effects (within-person) and "partner" effects (between-person).

This model elegantly separates several distinct mediating pathways:

Actor-Mediated Effect: The path from a person's own stress to their own distress, via their own disclosure. In the diagram, this is Perceived Stress (Patient) -> Distress-disclosure (Patient) -> Emotional Distress (Patient).
Partner-Mediated Effect: The path from one person's stress to their partner's distress, mediated by the partner's mechanism. In the diagram, this is Perceived Stress (Patient) -> Distress-disclosure (Caregiver) -> Emotional Distress (Caregiver).

Applying this to our triad, we can hypothesize and model several distinct mediation pathways stemming from the single Living_Arrangement treatment:

Actor-Mediated: LA -> Autonomy_A -> Satisfaction_A (How the LA affects A's satisfaction through A's own sense of autonomy).
Partner-Mediated (Spillover): LA -> Autonomy_B -> Satisfaction_A (How the LA affects A's satisfaction through B's sense of autonomy).
Spillover Mediated by Actor's Mechanism: LA -> Autonomy_A -> Satisfaction_B (How the LA affects B's satisfaction through A's sense of autonomy).

By constructing our DAG with these specific pathways, we can formulate highly nuanced research questions about the precise causal channels at play within the triadic system.

5. Multiple and Serial Mediators

Finally, the real world is rarely so simple as to have only one mediating mechanism. Our causal model can, and should, accommodate multiple mediators. These can be independent or causally linked themselves.

DAGs illustrating multiple mediators. (a) Independent mediators, where the treatment affects two separate mechanisms. (b) Dependent or serial mediators, where one mechanism triggers another in a causal chain.

For our vee-triad, this allows for richer theoretical models:

Parallel Mediation: The Living_Arrangement might simultaneously and independently increase Perceived_Autonomy and Perceived_Security, both of which then contribute to Satisfaction.
Serial Mediation: An increase in Perceived_Autonomy (Mediator 1) might lead to a decrease in Felt_Jealousy (Mediator 2), which in turn increases Satisfaction. This creates a causal chain: LA -> Autonomy -> Jealousy -> Satisfaction.

Your expertise in relationship dynamics is critical for hypothesizing these chains. Building them into the DAG makes your theoretical assumptions explicit and testable.

Conclusion

In this lesson, we have added a crucial layer of detail to our causal model by incorporating mediating pathways. This moves our inquiry from correlation or total effects to the actual mechanisms of change.

Key Takeaways:

Mediating pathways explain how and why a treatment affects an outcome by modeling the role of intermediate variables (mediators).
Adding a mediator allows us to decompose the Total Effect into a Direct Effect (unmediated) and an Indirect Effect (transmitted through the mediator).
Graphically, a mediator is a node on a causal path between the treatment and outcome. Formally, estimating its effect relies on the strong, untestable assumption of sequential ignorability (no unobserved confounding of the mediator-outcome link).
In interdependent systems like our vee-triad, the APIMeM framework provides a powerful way to model mediated pathways for both within-person (actor) and between-person (partner/spillover) effects.
Our models can also include multiple mediators acting in parallel or in a causal series, allowing for more nuanced theories.

With this new component, our DAG is becoming a powerful tool for representing a sophisticated theory of the vee-triad's dynamics. In our next lesson, we will confront another major challenge in modeling interdependent systems. We will address the learning outcome: Identify collider structures that introduce bias when conditioning on outcomes of interconnected individuals (e.g., the pivot's satisfaction).

Recommended resources