Building DAGs for Vee System Interference

1. From SUTVA Violation to Explicit DAGs

In traditional causal inference, we often rely on the Stable Unit Treatment Value Assumption (SUTVA), which posits that one unit's outcome is unaffected by the treatment assigned to other units. In our vee-triad, this assumption is clearly violated—the very nature of the relationship means the members are deeply interconnected. This interconnectedness is known as interference.

To build a valid causal model, we cannot simply ignore interference; we must represent it explicitly. Directed acyclic graphs provide the perfect tool for this. Instead of being a methodological problem to be assumed away, interference becomes a set of causal pathways to be drawn and analyzed.

The foundational paper by Ogburn and VanderWeele provides a clear taxonomy of interference.

In essence, "direct interference" would involve one person's treatment directly impacting another's outcome, while "interference by contagion" involves one person's outcome (or an intermediate outcome) impacting another's outcome. In our vee system, where the individuals' satisfaction levels and psychological states are interdependent, we can expect significant interference by contagion.

2. Basic Structure of a Network DAG

Before building our specific DAG, let's look at a generic example of how interference is represented. In a system with interference, we must draw arrows not only from an individual's own treatment to their outcome (T_i -> Y_i) but also from their partners' treatments to their outcome (T_j -> Y_i). These cross-unit arrows are the graphical signature of interference, often called "peer effects."

The paper "Estimating Causal Effects on Networked Observational Data" provides a clear, simple illustration of this for a three-node system.

This simple structure, where an individual's outcome y is affected by their own features and treatment (x_i, t_i) as well as their neighbors' (x_j, t_j), is the fundamental template we will now adapt and enrich for our vee-triad experiment.

3. Constructing the Vee-Triad DAG

Let's build our DAG step-by-step, incorporating the variables we defined during our experimental design phase. We'll denote the three individuals as P (the pivot), A (Partner A), and B (Partner B).

Step 1: Defining the Nodes

First, we list the key variables that will be the nodes in our graph.

• Exogenous Variables:
  • T: The single treatment variable, Living Arrangement ('Nesting' vs. 'Separate A'). This is a block-level treatment that affects the entire triad.
  • U_HS: Household Stress, an unmeasured shared confounder that we assume affects the satisfaction of all three members.
• Endogenous (Mediating) Variables:
  • Sec_P, Sec_A, Sec_B: Each individual's perceived Security.
  • Aut_P, Aut_A, Aut_B: Each individual's perceived Autonomy.
• Outcome Variables:
  • Y_P, Y_A, Y_B: Each individual's Relationship Satisfaction.

Step 2: Drawing the Non-Interference Pathways

Let's start by drawing the causal paths as if the individuals were independent. This gives us a baseline model.

1. The treatment T is a direct cause of each person's psychological state. So, we draw arrows from T to all six mediator nodes: {Sec_P, Sec_A, Sec_B, Aut_P, Aut_A, Aut_B}.
2. Each individual's psychological state affects their own satisfaction. This gives us the mediation paths: Sec_i -> Y_i and Aut_i -> Y_i for each individual i in {P, A, B}.
3. The shared confounder, Household Stress U_HS, affects everyone's final outcome. We draw arrows from U_HS to {Y_P, Y_A, Y_B}.

At this stage, our graph consists of three separate, parallel mediation models, plus a shared confounder.

Step 3: Drawing the Interference Pathways

This is the crucial step where we model the interdependencies. We will model interference as a form of "contagion," where the psychological state of one individual causally influences the satisfaction of another. These arrows represent the "peer effects."

Consider the plausible causal flows:

• Partner A's sense of security (Sec_A) likely has a direct impact on the Pivot's satisfaction (Y_P). A secure partner contributes to a more stable and satisfying dynamic for the pivot. This gives us an arrow: Sec_A -> Y_P.
• Similarly, Partner A's security could affect Partner B's satisfaction (Y_B), perhaps by reducing household tension. This gives us: Sec_A -> Y_B.
• The Pivot's autonomy (Aut_P) could influence Partner A's satisfaction (Y_A). For instance, if the pivot feels more autonomous, it might reduce pressure on Partner A, increasing their satisfaction. This gives us: Aut_P -> Y_A.

By extending this logic, we can posit a fully interconnected set of interference pathways. For simplicity and clarity, let's assume that each person's mediating states (Sec_i, Aut_i) are potential causes of the other two partners' final satisfaction (Y_j, Y_k).

This leads to the following set of interference arrows:

• From Sec_A and Aut_A to Y_P and Y_B.
• From Sec_B and Aut_B to Y_P and Y_A.
• From Sec_P and Aut_P to Y_A and Y_B.

Combining these with the pathways from Step 2 gives us our complete DAG. Notice that all arrows flow from the treatment (T) through mediators (Sec, Aut) to outcomes (Y), preserving the acyclic nature of the graph.

This is a simplified representation of the DAG for our vee-triad system. It shows the treatment (T), one mediator for each person (M_i), and the final outcome for each person (Y_i). The blue arrows represent the primary mediation pathways for each individual. The red arrows are the crucial interference pathways, showing how one person's state (e.g., M_A) causally affects the outcomes of their partners (Y_P, Y_B). A shared confounder (U) is also shown.

This graph operationalizes our theory of the vee system. A path like T -> Sec_A -> Y_P represents a causal pathway where the living arrangement affects Partner A's security, which in turn affects the Pivot's satisfaction. This is a quantifiable spillover effect, and our DAG now gives us a precise map to identify and estimate it.

This process of explicitly mapping peer effects is central to network causal inference.

4. Reviewing Key DAG Concepts in the Context of Interference

With our complex graph constructed, it's worth briefly refreshing two key concepts that will be vital for analysis in later lessons: backdoor paths and colliders. Mistaking these can lead to serious bias, especially in a dense graph like ours.

Conclusion

In this lesson, we have formally translated our abstract understanding of the vee-triad's interconnectedness into a concrete scientific object: a Directed Acyclic Graph. By moving beyond the limitations of SUTVA and explicitly drawing the pathways of interference, we have created a rich causal map of the system.

Key Takeaways:

• Interference, or the violation of SUTVA, is not just a nuisance but a core feature of an interconnected system that can be explicitly modeled in a DAG.
• We can represent interference by drawing "cross-unit" causal arrows. In our model, we specified these as paths where one person's psychological state (a mediator) causally affects another's satisfaction (an outcome).
• The resulting DAG is a powerful tool. It makes our theoretical assumptions about the system's dynamics transparent and falsifiable.
• Each path in the DAG corresponds to a specific causal mechanism. For example, paths that cross from one individual to another, like T -> Sec_A -> Y_P, represent the spillover effects (or peer effects) we aim to measure.

This DAG will be the foundation for the rest of this module. In our next lesson, we will use this graph to begin tackling the problem of causal identification, starting with how shared confounders can create spurious associations between the members of the triad.