Backdoor Criterion with Interference Pathways

Introduction

In our last lesson, we explored Simpson's paradox and saw how easily statistical analyses can be misled by aggregation. The paradox highlighted the critical danger of ignoring confounders, reinforcing that the choice of which variables to control for is a causal decision, not a statistical one. The key to resolving such paradoxes and identifying true causal effects is to block non-causal "backdoor" pathways in our causal model.

This lesson builds directly on that foundation. We will now formalize the process of selecting the right control variables using one of the most fundamental tools in causal inference: the backdoor criterion. Your existing familiarity with DAGs provides a strong starting point, but we will move into more complex territory. Our primary goal is to adapt the standard backdoor criterion, which assumes independent units, to the interconnected reality of your vee-triad study, where one person's experiences and choices invariably affect the others. By the end of this lesson, you will be able to identify a valid set of variables to control for in a DAG that explicitly models these interference pathways.

1. Review: The Backdoor Criterion for Independent Units

Before we tackle interference, let's solidify our understanding of the backdoor criterion in its standard form. At its heart, the criterion provides a graphical rule for selecting a set of covariates to condition on (i.e., "adjust for") to remove confounding bias and identify a causal effect.

An intuitive way to think about this is by separating all the paths between a treatment $T$ and an outcome $Y$ into two types:

Front-door paths: Causal pathways that start at $T$ and move forward to $Y$ . These represent the causal effect we want to measure.
Backdoor paths: Spurious, non-causal pathways that have an arrow pointing into $T$ . These paths are the source of confounding.

To isolate the causal effect, we must "close" all backdoor paths while keeping all front-door paths open. We close a path by conditioning on a variable that lies on it.

The following video provides an excellent visual explanation of this process.

Causality: Closing Back Doors

This video from "Econometrics, Causality, and Coding with Dr. HK" offers a clear, intuitive walkthrough of identifying and closing backdoor paths.

Please watch from the beginning until the main explanation. Focus on: The distinction between "front door" and "back door" paths (this section). The process of listing all paths and identifying a set of control variables to close the back doors (this example).

The video mentions two important caveats: controlling for a collider can open a path, and controlling for a mediator (a variable on a front-door path) will block part of the causal effect you want to measure. We will revisit these concepts later, but for now, the main principle is to find a set of non-descendant covariates that block all backdoor paths from treatment to outcome.

This graphical rule has a formal basis in the potential outcomes framework you're familiar with. Achieving the "no unmeasured confounding" assumption, also known as ignorability ( $Y (t) ⊥ T ∣ X$ ), is precisely what the backdoor criterion accomplishes graphically. When it holds, we can use the adjustment formula to identify the causal effect from observational data:

P (Y ∣ d o (T = t)) = x \sum P (Y ∣ T = t, X = x) P (X = x)

Here, $X$ is a set of covariates that satisfies the backdoor criterion. You've encountered this logic in your statistics background; the backdoor criterion gives us a rigorous, graphical method to determine what goes into $X$ .

2. Modeling Interference: Expanding the DAG

The standard backdoor criterion works beautifully when the units of analysis are independent. However, it breaks down in our triadic system, where the Stable Unit Treatment Value Assumption (SUTVA) is violated by design. One person's outcome is a function of the entire group's treatment vector.

To handle this, we must shift from drawing a DAG for a single individual to drawing one that represents the entire interacting system—the "block." In your case, the block is the triad.

Let's denote the three individuals as $A$ (Pivot), $B$ , and $C$ . Let their treatments be $T_{A}, T_{B}, T_{C}$ and their outcomes be $Y_{A}, Y_{B}, Y_{C}$ . The defining feature of interference is the presence of causal arrows between one person's treatment and another's outcome. For example, an arrow $T_{B} \to Y_{A}$ represents the "spillover" effect of Partner B's treatment on the Pivot's satisfaction.

The following seminal paper by Ogburn and VanderWeele provides the formal graphical language for this.

Causal Diagrams for Interference - Project Euclid

This paper is the key resource for adapting graphical models to interference. We will focus on the section that explains how to select covariates in this new context.

First, read the brief subsection 4.1 Direct Interference to understand the simplest form of interference, where one unit's treatment directly affects another's outcome. Next, and most importantly, read all of section 3. Covariate Control. This is the core of today's lesson. Pay close attention to the series of DAGs in Figure 5 and the accompanying explanations. The text describes which covariates are necessary to achieve exchangeability (i.e., satisfy the backdoor criterion) under different confounding structures.

The crucial insight from the paper is this: the logic of the backdoor criterion remains the same, but we must now apply it to a more complex graph. The "treatment" is the full vector of assignments $T = (T_{A}, T_{B}, T_{C})$ , and we seek to identify the effect of this vector on a specific outcome, say $Y_{A}$ . We must find a set of covariates that blocks all backdoor paths from any component of $T$ to $Y_{A}$ .

3. Adapting the Backdoor Criterion: Key Scenarios

Let's dissect the implications of the "Covariate Control" section from the Ogburn and VanderWeele paper, applying it to our triad. We'll use the notation $(A, B)$ instead of $(i, j)$ for clarity.

Scenario 1: Partner's Covariates as Confounders

Consider Figure 5(a) from the paper. In our context, this translates to a situation where a covariate of Partner B, let's call it $X_{B}$ (e.g., their individual work stress), is a cause of both their own treatment assignment $T_{B}$ and the Pivot's outcome $Y_{A}$ .

$X_{B} \to T_{B}$ : High work stress might make Partner B less likely to agree to a certain living arrangement.
$X_{B} \to Y_{A}$ : Partner B's high work stress might reduce the quality of their interactions with the Pivot, directly impacting the Pivot's satisfaction $Y_{A}$ .

The DAG would include the paths $T_{B} \leftarrow X_{B} \to Y_{A}$ . This is a classic backdoor path from $T_{B}$ to $Y_{A}$ . To estimate the spillover effect of $T_{B}$ on $Y_{A}$ , we must condition on Partner B's covariate $X_{B}$ . Controlling only for the Pivot's own covariates $X_{A}$ would be insufficient.

Scenario 2: Shared Block-Level Confounders

Now consider Figure 5(e). This depicts a block-level characteristic $D$ that is a common cause for everyone. In your study, a perfect example is Household Financial Stress.

$D \to (T_{A}, T_{B}, T_{C})$ : High financial stress might make certain living arrangements (e.g., living separately) infeasible, influencing the treatment vector for the whole triad.
$D \to (Y_{A}, Y_{B}, Y_{C})$ : High financial stress is likely to decrease the satisfaction of all three members, regardless of the living arrangement.

This creates backdoor paths like $T_{B} \leftarrow D \to Y_{A}$ . To identify the causal effect of the living arrangement on anyone's satisfaction, the shared confounder $D$ must be included in the adjustment set.

This concept is so central that it's often referred to as Network Conditional Ignorability, as highlighted in another applied paper on interference.

Causal Inference in the Presence of Interference in Sponsored ...

This paper applies the same principles in a different domain (online advertising), but its formal statement of the identification assumption is very clear and relevant.

Please read the subsection 4.2. Identification Assumptions. Notice their definition of network conditional ignorability: Y_i(\mathbf{a}) \perp \mathbf{A} | \mathbf{X}. This is simply the ignorability assumption extended to the interference setting, where \mathbf{A} is the treatment vector and \mathbf{X} is the set of all relevant covariates for the block. The paper states that this can be verified using d-separation on the graph, which is exactly what the backdoor criterion does.

The core message is consistent across both resources: when interference is present, the set of potential confounders expands. Your adjustment set must account for the covariates of all interacting units as well as any confounders shared by the group.

4. Application to the Vee-Triad Model

Let's synthesize these ideas and apply them to a simplified causal model for your vee-triad experiment. Our goal is to find a valid adjustment set $Z$ to estimate the causal effect of the living arrangement $T = (T_{A}, T_{B}, T_{C})$ on the Pivot's satisfaction $Y_{A}$ .

Consider the following causal structure:

A simplified DAG for the vee-triad, illustrating interference. 'Restrictions' represents the living arrangement (treatment), and 'Z' represents individual-level confounders. The index 't' denotes the current time, 't-1' the previous period, etc. Notice the spillover effects (e.g., Restrictions_t-1 -> Outcome_t) and shared influences.

This diagram is a bit abstract, so let's translate it. Imagine:

Restrictions_t is our treatment vector $T$ .
Outcome_t is an outcome like $Y_{A}$ .
Z_t represents a time-varying individual covariate for the Pivot, say Pivot_Anxiety_t.
A shared, unobserved confounder like Household_Stress might be influencing all Z, Restrictions, and Outcome nodes simultaneously (this is a common simplification when drawing these temporal graphs).

To identify the effect of $T$ on $Y_{A}$ , we must block all backdoor paths. Let's identify a few:

Path through Pivot's Own Covariate: The path $T_{A} \leftarrow Pivot_Anxiety_{A} \to Y_{A}$ is a standard backdoor path. We must control for $Pivot_Anxiety_{A}$ .
Path through Partner's Covariate: Suppose Partner B's work schedule ( $X_{B}$ ) affects their availability ( $T_{B}$ ) and also directly impacts the Pivot's satisfaction ( $Y_{A}$ ). This creates the backdoor path $T_{B} \leftarrow X_{B} \to Y_{A}$ . To block it, we must control for $X_{B}$ .
Path through Shared Confounder: Let's say overall household financial health ( $D$ ) affects the chosen living arrangement ( $T$ ) and everyone's satisfaction ( $Y_{A}, Y_{B}, Y_{C}$ ). This opens backdoor paths like $T_{C} \leftarrow D \to Y_{A}$ . To block these, we must control for $D$ .

Therefore, a valid adjustment set $Z$ to identify the effect of $T$ on $Y_{A}$ must include, at a minimum:

The Pivot's own relevant covariates ( $X_{A}$ ).
Any covariates of partners ( $X_{B}, X_{C}$ ) that are also direct causes of the Pivot's outcome.
Any shared, block-level confounders ( $D$ ) that affect both treatment and outcomes within the triad.

Failing to account for the full set of covariates $(X_{A}, X_{B}, X_{C}, D)$ would leave backdoor paths open, leading to biased estimates of the causal effects—precisely the kind of error that gives rise to phenomena like Simpson's paradox.

Conclusion

Today, we have extended the fundamental backdoor criterion to the more complex, but more realistic, setting of causal interference. By moving from an individual-level DAG to a system-level DAG, we can now formally reason about confounding in interconnected units.

Key Takeaways:

The backdoor criterion's goal is to find an adjustment set that blocks all non-causal paths from treatment to outcome.
In the presence of interference, the DAG must represent the entire interacting block (e.g., the triad).
To identify causal effects (including direct and spillover effects), the adjustment set must block all backdoor paths from the entire treatment vector to the outcome of interest.
This often means that a valid adjustment set must include not only an individual's own covariates but also the covariates of their partners and any confounders shared by the group.

In our next lesson, we will broaden our toolkit for causal identification. While the backdoor criterion is powerful, it is not always applicable (e.g., if a necessary confounder is unobserved). We will learn the three rules of do-calculus, a complete system for determining identifiability from a DAG, even in cases where the backdoor criterion fails.

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