Distinguishing Direct and Spillover Treatment Effects on Satisfaction

Introduction

In our previous lesson, we established Judea Pearl's do-calculus as a complete and powerful system for determining causal identifiability. We saw how its rules allow us to transform interventional distributions into estimable quantities, even in complex cases with unobserved confounders where the backdoor criterion alone would fail.

This lesson puts that machinery to a specific and crucial task: disentangling direct effects from spillover effects. In any interconnected system like your vee triad, an intervention on one person has ripple effects. The central question for this lesson is: Can we separately identify the causal effect of a treatment on the person who receives it (the direct effect) from the effect that their partners' treatments have on them (the spillover effects)? Answering this requires us to formally define these effects and then apply our graphical identification criteria to determine if they are computable from data.

1. Defining Direct and Spillover Effects

Before we can identify these effects, we must define them precisely. In the context of interference, where one person's outcome depends on the treatment status of others, we must move beyond individual-level counterfactuals like $Y_{i} (a)$ . Instead, we consider the outcome for individual $i$ as a function of the entire treatment vector for their group, $A$ .

The paper "Causal Diagrams for Interference" provides a clear framework for defining these effects.

[PDF] Causal Diagrams for Interference - arXiv

This paper by Ogburn and VanderWeele formalizes the definitions of different causal effects in the presence of interference. This provides the rigorous foundation we need.

First, read the brief introductory section, "Review of Identification...", to familiarize yourself with the counterfactual notation under interference, like Yki(ak). The key idea is that the outcome for person i in block k depends on the full treatment vector ak. Next, focus on the subsection Causal Effects. Pay close attention to the definitions of the unit-level effect and the spillover effect. Note the authors' comment that these are often called direct and indirect effects, respectively.

Let's distill these definitions for our triad (let's call the members A, B, and C). Let $T = (T_{A}, T_{B}, T_{C})$ be the vector of treatments. The potential outcome for person A is $Y_{A} (T) = Y_{A} (T_{A}, T_{B}, T_{C})$ .

Direct Effect (Unit-Level Effect): This is the effect of person A's own treatment on their satisfaction, holding their partners' treatments fixed. For example, we could compare A's satisfaction when they are treated versus not treated, while B and C's treatments are held constant (e.g., at 'control').
$Direct Effect on A = E [Y_{A} (T_{A} = 1, T_{B} = 0, T_{C} = 0)] - E [Y_{A} (T_{A} = 0, T_{B} = 0, T_{C} = 0)]$
Spillover Effect (Indirect Effect): This is the effect of the partners' treatments on person A's satisfaction, holding A's own treatment fixed. For example, we could compare A's satisfaction when B is treated versus not, while A and C are held at 'control'.
$Spillover Effect from B on A = E [Y_{A} (T_{A} = 0, T_{B} = 1, T_{C} = 0)] - E [Y_{A} (T_{A} = 0, T_{B} = 0, T_{C} = 0)]$

Your background in statistics might bring to mind the Actor-Partner Interdependence Model (APIM), which uses regression to estimate similar effects.

This APIM diagram shows how a patient's and caregiver's anxiety influences their own and each other's quality of life. The "Actor" path corresponds to a direct effect (one's own anxiety on one's own quality of life), while the "Partner" path corresponds to a spillover effect (one's partner's anxiety on one's own quality of life).

The APIM provides a statistical description, but our goal here is to establish the causal identifiability of these effects using graphical models. The question is whether the coefficients in a model like APIM can be interpreted causally.

2. The Identification Challenge

The ability to separately identify direct and spillover effects hinges on our ability to identify the full interventional distribution, $P (Y_{i} ∣ do (T = t))$ . If we can derive an estimable expression for this quantity for any treatment vector $t$ , then we can calculate the expectation under any specific treatment combination and compute the contrasts that define our direct and spillover effects.

The problem, therefore, reduces to applying the identification rules we've learned to the system-level DAG representing the entire triad. Let's construct a plausible DAG for the vee relationship and analyze it.

Consider the following causal graph for member A's satisfaction ( $Y_{A}$ ):

Treatments: $T_{A}, T_{B}, T_{C}$ (e.g., aspects of the living arrangement).
Outcome: $Y_{A}$ .
Observed Confounders: $C_{A}, C_{B}, C_{C}$ (e.g., pre-existing individual traits like attachment style) that influence each person's treatment assignment and outcome.
Unobserved Confounder: $U$ (e.g., overall household financial stress) that affects everyone's treatment and satisfaction.
Causal Paths:
- Direct Effect Path: $T_{A} \to Y_{A}$
- Spillover Paths: $T_{B} \to Y_{A}$ , $T_{C} \to Y_{A}$
- Confounding Paths, e.g.: $T_{A} \leftarrow C_{A} \to Y_{A}$ , and $T_{A} \leftarrow U \to Y_{A}$ .
- Interference Confounding Paths, e.g.: The effect of $T_{B}$ on $Y_{A}$ is confounded by $T_{B} \leftarrow C_{B} \to Y_{B} \to Y_{A}$ (if satisfaction is contagious) and $T_{B} \leftarrow U \to Y_{A}$ .

The crucial insight is that direct and spillover effects can be confounded with each other. For instance, the unobserved variable $U$ creates backdoor paths from all treatments ( $T_{A}, T_{B}, T_{C}$ ) to the outcome $Y_{A}$ . If we can't block these paths, we can't identify the effect of $T$ on $Y_{A}$ , and therefore we can't separate the direct component from the spillover components.

The following reading explains how to approach covariate control on such system-level graphs.

[PDF] Causal Diagrams for Interference - arXiv

This section uses DAGs to illustrate how confounding works in the presence of interference and what needs to be controlled for.

Read Section 3, "Covariate Control". Focus on the logic behind drawing the DAG for an entire block and the principles of blocking backdoor paths. Pay special attention to the examples in Figure 5 and the associated text, which describe scenarios where failing to account for the full system structure leads to incorrect conclusions. For example, Figure 5(a) shows that even if T_j is randomized, you might still need to control for C_j to identify the spillover effect of T_j on Y_i.

This leads to a clear conclusion: if the joint causal effect of the entire treatment vector $T$ on an outcome $Y_{i}$ is identifiable (via backdoor, front-door, or the full power of do-calculus), then the specific direct and spillover effects are also identifiable, as they are simply specific contrasts of the identified interventional distribution. If the joint effect is not identifiable (e.g., due to an unblockable backdoor path from $U$ ), then you generally cannot separate the effects.

For example, to identify the direct effect of $T_{A}$ on $Y_{A}$ , you must be able to block all non-causal paths between them. One such path is the backdoor path $T_{A} \leftarrow U \to Y_{A}$ . If $U$ is unobserved, this path is open. Similarly, for the spillover effect of $T_{B}$ on $Y_{A}$ , the path $T_{B} \leftarrow U \to Y_{A}$ is also open. The unobserved confounder $U$ mixes the apparent effects of all treatments, making it impossible to attribute observed changes in $Y_{A}$ to a specific source.

3. A Practical Approach: Exposure Mappings

While thinking in terms of the full treatment vector $T$ is formally correct, it can be unwieldy. A highly effective practical strategy is to simplify the problem using exposure mappings. This approach, common in network causal inference, involves defining a limited set of qualitatively different "exposure conditions" that a unit can experience.

This approach is detailed in the next reading, which provides a design-based perspective on estimation.

[PDF] 16. Estimating causal effects in experiments with network spillovers

This chapter from a handbook on computational social science provides a clear, practical guide to defining and estimating effects under interference using exposure mappings and inverse probability weighting.

Read the sections titled "Network Experimental Setting" and "Exposure Mappings and Causal Effects". The key idea is to define a function f(\mathbf{T}) that maps the complex treatment vector \mathbf{T} to a simpler, categorical exposure variable d. Then, skim the section "Design-Based Inference". The main takeaway is that even with a randomized experiment, the probability of receiving a certain exposure is non-random (it depends on network position). This induces confounding, which can be corrected using Inverse Probability Weighting (IPW), a technique you'll be familiar with from your statistics background.

Let's apply this to your triad. For person A, instead of considering all $2^{3} = 8$ possible treatment vectors, we could define a simpler set of exposures:

d=1 (Isolated Direct): $T_{A} = 1, T_{B} = 0, T_{C} = 0$ .
d=2 (Pure Spillover): $T_{A} = 0$ and ( $T_{B} = 1$ or $T_{C} = 1$ ).
d=3 (Direct + Spillover): $T_{A} = 1$ and ( $T_{B} = 1$ or $T_{C} = 1$ ).
d=4 (No Exposure): $T_{A} = 0, T_{B} = 0, T_{C} = 0$ .

The causal estimands are now contrasts between the average potential outcomes for these exposure categories:

Average Direct Effect (in isolation): $E [Y_{A} (d = 1)] - E [Y_{A} (d = 4)]$
Average Spillover Effect (in isolation): $E [Y_{A} (d = 2)] - E [Y_{A} (d = 4)]$

The identification problem is now framed as: can we obtain an unbiased estimate for each $E [Y_{A} (d)]$ ? As the reading explains, we can use an IPW estimator, such as the Horvitz-Thompson or Hajek estimator. The probability weights, $π_{i} (d) = P (unit i receives exposure d)$ , are calculated based on the experimental design (the randomization probabilities for $T$ ) and the triad structure. These weights adjust for the fact that a person's "exposure" is confounded by the structure of their relationships.

This method effectively allows you to estimate direct and spillover effects as long as two conditions are met:

Positivity: Every unit has a non-zero probability of being in every exposure condition, $P (d_{i} = d) > 0$ .
Consistency: The potential outcome under a given exposure is well-defined.

This shifts the problem from complex graphical manipulations to a more straightforward estimation problem, provided a randomized experiment is conducted and the exposure probabilities can be computed.

Conclusion

In this lesson, we have established a formal process for determining the identifiability of direct and spillover effects within an interconnected system.

Key Takeaways:

Formal Definitions: Direct effects relate to a person's own treatment, while spillover effects relate to their partners' treatments. Both are defined as contrasts between potential outcomes under different configurations of the full treatment vector.
Identification is Joint: The ability to separate direct from spillover effects is contingent on the identifiability of the joint interventional distribution $P (Y_{i} ∣ do (T))$ . If the overall causal effect of the treatment vector is confounded by unobserved variables, its components generally cannot be disentangled.
Graphical Criteria Apply: We use the same graphical criteria (backdoor, do-calculus) on the full system-level DAG to check for identifiability. The challenge lies in the complexity of the graph, not in new identification rules.
Exposure Mappings as a Practical Tool: For estimation, we can simplify the problem by defining a set of "exposure conditions." This reframes the problem into estimating average potential outcomes for these conditions, which can be achieved with methods like Inverse Probability Weighting (IPW) in the context of a randomized experiment.

In our next lesson, we will continue our exploration of identification by focusing on a different type of causal path. We will apply the front-door criterion to identify a mediated effect, where we are interested in the pathway through which a treatment exerts its influence, such as how a living arrangement affects satisfaction via its impact on a person's sense of autonomy.

Recommended resources

Introduction

1. Defining Direct and Spillover Effects

2. The Identification Challenge

3. A Practical Approach: Exposure Mappings

Conclusion