Welcome back. In our previous lesson, we established a formal language for causal inference in your triadic context by defining direct, indirect, total, and overall causal estimands. This allowed us to specify precisely what we want to measure when one person's actions can affect another's outcomes. We acknowledged that interference, the violation of the "no interference" assumption, is the central dynamic to be studied, not a nuisance to be avoided.
Today, we shift from the "what" to the "how." Our focus will be on describing the pathways through which one person's 'treatment' (e.g., an activity) can affect another person's outcome. We will explore the concrete mechanisms of this spillover effect. Instead of simply stating that the husband's potential outcome for jealousy, , is a function of the entire treatment vector, , we will unpack how the treatments of the wife () and bull () transmit their influence to the husband. This will provide the conceptual foundation for the modeling strategies we will discuss later in the course.
The "No Interference" Assumption: A Focused Review
Before we explore the mechanisms of interference, let's briefly review the assumption being violated. The "no interference" assumption states that any individual's potential outcome is only a function of their own treatment assignment.
2 - Potential Outcomes (Week 2)
This short clip from Brady Neal's Causal Inference course provides a clear and concise graphical explanation of the no interference assumption.
Please watch the segment on no interference. The core idea is that an individual's potential outcomes, Y_i, are written as Y_i(t_i), depending only on their own treatment, t_i, and not the treatments of others in the population.
In your triadic research setting, this assumption is implausible. The husband's emotional state is unlikely to be independent of whether the wife and bull are engaged in a specific activity, even if the husband's own assigned "treatment" is held constant. Our task is to describe the causal pathways that explain this dependence.
Mechanisms of Interference in the Triad
Interference in a social system like your triad is not random noise; it is a structured process that operates through specific, identifiable mechanisms. Understanding these mechanisms is critical for building a valid causal model. Let's consider two primary pathways relevant to your research.
1. Rivalry for Shared Resources
Your background in operations research provides a powerful lens for understanding one of the most fundamental mechanisms of interference: competition for rivalrous resources. A resource is rivalrous if its consumption by one agent prevents its consumption by another. In the context of a relationship, key resources like time, attention, emotional energy, and social capital are inherently rivalrous.
When the wife and bull engage in an activity (a "treatment"), they are consuming these finite resources. That consumption directly creates an effect on the husband, because those resources are now unavailable to the husband-wife dyad. This is not merely a "side effect"; it is a direct causal pathway of interference.
The paper by Jahani and Eckles, "Rivalrous Resource Sharing in Networks," offers a formal game-theoretic perspective on this process. While its focus is on economic inequality, the underlying model of strategic sharing of rivalrous resources is directly applicable.
[PDF] Rivalrous Resource Sharing in Networks can Exacerbate Existing ...
This paper models how strategic behavior around rivalrous resources in a network can lead to specific outcomes. We will use its conceptual framework to understand how the allocation of time and attention in the triad acts as a mechanism for interference.
Please read two parts of this paper: In the Introduction, focus on the argument for moving beyond simple structural accounts of network effects. Read the paragraph that explains the limitations of a purely structural view and introduces the importance of strategic behavior and rivalry. Next, read Section 2.1, "Game Setup". This section describes a formal game where players decide whether to share a rivalrous resource. As you read the description of the game's structure, map the concepts to your triad: The "rivalrous resource" can be seen as the wife's time, attention, or emotional availability. An agent "receiving the resource" and deciding to "share it with its network contacts" is analogous to the wife choosing to spend time on an activity with the bull. The consequence is that this shared resource is now divided, and its utility for others (i.e., the husband) is diminished.
This "rivalrous resource" framework provides a clear mechanism for how a treatment applied to the wife-bull dyad () directly affects the husband's potential outcomes, even when his own treatment status is unchanged (). The activity alters the system's resource allocation, which is a causal effect.
2. Information, Communication, and Disclosure
A second primary mechanism of interference is the flow of information. An individual's emotional response is often not to an event itself, but to the knowledge of that event. In your context, the husband's jealousy may be triggered or modulated by what he is told about an activity between his wife and the bull.
Consider these scenarios:
- No Information: The wife and bull have a date, and the husband is unaware. Interference may still occur through the "rivalry" channel (e.g., he notices his wife is less available), but the informational channel is closed.
- Partial Information: The husband is told a date occurred, but with no details.
- Full Information (Disclosure): The husband is told the details of the date.
The husband's emotional outcome will likely be different in each case. The "treatment" (the date) is the same, but the information flow creates different causal effects. Therefore, communication and disclosure are not just contextual factors; they are integral parts of the causal pathway of interference.
Formalizing Interference: The Concept of Exposure Mappings
We have described the mechanisms; now let's connect them to the formal potential outcomes framework. The key is to recognize that one person's treatment does not just affect another's outcome directly; it places them into a specific exposure condition.
The paper by Aronow and Samii provides a robust framework for this, distinguishing between a unit's assigned treatment and their resulting exposure.
estimating average causal effects under general interference, with ...
This paper develops a general framework for causal inference under interference. Its central contribution is the idea of an "exposure mapping," which formally connects a vector of treatment assignments to the specific type of exposure a unit receives.
Please read the following sections: First, review Section 1, "Introduction," which sets up the problem of interference in social networks and the violation of SUTVA. Next, read Section 3, "Treatment assignment and exposure mappings," carefully. This is the core concept for this lesson. Focus on the formal distinction between treatment assignments and exposures. The exposure mapping, f(\mathbf{z}, \theta_i), is a function that takes the entire treatment vector for the group (\mathbf{z}) and unit-specific traits (\theta_i) and maps them to a specific exposure condition for unit i.
Let's apply this to your triad. The husband's potential outcome for jealousy, , does not depend on the raw treatment vector in some amorphous way. It depends on the exposure condition this vector creates for him.
Let's define a simple exposure mapping for the husband, where :
- No Exposure (): This occurs when and . This is the baseline condition.
- Indirect Exposure without Disclosure (): This occurs if , and there is no communication about the event to the husband. The causal pathway here is primarily resource rivalry.
- Indirect Exposure with Disclosure (): This occurs if , and the husband is informed about the event. The causal pathways here include both resource rivalry and information transfer.
Under this framework, the "no interference" assumption is violated because the husband's potential outcome is not just a single . Instead, he has multiple potential outcomes corresponding to his different exposure conditions, all while his own treatment assignment is :
- : Potential jealousy under no exposure.
- : Potential jealousy under indirect exposure without disclosure.
- : Potential jealousy under indirect exposure with disclosure.
The difference, for example, between and is the causal effect of the wife-bull activity (with disclosure) on the husband. This is a precisely defined indirect, or spillover, effect. The exposure mapping gives us a formal way to describe how the treatment assigned to the wife and bull affects the husband's outcome.
Conclusion
In this lesson, we have moved from simply acknowledging interference to describing its underlying mechanisms. We have seen that the violation of the "no interference" assumption is not a statistical problem to be fixed, but a substantive causal process to be understood.
Key Takeaways:
- Interference operates through concrete mechanisms. In your triadic context, two key mechanisms are rivalry for shared resources (time, attention) and the flow of information (communication, disclosure).
- One person's "treatment" affects another person's outcome by placing them into a specific exposure condition.
- The concept of an exposure mapping, , provides a formal tool to link the full vector of treatment assignments in a group to the specific exposures experienced by each individual.
- This allows us to write potential outcomes as a function of exposure, e.g., , which makes the nature of the interference explicit and prepares us for estimation.
Having defined our estimands (Lesson 8.1) and described the mechanisms of interference (this lesson), we are now equipped to tackle the challenges of estimation. In the next lesson, we will address a core problem in any causal analysis: confounding. We will explore how to adapt classical adjustment strategies, like the backdoor and frontdoor criteria, to account for confounders that operate across the multiple, interdependent perspectives of the triad.