Causal Effects Under Network Interference

1. Expanding the Definition of an Effect

The Stable Unit Treatment Value Assumption (SUTVA) allows us to define a simple causal effect for an individual $i$ as the difference between two potential outcomes: $Y_i(1)-Y_i(0)$. However, as we've established, interference violates this. The outcome for individual $i$ depends not just on their own treatment, $T_i$, but on the entire vector of treatments across the triad, $\mathbf{T}$.

This means we must expand our schedule of potential outcomes to account for the treatment status of others. A clear and simple illustration of this principle is presented in the following resource, which uses a household as an analogy for an interconnected group.

As the presentation shows, once you acknowledge interference, you are forced to be more specific about the causal effect you want to estimate. Is it the effect of your treatment when your partner is untreated? Or the effect of your partner's treatment on you? These are different questions with different answers.

2. A Formal Typology of Causal Effects under Interference

The household example provides the intuition. Now, let's formalize these new estimands. In the context of interference, we typically decompose the overall impact of a treatment program into several components. The following presentation slides offer a standard and widely used set of definitions.

Let's summarize and contextualize these definitions:

• Direct Causal Effect: This measures the effect of an individual receiving the treatment themselves, holding the treatment conditions of their partners constant. It answers the question: "What is the effect of the treatment on me?" For example, $\bar{y}(1,\alpha )-\bar{y}(0,\alpha )$ compares the average outcome for treated individuals to untreated individuals, within the same spillover environment (defined by treatment strategy $\alpha$).

• Indirect (Spillover) Causal Effect: This measures the effect of some individuals being treated on the outcomes of other individuals who are not treated. It answers the question: "What is the effect of my partners' treatment on me?" For example, $\bar{y}(0,{\alpha }_1)-\bar{y}(0,{\alpha }_0)$ compares the average outcome for untreated individuals under one treatment strategy (${\alpha }_1$) versus another (${\alpha }_0$).

• Total Causal Effect: This captures the combined influence of receiving the treatment yourself while also being exposed to the spillovers from a treatment program. It compares an individual who is treated under a treatment strategy to an individual who is not treated under a control strategy: $\bar{y}(1,{\alpha }_1)-\bar{y}(0,{\alpha }_0)$. As the slides note, this can be decomposed: Total Effect = Direct Effect + Indirect Effect.

• Overall Causal Effect: This measures the population-level impact of implementing a treatment strategy versus a control strategy, averaging over both direct and indirect effects for everyone. It compares $\bar{y}({\alpha }_1)-\bar{y}({\alpha }_0)$.

This typology gives us a precise language to describe the different ways an intervention can influence outcomes in an interconnected system. The image below, adapted from Halloran & Struchiner (1991), provides a useful visual summary of these concepts in a clustered setting, which is analogous to our triad.

3. Application to the Vee Triad

Let's apply these definitions to our vee triad (A: pivot; B, C: wings) and the daily mindfulness treatment example from the last lesson. Recall our exposure mapping for member B, which simplified the 8 treatment vectors into four potential outcomes based on B's exposure:

• $Y_B(d_{11})$: B is treated, at least one partner is treated.
• $Y_B(d_{10})$: B is treated, no partners are treated.
• $Y_B(d_{01})$: B is not treated, at least one partner is treated.
• $Y_B(d_{00})$: No one is treated.

Using this framework, we can now define highly specific causal questions about B's satisfaction:

• Isolated Direct Effect: $Y_B(d_{10})-Y_B(d_{00})$

  • Interpretation: What is the effect of B practicing mindfulness when their partners are not, compared to no one practicing? This isolates the personal benefit (or cost) of the practice itself.

• Spillover Effect on an Untreated Individual: $Y_B(d_{01})-Y_B(d_{00})$

  • Interpretation: What is the effect on B's satisfaction when a partner starts practicing mindfulness but B does not? This captures the positive (or negative) externality of the treatment.

• Direct Effect in the Presence of Spillover: $Y_B(d_{11})-Y_B(d_{01})$

  • Interpretation: What is the effect of B starting the practice, given that a partner is already practicing? This measures the additional benefit of B's personal adoption on top of the existing spillover environment. This might be different from the isolated direct effect due to interactive or synergistic effects.

• Spillover Effect on a Treated Individual: $Y_B(d_{11})-Y_B(d_{10})$

  • Interpretation: For an individual B who is already practicing mindfulness, what is the additional effect of a partner also starting the practice? This measures the synergistic effect of mutual practice.

Notice that the "Total Effect" of going from a state of no treatment to full treatment, $Y_B(d_{11})-Y_B(d_{00})$, can be decomposed in two ways:

1. $[Y_B(d_{10})-Y_B(d_{00})]+[Y_B(d_{11})-Y_B(d_{10})]$ = (Isolated Direct Effect) + (Spillover on Treated)
2. $[Y_B(d_{01})-Y_B(d_{00})]+[Y_B(d_{11})-Y_B(d_{01})]$ = (Spillover on Untreated) + (Direct Effect with Spillover)

Your background in experimental design and analysis highlights why this is so powerful. These are not just notational tricks; they are distinct, estimable quantities that correspond to different causal pathways. The choice of which to prioritize depends entirely on the research question.

The following video provides an excellent high-level discussion on this point, emphasizing that the move away from SUTVA is fundamentally about being more explicit about the causal question you want to answer.

Conclusion

In this lesson, we have built upon the concept of exposure mappings to give ourselves a richer, more precise causal vocabulary. By acknowledging interference, we are forced to move beyond a single, often ambiguous, average treatment effect and instead specify exactly which causal pathway we wish to investigate.

The key takeaways are:

• When interference is present, we must expand the schedule of potential outcomes for each individual to depend on the treatment status of their network partners.
• This allows us to define distinct causal estimands:
  • Direct effects capture the impact of an individual's own treatment.
  • Indirect (spillover) effects capture the impact of their partners' treatment.
  • Total effects capture a combination of the two.
• The choice of estimand is not merely a statistical technicality but a core part of defining the substantive research question. It forces us to be explicit about the mechanisms of influence we want to understand.

You are now in a position to translate the broad goal of "maximizing satisfaction" into a set of precise, answerable questions. In our next lesson, we will do exactly that, formulating specific research questions for the vee triad experiment using the network-aware causal estimands we've defined today.