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Higher-order Spillover Effects Under Partial Interference

This paper proposes a generalized interference framework and develops new Horvitz-Thompson, Hajek, and weighted regression estimators to accurately quantify higher-order spillover effects from specific network distances, addressing the limitations of traditional neighborhood interference assumptions through theoretical bias analysis, simulations, and an application to a randomized trial in Honduras.

Original authors: Qixiang Xu, Laura Forastiere

Published 2026-07-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Qixiang Xu, Laura Forastiere

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you're trying to figure out why a rumor spreads through a high school. You know that if your best friend hears a juicy secret, they might tell you. That's easy to track. But what if your friend's friend (who you don't know) tells your friend, who then tells you? Or what if the rumor travels through three or four people before it reaches your desk?

For a long time, scientists studying how things spread—like diseases, ideas, or health programs—have used a rule of thumb called "neighborhood interference." It's like saying, "Hey, only your direct friends can influence you." The authors of this paper, Qixiang Xu and Laura Forastiere, think that rule is way too simple. They argue that in the real world, influence can travel much further, like a ripple in a pond that doesn't stop at the first rock it hits.

The Problem with the "Direct Neighbor" Rule
The paper argues that many previous studies made a big mistake by assuming that only your immediate neighbors (people directly connected to you) affect your outcome. They call this the "neighborhood interference assumption." The authors show through simulations (computer experiments) that if you assume the influence stops at your direct friends, but it actually travels further, your math gets messed up. It's like trying to measure how much rain fell on your roof by only looking at the water dripping from the gutter, ignoring the water running down the side of the house.

They also point out that researchers often guess how the influence works. Maybe they assume it's just the number of treated friends, or the percentage. The paper shows that if you guess the wrong formula for how influence works, your results are biased (wrong), even if you have the right data.

The New Solution: The "Conservative" Approach
Instead of guessing exactly how far the influence goes or exactly how it works, the authors propose a "conservative" approach. Imagine you are trying to catch a fish, but you don't know how deep the water is. Instead of guessing a specific depth, you cast a net that covers the entire possible area where the fish could be.

In their method, they define a "conservative interference set." This is a group of people that is guaranteed to include everyone who could possibly influence a person, even if we don't know exactly who those people are or how they are connected. They don't need to know the exact "exposure mapping" (the specific formula for how influence travels). They just need to know the group is big enough to catch all the ripples.

How They Measure It
To measure the effect of these distant ripples, the authors invented a new way of looking at the data. They imagine two different scenarios happening at the same time:

The "H-Order" Neighborhood: They pretend that people at a specific distance (say, 2 steps away) are treated with a certain probability (like a 60% chance).
The Rest of the Group: They pretend everyone else in that big "conservative" group is treated with a different probability (like a 40% chance).

By comparing what happens when they tweak these probabilities, they can isolate the "spillover effect" from that specific distance. They developed three new mathematical tools (estimators) to do this:

Horvitz-Thompson: A direct counting method. It's accurate but can be shaky if the numbers get extreme (like having a tiny chance of something happening).
Hajek: A refined version that smooths out the bumps, making it more stable.
Weighted Least Squares (WLS): A method that uses a simple line to fit the data, which is very efficient if the relationship is simple, but might be wrong if the relationship is complex.

What the Simulations Showed
The authors ran thousands of computer simulations to test their ideas.

They found that the old "naive" methods (like simple regression) were often biased. If they ignored the second or third layer of friends, the results were wrong.
Their new methods (Hajek and WLS) stayed unbiased (correct) even when the interference was complex or when the interference set was huge.
They also showed that the "Horvitz-Thompson" method can have high variance (it jumps around a lot) when the probabilities are extreme, while the Hajek and WLS methods are much more stable.

The Real-World Test: Honduras
To see if this works in the real world, they applied their method to a real study in Honduras about a maternal and child health program. In this study, some households got health education, and the researchers wanted to see if this knowledge "spilled over" to neighbors.

Using their new tools, they found some fascinating patterns:

For treated people (those who got the health education): They saw a "spillover" effect. If their first-degree neighbors (direct friends) were also treated, these people learned even more. The effect was strongest when about half the village was treated, but then it dropped off if everyone was treated (maybe because there was too much information, or "saturation").
For untreated people: They didn't see much of a spillover effect. The authors suggest this might be because untreated people lacked the background knowledge to understand the information their treated friends were sharing.
Second-order effects (friends of friends): Surprisingly, for treated people, the influence from "friends of friends" was sometimes just as strong as direct friends, especially when the treatment was rare. The authors suggest this might be because information from a friend of a friend gets "filtered" and distilled into the most important points, making it easier to digest.

The Bottom Line
The paper doesn't claim to have solved every mystery of how influence spreads. Instead, it offers a safer, more flexible way to measure it. It argues that we shouldn't force our data into a box that says "only direct neighbors matter." By using a wider net and not guessing the exact formula for influence, we can get a truer picture of how treatments ripple through a community. The authors suggest that while their method might be a bit less precise than older, simpler methods (because it makes fewer assumptions), it is much more reliable because it doesn't fall apart when the real world is messy and complex.

Technical Summary: Higher-order Spillover Effects Under Partial Interference

Problem Statement

Causal inference in networked settings is complicated by interference, where a unit's outcome depends on the treatments of other units. While existing literature often relies on the neighborhood interference assumption (limiting interference to direct neighbors) or the partial interference assumption (limiting interference to pre-defined clusters), these approaches face significant limitations:

Restrictive Scope: In scenarios involving outcome diffusion (e.g., disease transmission, information spread), interference often extends beyond direct neighbors to higher-order network distances.
Misspecification Risks: Standard methods require specifying an exposure mapping function (a summary of treatments in an interference set, e.g., proportion of treated neighbors). If the interference set or the mapping function is misspecified (e.g., assuming only first-order neighbors matter when second-order effects exist, or using a binary proxy for a continuous exposure), estimators become biased.
Lack of Higher-Order Disentanglement: Current methods struggle to disentangle spillover effects originating from specific network distances (e.g., distinguishing first-order from second-order neighbors) without relying on strong, potentially incorrect parametric assumptions about the interference mechanism.

Methodology

1. Generalized Interference Assumption

The authors propose a Generalized Partial Interference Assumption (Assumption 4). Unlike traditional assumptions that restrict interference to direct neighbors or entire clusters, this framework allows a unit's potential outcome to depend on treatments within a conservative interference set $I_{ij}$ . Crucially, this approach does not require specifying a summarizing exposure mapping function $g(\cdot)$ to reduce the treatment vector to a scalar or low-dimensional vector. This avoids bias arising from incorrect functional forms of interference.

2. Causal Estimands: $h$ -Order Spillover Effects

The paper defines new causal estimands to quantify spillover effects from units at a specific network distance $h$ .

Hypothetical Assignment: The authors introduce a hypothetical Bernoulli treatment allocation where:
- Units in the $h$ -order neighborhood ( $C^h_{ij}$ ) are assigned treatment with probability $\alpha_h$ .
- Units in the remainder of the interference set ( $C_{i\setminus(j, C^h_{ij})}$ ) are assigned treatment with probability $\alpha$ .
Definition: The $h$ -order spillover effect, $SE^h(\alpha_h, \alpha'_h; a, \alpha)$ , is defined as the difference in population average potential outcomes when the assignment probability for the $h$ -order neighborhood changes from $\alpha'_h$ to $\alpha_h$ , while holding individual treatment $a$ and the rest of the interference set assignment $\alpha$ fixed.

3. Estimators

To estimate these effects without relying on exposure mapping functions, the authors develop three non-parametric and semi-parametric estimators under a two-stage randomized design:

Horvitz-Thompson (HT) Estimator: A weighted average of observed outcomes using inverse probability weights. It is unbiased but can suffer from high variance due to extreme weights.
Hajek Estimator: A ratio estimator (normalized HT) that stabilizes variance by dividing the weighted sum of outcomes by the weighted sum of weights. It is consistent and asymptotically unbiased.
Weighted Least Squares (WLS) Estimator: A regression-based approach that models the relationship between the outcome and the hypothetical probability $\alpha_h$ (e.g., a linear marginal structural model). It uses HT weights to correct for selection bias but gains efficiency by modeling the outcome structure.

4. Bias Analysis of Misspecification

The paper derives theoretical bias formulas (Proposition 2 and Corollaries 1 & 2) for estimators that rely on misspecified interference sets or exposure mapping functions.

It demonstrates that using an incorrect interference set (e.g., restricting to first-order neighbors when interference is second-order) or an incorrect mapping function leads to the identification of a weighted average of potential outcomes under the true structure, rather than the true causal effect.
Specifically, omitting higher-order terms in a regression (OLS) induces omitted variable bias due to the correlation between exposure metrics at different network distances, often exacerbated by the two-stage randomization design (Simpson's Paradox).

Key Results

Simulation Study

The authors conducted Monte Carlo simulations across four scenarios (no interference, first-order, second-order, and second-order with interaction) and various network structures (Erdős–Rényi and regular graphs).

Bias: The proposed HT, Hajek, and WLS estimators remained unbiased across all scenarios, including those with higher-order interference and interactions. In contrast, standard OLS estimators relying on restrictive interference sets (e.g., $OLS(G_1)$ ) or misspecified exposure mappings exhibited significant bias.
Variance: The HT estimator showed the highest variance, particularly with large interference sets. The Hajek and WLS estimators demonstrated lower variance, with WLS generally being the most efficient due to parametric smoothing, provided the model specification was reasonable.
Network Density: The variance of the Hajek estimator was sensitive to network density ( $p$ ). For first-order effects, variance increased with density; for second-order effects, variance peaked at intermediate densities and decreased as networks became saturated.

Empirical Application: Honduras Maternal and Child Health Trial

The methods were applied to a two-stage randomized trial in Honduras involving 176 villages and a maternal/child health intervention.

First-Order Spillovers: For treated individuals, first-order spillover effects followed an inverted U-shape, peaking at moderate treatment saturation ( $\alpha \approx 0.5$ ) and declining at higher saturation (suggesting information saturation). Untreated individuals showed negligible spillover effects.
Second-Order Spillovers: Second-order effects were strongest when treatment was scarce (low $\alpha$ ) and diminished as treatment became prevalent, suggesting a substitution mechanism where direct neighbors fill information gaps.
Heterogeneity: Treated individuals benefited significantly more from peer spillovers than untreated individuals, attributed to higher absorptive capacity and baseline preparedness.

Significance and Claims

The paper claims to provide a robust framework for estimating higher-order spillover effects without relying on restrictive assumptions about the interference mechanism or the specific functional form of exposure.

Robustness to Misspecification: By avoiding exposure mapping functions, the proposed estimators avoid the bias inherent in "naive" approaches that incorrectly specify the scope or mechanism of interference.
Flexibility: The framework allows researchers to define interference sets conservatively (e.g., all units reachable via finite paths) and still estimate well-posed causal effects at specific network distances.
Practical Utility: The empirical application demonstrates that ignoring higher-order interference or using restrictive assumptions can obscure complex patterns, such as the non-monotonic relationship between treatment saturation and spillover magnitude.
Trade-off: The authors acknowledge that while their conservative approach ensures robustness, it may come at the cost of efficiency (wider confidence intervals) compared to methods that correctly specify the interference structure. They recommend this approach when the complexity of social processes makes it imprudent to commit to restrictive exposure mappings.