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Neighborhood Adaptive Estimators for Causal Inference under Network Interference

Alexandre Belloni, Fei Fang, Alexander Volfovsky

TL;DR

This work addresses causal inference under network interference when the interference radius can vary across units and is unknown. It introduces neighborhood-adaptive estimators that learn unit-specific interference patterns by aggregating across $m_i$-hop neighborhoods and selecting patterns with a Lepski-type rule, along with a synthetic-treatment decoupling mechanism to enable valid inference from ML-driven feature engineering. The authors establish an oracle inequality for the interference estimator and distributional results for the AIPW estimator, plus convergence rates for the outcome-regression estimator under adaptive interference. Empirical results from simulations and a real data application show the proposed methods effectively estimate the average direct treatment effect on the treated (ADTT) under complex interference, with practical guidance for variance estimation and inference in network settings.

Abstract

Estimating causal effects has become an integral part of most applied fields. In this work we consider the violation of the classical no-interference assumption with units connected by a network. For tractability, we consider a known network that describes how interference may spread. Unlike previous work the radius (and intensity) of the interference experienced by a unit is unknown and can depend on different (local) sub-networks and the assigned treatments. We study estimators for the average direct treatment effect on the treated in such a setting under additive treatment effects. We establish rates of convergence and distributional results. The proposed estimators considers all possible radii for each (local) treatment assignment pattern. In contrast to previous work, we approximate the relevant network interference patterns that lead to good estimates of the interference. To handle feature engineering, a key innovation is to propose the use of synthetic treatments to decouple the dependence. We provide simulations, an empirical illustration and insights for the general study of interference.

Neighborhood Adaptive Estimators for Causal Inference under Network Interference

TL;DR

This work addresses causal inference under network interference when the interference radius can vary across units and is unknown. It introduces neighborhood-adaptive estimators that learn unit-specific interference patterns by aggregating across -hop neighborhoods and selecting patterns with a Lepski-type rule, along with a synthetic-treatment decoupling mechanism to enable valid inference from ML-driven feature engineering. The authors establish an oracle inequality for the interference estimator and distributional results for the AIPW estimator, plus convergence rates for the outcome-regression estimator under adaptive interference. Empirical results from simulations and a real data application show the proposed methods effectively estimate the average direct treatment effect on the treated (ADTT) under complex interference, with practical guidance for variance estimation and inference in network settings.

Abstract

Estimating causal effects has become an integral part of most applied fields. In this work we consider the violation of the classical no-interference assumption with units connected by a network. For tractability, we consider a known network that describes how interference may spread. Unlike previous work the radius (and intensity) of the interference experienced by a unit is unknown and can depend on different (local) sub-networks and the assigned treatments. We study estimators for the average direct treatment effect on the treated in such a setting under additive treatment effects. We establish rates of convergence and distributional results. The proposed estimators considers all possible radii for each (local) treatment assignment pattern. In contrast to previous work, we approximate the relevant network interference patterns that lead to good estimates of the interference. To handle feature engineering, a key innovation is to propose the use of synthetic treatments to decouple the dependence. We provide simulations, an empirical illustration and insights for the general study of interference.
Paper Structure (24 sections, 19 theorems, 166 equations, 4 figures, 4 tables)

This paper contains 24 sections, 19 theorems, 166 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Recent development beyond neighborhood interference
  3. Our contribution
  4. Setup
  5. Estimators
  6. Interference Estimator
  7. Decoupling feature engineering based on Synthetic Treatments
  8. Theoretical Results
  9. Oracle Inequality for Interference Function Estimator
  10. Distributional results for the AIPW estimator
  11. Rates for the Outcome Regression Estimator
  12. Experimental Performance of the AIPW Estimator
  13. Real data application
  14. Additional Discussion and Extensions
  15. Proofs for section \ref{['The_Res']}
  16. ...and 9 more sections

Key Result

Theorem 4.2

Suppose that Assumption assump:outc_mod holds. Then, with probability at least $1-\delta$, uniformly over all nodes $i\in [n]$ we have where $V_g := \{ i \in [n] : Z_i=0, \exists m \in [n] \hbox{s.t.} \ g=\gamma_0( G^Z_i(m) )\}$ as in Step 1 of Algorithm 1.

Figures (4)

  • Figure 1: The depth of the tree corresponds to the number of hops in the network. The leaves of the context tree correspond to the relevant interference patters (each will be allowed to have its own interference intensity). The panel on the left represents one interference function that depends only if at least one immediate neighbor was treated or not. The panel on the right represents an interference function that when no friends were treated, it depends if any friends of friends were treated or not. Note that a tree with only a root node represents a setting where there is no interference.
  • Figure 2: Structure of $\widehat{\mathcal{K}}$, interference estimators, and control unit counts at tree nodes generated by algorithm $2$ ( node size $\geq 5$)
  • Figure 3: Structure of $\widehat{\mathcal{K}}$, interference estimators, and control unit counts at tree nodes generated by algorithm $2$ ($\lambda = 0.4$, node size $\geq 5$)
  • Figure 4: Structure of $\widehat{\mathcal{K}}$, interference estimators, and control unit counts at tree nodes generated by algorithm $2$ ($\lambda = 0.7$, node size $\geq 5$)

Theorems & Definitions (47)

  • Example 1: Vaccine efficacy in an enclosed population
  • Example 2: After school programs on child outcome
  • Example 3: Number of treated neighbors
  • Example 4: Generalized Number of treated neighbors
  • Example 5: Isomorphic patterns
  • Example 6: Linear-in-means
  • Definition 4.1: Interference approximation error
  • Theorem 4.2
  • Example 7
  • Example 8
  • ...and 37 more