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Treatment effect estimation under convergent network interference

Bryan Park, Stefan Wager

Abstract

Under network interference, the treatment given to one unit may also affect the outcomes of its neighboring units in an exposure graph. Existing large-sample theory has focused on settings where either the exposure graph is sparse, or the exposure graph is randomly generated using a random graph model. The question of how to analyze treatment effect estimation in network interference models with dense, non-random exposure graphs has remained open to date. Here, we address this gap and prove a central limit theorem for possibly dense, non-random models by extending the graph limit framework pioneered by Lovász and Szegedy to the setting of causal inference under network interference. Our result implies that the uncertainty for average direct effect estimation is to first-order driven by random treatment assignment, and so asymptotic results derived under the random graph model correctly predict statistical behavior in non-random network interference designs.

Treatment effect estimation under convergent network interference

Abstract

Under network interference, the treatment given to one unit may also affect the outcomes of its neighboring units in an exposure graph. Existing large-sample theory has focused on settings where either the exposure graph is sparse, or the exposure graph is randomly generated using a random graph model. The question of how to analyze treatment effect estimation in network interference models with dense, non-random exposure graphs has remained open to date. Here, we address this gap and prove a central limit theorem for possibly dense, non-random models by extending the graph limit framework pioneered by Lovász and Szegedy to the setting of causal inference under network interference. Our result implies that the uncertainty for average direct effect estimation is to first-order driven by random treatment assignment, and so asymptotic results derived under the random graph model correctly predict statistical behavior in non-random network interference designs.

Paper Structure

This paper contains 21 sections, 8 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. The Model and Asymptotic Regularity
  4. Problem Setting
  5. Convergence of Experiments under Network Interference
  6. A Central Limit Theorem
  7. Proof of Main Result
  8. The Explicit Coupling
  9. Exploiting the Coupling
  10. Transferring Asymptotic Normality
  11. Numerical Illustrations
  12. Discussion
  13. Appendix
  14. Simulation Details for Section \ref{['Numeric']}
  15. Technical Lemmas for Degrees
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\{(G_n, v_n)\}_{n=1}^\infty$ be a deterministic sequence of exposure graphs and potential outcome functions modeling anonymous interference. Let $\mathcal{L}$ be a kernel and $\ell: [0,1]\to \mathcal{F}$ be measurable. Finally, let $\{\rho_n\}_{n=1}^\infty$ denote a sequence of scale factors in then we have with asymptotic variance given by where $U_i\overset{\text{i.i.d.}}{\sim} U[0,1]$ an

Figures (2)

  • Figure 1: Experiments in Dense Regime.
  • Figure 2: Experiments in Sparse Regime.

Theorems & Definitions (18)

  • Definition 1: Cut norm
  • Definition 2: Graph Convergence
  • Definition 3: Convergence of $(G_n,v_n)$ at scale $\rho_n$
  • Theorem 1
  • Proposition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • ...and 8 more