New Estimands for Experiments with Strong Interference

Abstract

In experiments that study social phenomena, such as peer influence or herd immunity, the treatment of one unit may influence the outcomes of others. Such “interference between units” violates traditional approaches for causal inference, so that additional assumptions are often imposed to model or limit the underlying social mechanism. For binary outcomes, we propose new estimands that can be estimated without such assumptions, allowing for interval estimates that assume only the randomization of treatment. However, the causal implications of these estimands are more limited than those attainable under stronger assumptions. The estimand shows whether the treatment effects under the observed assignment varied systematically as a function of each unit’s direct and indirect exposure to treatment, while also lower bounding the number of units affected. Supplementary materials for this article are available online.

Keywords:

• Causal inference
• Convex optimization
• Interference between units
• Social networks
• SUTVA

Supplementary Materials

The supplement contains additional data examples; proof of (10) and (25); asymptotic results regarding consistency and interval coverage for regression estimands (including Proposition 1); and data summaries.

Acknowledgments

The author wishes to acknowledge helpful discussions with Brian Kovak, Dan Nagin, Beka Steorts, Eric Tchetgen Tchetgen, and Stefan Wager, as well as detailed and insightful feedback from the editors and reviewers

Disclosure Statement

The author reports there are no competing interests to declare.

Notes

1 In the terminology of Imbens and Manski (2004), ${\sigma }_{\theta }^2$ has identification set $[0, 1/4]$, inducing an identification set for the prediction interval of ${\widehat{\tau }}_1-{\tau }_1$.

2 Under conditions where ${\widehat{\tau }}_1$ is consistent for τ₁, such as those of Proposition 1, assumptions that imply consistency of ${\widehat{\tau }}_1$ for the EATE also imply convergence of τ₁ to the EATE.

3 To see this, consider that if Y is deterministic, then $\mathbb{E}{\widehat{\tau }}_1=0$ and a confidence interval for $\mathbb{E}{\widehat{\tau }}_1$ is given by ${\widehat{\tau }}_1\pm z_{1-\alpha }\sqrt{\frac{N}{N-1}\frac{N}{N_1{N}_0} {\sigma }_Y^2}$, where ${\sigma }_Y^2$ is the variance of $Y_1,\dots ,Y_N$. As ${\sigma }_Y^2$ is upper bounded by 1/4, this interval is contained by the prediction interval of Proposition 1.

4 As $-U\le -w^T\theta \le -L$ holds with probability $1-\alpha$, adding w^TY implies $w^{T}Y-U\le w^{T}A\le w^{T}Y-L$ holds with the same probability.

5 Data available at https://www.aeaweb.org/articles?id=10.1257/app.20130442.

6 Available in the R package inferference.