Identification and Estimation of Treatment and Interference Effects in Observational Studies on Networks

Abstract

Abstract–Causal inference on a population of units connected through a network often presents technical challenges, including how to account for interference. In the presence of interference, for instance, potential outcomes of a unit depend on their treatment as well as on the treatments of other units, such as their neighbors in the network. In observational studies, a further complication is that the typical unconfoundedness assumption must be extended—say, to include the treatment of neighbors, and individual and neighborhood covariates—to guarantee identification and valid inference. Here, we propose new estimands that define treatment and interference effects. We then derive analytical expressions for the bias of a naive estimator that wrongly assumes away interference. The bias depends on the level of interference but also on the degree of association between individual and neighborhood treatments. We propose an extended unconfoundedness assumption that accounts for interference, and we develop new covariate-adjustment methods that lead to valid estimates of treatment and interference effects in observational studies on networks. Estimation is based on a generalized propensity score that balances individual and neighborhood covariates across units under different levels of individual treatment and of exposure to neighbors’ treatment. We carry out simulations, calibrated using friendship networks and covariates in a nationally representative longitudinal study of adolescents in grades 7–12 in the United States, to explore finite-sample performance in different realistic settings. Supplementary materials for this article are available online.

KEYWORDS:

• Causal inference
• Generalized propensity scores
• Network interference
• Potential outcomes
• Subclassification
• Unconfoundedness

Supplementary Materials

In Appendix A we we discuss in details the plausibility of the neighborhood interference assumption (Assumption 3) in network setting. In Appendix B, we develop propensity score-based estimators for conditional effects, that is, main and spillover effects among units that are observed under specific values of individual and neighborhood treatments. We detail the data generating model for the simulation study in Appendix C. Details of the proposed estimator, the specific models used for the simulation study, as well as a proposed approach on how to conduct statistical inference, are presented in Appendix D. Proofs of the theorems and corollaries are reported in Appendix E.

Acknowledgments

We thank Guido W. Imbens, Donald B. Rubin, the associate editor, and two anonymous reviewers, for detailed comments that helped improve this article.

Notes

1 We prefer the terms individual treatment effect and main effect over the term direct effect, because the latter may be confused with the direct effect of the mediation literature. The term individual treatment effect highlights the level of treatment assignment (the individual) and the term main effect stresses that this is usually the main effect of interest.

2 In principle, we could also define (and estimate) spillover effects by contrasting two arbitrary levels of the neighborhood treatment, g and $g\prime$. Here we view the case of the neighborhood treatment G_i = 0 as the standard scenario where the unit is not exposed to the treatment of other units. Therefore, we focus on contrasts between $g\in \mathcal{G}$ and 0.

3 In the case of continuous neighborhood treatment $\psi (z;g;x)$ is the probability density function. Throughout we will focus on probability mass functions $P(·)$ for discrete neighborhood treatments, although the extension to the continuous case is straightforward.

4 This property is similar to the individualistic property of the assignment mechanism in Imbens and Rubin (2015). However, here it is defined on the extended assignment mechanism defined on both the individual and the neighborhood treatment and the vector of covariates ${\mathbf{X}}_i$ include neighbors’ characteristics.

5 Note that we included in the joint propensity score the set of covariates that is sufficient to satisfy the unconfoundedness assumption. Oftentimes, the probability of unit i being exposed to a certain neighborhood treatment does depend on nonneighboring characteristics, given that the probability of their neighbors being assigned to treatment might depend on their neighbors’ covariates. In this case the joint propensity score will not coincide with the unit-level assignment mechanism.

Additional information

Funding

This work was supported, in part, by NSF awards CAREER IIS-1149662 and IIS-1409177, by ONR awards YIP N00014-14-1-0485 and N00014-17-1-2131, and by the “Dipartimenti Eccellenti 2018–2022” Italian Ministerial Funds.