Many causal questions involve interactions between units, also known as interference, for example between individuals in households, students in schools, or firms in markets. In this paper we formalize the concept of a conditioning mechanism, which provides a framework for constructing valid and powerful randomization tests under general forms of interference. We describe our framework in the context of two-stage randomized designs and apply our approach to a randomized evaluation of an intervention targeting student absenteeism in the School District of Philadelphia. We show improvements over existing methods in terms of computational and statistical power. arXiv:1709.08036v3 [stat.ME]

Two-stage randomization is a powerful design for estimating treatment effects in the presence of interference; that is, when one individual's treatment assignment affects another individual's outcomes. Our motivating example is a two-stage randomized trial evaluating an intervention to reduce student absenteeism in the School District of Philadelphia. In that experiment, households with multiple students were first assigned to treatment or control; then, in treated households, one student was randomly assigned to treatment. Using this example, we highlight key considerations for analyzing two-stage experiments in practice. Our first contribution is to address additional complexities that arise when household sizes vary; in this case, researchers must decide between assigning equal weight to households or equal weight to individuals. We propose unbiased estimators for a broad class of individual-and household-weighted estimands, with corresponding theoretical and estimated variances. Our second contribution is to connect two common approaches for analyzing two-stage designs: linear regression and randomization inference. We show that, with suitably chosen standard errors, these two approaches yield identical point and variance estimates, which is somewhat surprising given the complex randomization scheme. Finally, we explore options for incorporating covariates to improve precision. We confirm our analytic results via simulation studies and apply these methods to the attendance study, finding substantively meaningful spillover effects.

Interference exists when a unit's outcome depends on another unit's treatment assignment. For example, intensive policing on one street could have a spillover effect on neighbouring streets. Classical randomization tests typically break down in this setting because many null hypotheses of interest are no longer sharp under interference. A promising alternative is to instead construct a conditional randomization test on a subset of units and assignments for which a given null hypothesis is sharp. Finding these subsets is challenging, however, and existing methods are limited to special cases or have limited power. In this paper, we propose valid and easy-to-implement randomization tests for a general class of null hypotheses under arbitrary interference between units. Our key idea is to represent the hypothesis of interest as a bipartite graph between units and assignments, and to find an appropriate biclique of this graph. Importantly, the null hypothesis is sharp within this biclique, enabling conditional randomization-based tests. We also connect the size of the biclique to statistical power. Moreover, we can apply off-the-shelf graph clustering methods to find such bicliques efficiently and at scale. We illustrate our approach in settings with clustered interference and show advantages over methods designed specifically for that setting. We then apply our method to a large-scale policing experiment 174

We consider the problem of how to assign treatment in a randomized experiment, in which the correlation among the outcomes is informed by a network available preintervention. Working within the potential outcome causal framework, we develop a class of models that posit such a correlation structure among the outcomes. Then we leverage these models to develop restricted randomization strategies for allocating treatment optimally, by minimizing the mean square error of the estimated average treatment effect. Analytical decompositions of the mean square error, due both to the model and to the randomization distribution, provide insights into aspects of the optimal designs. In particular, the analysis suggests new notions of balance based on specific network quantities, in addition to classical covariate balance. The resulting balanced, optimal restricted randomization strategies are still design unbiased, in situations where the model used to derive them does not hold. We illustrate how the proposed treatment allocation strategies improve on allocations that ignore the network structure, with extensive simulations.

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