A Graph-Theoretic Approach to Randomization Tests of Causal Effects under General Interference

Abstract

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

Policy Evaluation for Temporal and/or Spatial Dependent Experiments in Ride-sourcing Platforms

“…Second, the partial interference assumption has been widely used to restrict interference only in known and disjoint groups of units (Tchetgen Tchetgen and VanderWeele, 2012;Halloran and Hudgens, 2016). Third, the network-based interference assumption was introduced to deal with interference between connected units in an exposure graph or network (Bakshy et al, 2014;Puelz et al, 2019).…”

Multiple conditional randomization tests

“…In more sophisticated problems, H(z, z * ) may depend on z and z * in a nontrivial way and we call such hypothesis partially sharp. The concept of imputability has appeared before, though the property was tied to a test statistic instead of a null hypothesis , Puelz et al, 2019; see Definition 4 below.…”

“…Their key insight is that imputability mapping of the above form can be represented as a bipartite graph with vertex set V = [N ] ∪ Z and edge set Puelz et al [2019] referred to this as the null exposure graph. Then by using (9), we have…”

“…Therefore, the challenging problem of finding a good partition of Z is reduced to finding a collection of large bicliques {(V m , E m )} M m=1 in the graph such that {S m } M m=1 partitions Z (this was called a biclique decomposition by Puelz et al [2019]). They further described an algorithm to find a biclique decomposition by greedily removing treatment assignments in the largest biclique (some approximate algorithm is needed to find the largest biclique as it is generally an NP-hard problem).…”

“…Much of our theoretical development in Section 2 is inspired by an exploding literature on causal inference with interference [Hudgens and Halloran, 2008, Bowers et al, 2013, Aronow and Samii, 2017, Li et al, 2019, Puelz et al, 2019.…”