agraph-theoreticapproachtorandomizationtestsof
TRANSCRIPT
A Graph-Theoretic Approach to Randomization Tests ofCausal Effects Under General Interference∗
David Puelz† Guillaume Basse‡ Avi Feller§ Panos Toulis†
This Version: May 26, 2021
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 neighboring streets.Classical randomization tests typically break down in this setting because many null hypothesesof interest are no longer sharp under interference. A promising alternative is to instead constructa conditional randomization test on a subset of units and assignments for which a given nullhypothesis is sharp. Finding these subsets is challenging, however, and existing methods arelimited 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 interferencebetween units. Our key idea is to represent the hypothesis of interest as a bipartite graph betweenunits and assignments, and to find an appropriate biclique of this graph. Importantly, the nullhypothesis is sharp within this biclique, enabling conditional randomization-based tests. We alsoconnect the size of the biclique to statistical power. Moreover, we can apply off-the-shelf graphclustering methods to find such bicliques efficiently and at scale. We illustrate our approach insettings with clustered interference and show advantages over methods designed specifically for thatsetting. We then apply our method to a large-scale policing experiment in Medellín, Colombia,where interference has a spatial structure.
Keywords: randomization test, interference, causal inference, networks, biclique.
∗email: [email protected]. We thank Peng Ding, Connor Dowd, Colin Fogarty, Sam Pimentel,Stephen Raudenbush, Paul Rosenbaum, and Santiago Tobón, as well as conference participants at Polmeth, Advanceswith Field Experiments, Design and Analysis of Experiments, Atlantic Causal Inference, and seminar participants atthe University of Chicago for helpful comments and discussion. AF gratefully acknowledges support from a NationalAcademy of Education/Spencer Foundation postdoctoral fellowship. PT is grateful for the John E. Jeuck Fellowship atUniversity of Chicago, Booth School of Business.
†The University of Chicago, Booth School of Business‡Stanford University§The University of California, Berkeley
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1. Introduction
The assumption of “no interference” between units (Cox, 1958) underlies most classical approachesto causal inference. The key premise is that a unit’s treatment does not affect another unit’s outcome,so that each unit’s outcome depends only on its own treatment status. This is implausible in manysettings, however, as it precludes peer effects, treatment spillovers, and other forms of treatmentinterference (Halloran and Hudgens, 2016).
Classical approaches often break down under interference. The canonical Fisher RandomizationTest (FRT), for example, is valid for testing the global sharp null hypothesis of no treatment effectbut fails when testing non-sharp null hypotheses, such as tests of treatment spillovers. Several recentproposals address this issue by restricting the randomization test to a subset of units, often calledfocal units, and a subset of assignments (Aronow, 2012; Athey et al., 2018; Basse et al., 2019b). Thecentral idea is that, conditional on these subsets, the specified null hypothesis is sharp for every focalunit, and thus the conditional randomization test is valid. These randomization-based approaches havemany advantages over model-based alternatives (Manski, 2013; Graham, 2008; Jackson, 2010; Grahamand Hahn, 2005; Brock and Durlauf, 2001; Blume et al., 2015; Toulis and Kao, 2013; Bowers et al.,2013; Auerbach, 2016; Leung, 2015) because they make minimal assumptions and are typically morerobust. However, existing randomization-based methods are usually tailored to a specific interferencestructure, which limits their power and widespread adoption. For example, Basse et al. (2019b) developa permutation-based approach tailored to clustered interference, which cannot be easily generalized,and Athey et al. (2018) restrict the type of conditioning information used for constructing the test.
In this paper, we propose a general procedure for identifying subsets of units and assignments forwhich the null hypothesis of interest is sharp, and use it to develop a method for randomization testsunder arbitrary interference. Our key methodological contribution is the null exposure graph. This isa bipartite graph with units and assignments as the nodes, and an edge between any unit-assignmentpair if we observe a null exposure, i.e., a treatment exposure specified in the null hypothesis, for theunit and assignment in the pair. A biclique of the null exposure graph is a complete bipartite subgraphin which all units on one side are connected to all assignments on the other side. Crucially for testing,this biclique constitutes a subset of units and assignments for which the null hypothesis is sharp anda valid randomization test is possible.
Our proposed method offers three main benefits over existing approaches. First, it allows con-ditioning on the observed assignment, which can increase power over methods that suggest randomconditioning (Athey et al., 2018; Aronow, 2012). Second, since a null hypothesis uniquely determinesa null exposure graph, our method is constructive and defines, step by step, how to conduct ran-domization tests under general forms of interference. This is an improvement over methods that aretailored to specific patterns of interference (Basse et al., 2019b). Finally, our method translates ques-tions of computation and statistical power into properties and operations on the null exposure graph,separating these considerations from test validity. Our approach is therefore modular, and can benefitfrom separate advances in graph algorithms for biclique computation.
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To illustrate our method, we consider two natural structures of interference: clustered interferenceand spatial interference. Under clustered interference, units can be separated into well-defined clus-ters, such as households or classrooms, where we assume that units interact within clusters but notbetween clusters. Our motivating example here is a two-stage randomized trial of a student attendanceintervention in which households are assigned to treatment or control and then, within each treatedhousehold, one student is randomly treated; see Basse and Feller (2018). Under spatial interference,we assume that interactions “pass through” neighboring units, but without the simpler structure ofclustered interference. Here, we focus on re-analyzing a large-scale experiment in Medellín, Colombiastudying the impact of “hotspot policing” on crime (Collazos et al., 2019). Our analysis of spatialinterference considers hypotheses on any specified spillovers, which differs from other design-basedmethods that consider a marginal spillover effect over the design (Aronow et al., 2019). We also ex-tend our framework to test null hypotheses that restrict the level of interference between units, suchas the null hypothesis of no interference.
Our paper is structured as follows. Section 2 introduces the problem setup and all necessarynotation. Section 3 outlines our methodology, comprised of the null exposure graph (Section 3.1) andbiclique decompositions of the graph (Section 3.2). Section 4 presents the proposed randomizationtest. Section 5 gives the main results on statistical power. We then illustrate our method in twoapplications. Section 6 considers settings with clustered interference, and Section 7 considers settingswith spatial interference, specifically in the context of a large-scale policing experiment in Medellín,Colombia. Finally, Section 8 extends our results to complex null hypotheses. The Appendix containsadditional empirical and theoretical results, as well as the proofs.
2. Overview of causal inference under interference and problem setup
2.1. Setup and notation
Consider a finite population of N units indexed by i = 1, . . . , N . Let Zi denote unit i’s treatment,which we assume to be binary without loss of generality. Let Z = (Z1, Z2, . . . , ZN ) ∈ {0, 1}N denotethe population treatment assignment, and P (Z) its distribution according to the experiment design.We focus on experimental studies, and so P (Z) is known. Let Yi(z) ∈ R denote the potential outcomeof unit i under population assignment z ∈ {0, 1}N . For the observed quantities we use the modifier“obs” as a superscript for emphasis. Thus, Zobs ∈ {0, 1}N is the observed population treatment, andY obs = (Y1(Z
obs), . . . , YN (Zobs)) ∈ RN is the vector of observed outcomes. In the randomizationframework, Zobs is random according to the design, Zobs ∼ P (Zobs), whereas the potential outcomesare fixed. Let U = {1, . . . , N} denote the set of all units, and Z = {z ∈ {0, 1}N : P (z) > 0} denotethe set of all population treatment assignments supported by the design.
The main challenge is that causal inference is infeasible without restrictions on the potentialoutcomes; unrestricted, each unit has 2N possible potential outcomes. For instance, the common nointerference assumption states that the outcome for each unit depends only on its own treatmentassignment, so each unit has only two potential outcomes.
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When this assumption is implausible, one strategy is to define a treatment exposure, which is alow-dimensional summary of Z, such as “spillovers” or “peer effects". In particular, we assume a finiteset of possible treatment exposures, F = {a, b, ...}, and exposure mapping functions, fi : Z → F, foreach unit i ∈ U. Thus, the definition of fi is application-specific, as it needs to consider the interferencestructures that are appropriate for a given problem; see Aronow et al. (2017). To make this concrete,we briefly introduce three examples.
Example 1 (Clustered interference). Following the setting in Basse et al. (2019b), each unit belongsto a fixed cluster, such as individuals within households. The key assumption is that units interactwithin each cluster, but not between clusters, also known as partial interference (Sobel, 2006). Inthe experiment, clusters are assigned to treatment or control, and, within treated clusters, one unit israndomly assigned to treatment. Here, the exposure function may be defined as fi(z) = zi +
∑j∈[i] zj,
where [i] denotes all units in i’s cluster. Thus, F = {0, 1, 2}, and there are three exposure levels: fi = 0
denotes a control unit in a control cluster; fi = 1 denotes a control unit in a treated cluster; and fi = 2
denotes a treated unit in a treated cluster. We explore this setting further in Section 6.
Example 2 (Spatial interference). In this setting, units interact with one another locally throughshared space, like street segments in a city. The goal is to test whether the outcomes of untreatedunits close to treated units (i.e., spillover units) are affected by the treatment. For example, let grij =
1{d(i, j) < r} denote whether units i and j are within distance r of each other, where d(i, j) denotestheir spatial distance. We may define fi(z) = (wi, zi), where wi = 1{
∑j 6=i g
rijzj > 0} indicates whether
i is within a radius r of any other treated unit. Here, F = {(0, 0), (0, 1), (1, 0), (1, 1)}, and there arefour exposure levels. We explore this setting further in Section 7.
Example 3 (Extent of interference). In this setting, units are linked in a social network. Specifically,we assume an integer-valued function d(i, j) ∈ N+ that measures the distance between units i and j inthe network, such that d(i, j) is symmetric, d(i, i) = 0 for each i, and d(i, j) = ∞ if i and j are notconnected. For some integer k ≥ 0, let Gk = (gkij) be the social network such that gkij = I{d(i, j) ≤ k},and define fi(z) = Gki z, where Gki is the N × N diagonal matrix having as diagonal the i-th rowof matrix Gk. Here, the exposure for every unit is an N -length binary vector, and so F ⊆ {0, 1}N .With this setup, different values of k restrict the extent of interference. For example, when k = 0,the exposure fi(z) for some unit i depends only on individual treatment zi. This can describe settingswhere the extent of interference is minimal. However, when k = 1, fi(z) depends on the individualtreatment of i and the treatments of i’s immediate neighbors; when k = 2, fi(z) additionally dependson the treatments of second-order neighbors, and so on. Thus, for sufficiently large k, fi(z) coulddepend on the treatments of all units connected to i. We explore this setting further in Section 8.
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2.2. Null hypotheses on exposure mappings
Our primary goal is to test null hypotheses that specify relations between treatment assignmentsand potential outcomes. First, consider the following general hypothesis:
HF0 : Yi(z) = Yi(z′) for all i = 1, . . . , N, and any z, z′ ∈ Z such that fi(z), fi(z′) ∈ F ⊆ F. (1)
In words, HF0 states that for each unit the outcomes under any exposure in F are identical, regardlessof the particular population treatment assignment, z ∈ Z. The formulation in Equation (1) is quitegeneral, and can express different kinds of hypotheses under interference for various choices of F . Atone extreme, HF
0 is the “global null hypothesis” of no treatment effect. This is a sharp null hypothesisthat can be tested by the classical Fisher randomization test, since all potential outcomes can beimputed under the null. At the other extreme, H∅0 is degenerate and is fundamentally untestable.
An important special case is the singleton null hypothesis, H{a}0 , where F = {a}. Formally:
H{a}0 : Yi(z) = Yi(z
′) for all i = 1, . . . , N, and any z, z′ ∈ Z such that fi(z) = fi(z′) = a. (2)
This is a stability or exclusion restriction hypothesis: under H{a}0 , the potential outcomes for exposurefi(z) = a are only functions of exposure a and not the underlying population treatment assignment, z.Versions of this statement often appear as assumptions about properly specified exposure mappings;see Aronow et al. (2017); Basse et al. (2019a). By itself, this singleton hypothesis H{a}0 is not typicallytestable. However, it is the building block for many more general hypotheses. We focus on two classes:(1) contrast hypotheses that compare two exposures, e.g., H{a,b}0 ; and (2) intersection hypotheses thatconsider sets of singleton hypotheses, e.g.,
⋂a∈FH
{a}0 .
First, to compare two exposures, a and b, we set F = {a, b}, and refer to the resulting hypothesisH{a,b}0 as a contrast hypothesis:
H{a,b}0 : Yi(z) = Yi(z
′) for all i = 1, . . . , N, and any z, z′ ∈ Z such that fi(z), fi(z′) ∈ {a, b}. (3)
A subtle issue with such contrast hypotheses is that H{a,b}0 implies both H{a}0 and H
{b}0 . As such,
rejecting H{a,b}0 does not necessarily mean that treatment exposures a are b are different, but it couldbe that either H{a}0 or H{b}0 is not true. We will return to this issue of interpretation in the applicationof Section 7.3.
Example 1 (Clustered interference (cont.)). In this setting, we consider testing H{a,b}0 with a = 0
and b = 1; that is, we test whether there is a spillover effect on control units from a treated unit inthe same cluster.
Example 2 (Spatial interference (cont.)). In this setting, we consider testing H{a,b}0 with a = (0, 0)
and b = (1, 0) for some value of the distance threshold, r. That is, we test whether there is a spillovereffect on an untreated unit from having one or more treated units within the specified distance, r (e.g.,r = 125m in the application of Section 7).
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An alternative direction is to instead consider intersections of singleton hypotheses. For example,the intersection hypothesis of all possible singletons, namely
⋂a∈FH
{a}0 =: Hex
0 , is a full exclusionrestriction condition on the exposures:
Hex0 : Yi(z) = Yi(z
′) for all i = 1, . . . , N, and any z, z′ ∈ Z such that fi(z) = fi(z′). (4)
This covers many standard hypotheses under interference in the literature (Toulis and Kao, 2013;Bowers et al., 2013; Rosenbaum, 2007; Aronow, 2012; Basse et al., 2019a,b; Athey et al., 2018). InSection 8, we show that our main procedure can also test intersection null hypotheses of the formHF1
0 ∩HF20 , where F1 ∩ F2 = ∅.
Example 3 (Extent of interference). In the social network setting, we consider the null hypothesisthat for all i = 1, . . . , N :
Yi(z) = Yi(z′) for any z, z′ ∈ Z such that zj = z′j for all j = 1, . . . , N, for which d(i, j) ≤ k. (5)
This hypothesis posits that a unit’s outcome may depend only on treatments of units up to k hops awayin the social network but no further. This is standard in the literature; see, for example, Hypotheses2, 3, & 4 in Athey et al. (2018). The hypothesis in Equation (5) is equivalent to Hex
0 with an exposurefunction as described in Example 3 of Section 2.1 above.
As we discuss next, the main challenge in testing these hypotheses is that the treatment exposuresin F are not independently manipulated in the experiment—the design only determines the jointdistribution of the population treatment assignment, Z. This precludes a simple randomization testfor HF0 since we cannot impute the outcomes of units assigned to exposure levels other than those inF .1 Similar to other approaches discussed below, we propose to address this issue by appropriatelyconditioning the randomization test. A contribution of this paper is to constructively find suchconditioning for general null hypotheses. We give a general overview of conditional randomizationtests, and in Section 3 we describe our proposed conditioning method based on the concept of nullexposure graphs to test HF0 .
2.3. Conditional randomization tests under interference: A review
We briefly review the general framework proposed by Basse et al. (2019b) for valid randomizationtests under interference. This framework builds on the key insight first formulated by Aronow (2012)and developed by Athey et al. (2018) that, although the null hypothesis HF0 is not sharp in general,it can be “made sharp” if we restrict our attention to a well chosen subset of units, U ⊆ U, and subsetof assignments, Z ⊆ Z. Basse et al. (2019b) formalized the idea as sampling a conditioning event,
1In Appendix A, we discuss an equivalent formulation of this imputability problem. Specifically, we express HF0as a composite null hypothesis on the full schedule of potential outcomes. Then, the imputability problem becomesessentially a problem of identification, where several alternative hypotheses may lead to a randomization distributionthat is equal to the null.
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C = (U,Z), from a carefully constructed distribution P (C|Zobs), called a conditioning mechanism,and then running a test conditional on C. This requires the use of a test statistic restricted on C:
Definition 1 (Restricted test statistic). Let C = (U,Z) be a conditioning event, where U ⊆ U andZ ⊆ Z. A test statistic, t(z, y;C) : {0, 1}N × Rn → R is restricted on C iff:
t(z′, y′;C) = t(z, y;C), for all y, y′, and z, z′ ∈ Z, such that z′U = zU and y′U = yU .
Here, subscript “U ” denotes the corresponding subvector restricted only to units U in event C.
Theorem 1 (Basse et al. (2019b)). Let H0 be a null hypothesis. For some conditioning event C =
(U,Z), where U ⊆ U and Z ⊆ Z, let t(z, y;C) be a test statistic restricted on C. Suppose that the teststatistic is conditionally imputable under H0 given C, i.e., YU (z′) = YU (z) for any z, z′ ∈ Z such thatP (C|z) > 0 and P (C|z′) > 0. The p-value obtained from the following procedure:
1. Draw Zobs ∼ P (Zobs), and observe Y obs = Y (Zobs).
2. Draw C ∼ P (C|Zobs), and compute T obs = t(Zobs, Y obs;C).
3. Compute p-value = E[1{t(Z, Y obs;C) > T obs}|C
], where the expectation is with respect to the
correct randomization distribution, P (Z|C) ∝ P (C|Z)P (Z),
is valid conditionally and marginally.
Conditional validity means that, under any conditioning event C, the test in the third step ofTheorem 1 has the correct level (i.e., has Type-I error probability equal to α under the null) withrespect to the conditional randomization distribution of Zobs given C. Marginal validity means thatthe test has the correct level marginally over C, and is thus weaker than conditional validity. Indeed,it is possible that the the test does not have the correct level for some conditioning events, butnonetheless has correct Type-I error on average over C. On the other hand, if the test is conditionallyvalid then it is also marginally valid.
The conditioning event C is an abstract device used to construct valid randomization tests andis generated from the conditioning mechanism P (C|Zobs). Moreover, the analyst can construct theconditioning mechanism of their choosing so that C depends stochastically or deterministically onZobs. A key contribution of this paper is to propose an approach such that C is easily constructed.
We can see from Theorem 1 that there are two main challenges in constructing a conditioningmechanism that leads to valid conditional randomization tests. First, the test statistic should beimputable under the null hypothesis H0. Generally, this means that based only on the observedvalue of the outcomes, Y obs, we can compute the null distribution of the test statistic t(z, Y (z);C) =
t(z, Y obs;C) that is induced by the randomization distribution, P (Z|C). Second, we must be able todraw samples from this randomization distribution, given by its conditional-marginal decompositionP (Z|C) ∝ P (C|Z)P (Z) in the third step. Ensuring that this distribution is computationally tractablecan be challenging (Basse et al., 2019a,b).
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In addition, the null hypotheses of interest are defined for all units i = 1, . . . , N , even if theyare defined for a subset of assignments, e.g., fi ∈ F . As a result, rejecting a null hypothesis fora subset of units logically implies rejecting that same null hypothesis for all units. Thus, rejectinga null hypothesis with a conditional randomization test has the same interpretation as rejecting anull hypothesis with a (possibly infeasible) unconditional test. In addition, this interpretation holdsregardless of the specific conditioning event, although the power of the test might depend on theconditioning mechanism, as we discuss in Section 5.
To gain more intuition, we can also describe the approaches of Basse et al. (2019b), Athey et al.(2018) and Aronow (2012) within this framework, each corresponding to different choices of the con-ditioning mechanism. In particular, Basse et al. (2019b) propose a conditioning mechanism underclustered interference, such that the implied randomization distribution, P (Z|C), leads to a permu-tation test, which is easy to implement. However, their approach does not readily generalize to othersettings, such as spatial interference. The methods of Athey et al. (2018) and Aronow (2012) corre-spond to conditioning mechanisms of the form P (C|Z) = P (C), where conditioning is either random,or guided by known auxiliary information, but is not conditioned on the observed assignment. Incontrast to the approach of Basse et al. (2019b), these methods can be applied in general interferencesettings. However, they are usually underpowered because they do not use the observed assignmentto do the conditioning; as an extreme example, these approaches cannot test HF0 if there are no unitsin C exposed to any exposure value in F . This may happen because C is randomly sampled, and doesnot use the exposure information in Zobs. We discuss this issue further in Section 4.2.
In this paper, we propose a testing method that is both general and powerful. In the following sec-tions, we develop the core concepts of our approach, using the framework of conditioning mechanismspresented here. Our main contribution is an algorithm that automatically constructs a tractable con-ditioning mechanism, P (C|Z), through the concept of null exposure graph presented in the upcomingsection. Our proposed randomization test for HF0 is presented later in Section 4, and in Section 4.2we follow up on this discussion of related methods and describe the benefits of our approach.
3. The null exposure graph and bicliques
We now introduce some preliminary concepts underlying our test for HF0 in Equation (1). Thefirst key idea is to represent the imputability of outcomes under the null hypothesis through a graphbetween units and assignments, which we call the null exposure graph. The conditioning event Cwill then be taken to be a biclique in that graph, and the conditioning mechanism P (C|Zobs) willdetermine the biclique that contains Zobs. This transforms the analytical task of defining P (C|Zobs)
into a computational task.
3.1. The null exposure graph
The first component of our method is the null exposure graph, which is a graph that encodes theunits’ treatment exposures under different population treatment assignments. As we will show in the
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next section, the null exposure graph determines the appropriate conditioning for the randomizationtest of HF0 .
Definition 2 (Null exposure graph). Let U = {1, . . . , N} and Z = {z1, . . . , zJ} denote the sets ofunits and assignments, respectively. Define the vertex set as V = U ∪ Z, and the edge set as
E = {(i, z) ∈ U× Z : fi(z) ∈ F}. (6)
That is, an edge between unit i and assignment z exists if and only if i is exposed to F under z. Thereare no edges between units or between assignments. Then, GFf = (V,E) is the null exposure graph ofHF0 with respect to exposure mapping f and exposure set F .
The null exposure graph is a bipartite graph since there are not edges between units or betweenassignments. In order to visualize the null exposure graph in a concrete setting, we return to theexample of clustered interference (Example 1).
Example 1 (Clustered interference (cont.)). For simplicity, suppose we have four units in two clusters,namely {1, 2} and {3, 4}; and that we treat exactly one cluster leading to four possible assignmentvectors, depending on which unit is treated within the treated cluster. Following Section 2,
f1(z) = 2z1 + z2, f2(z) = 2z2 + z1;
f3(z) = 2z3 + z4, f4(z) = 2z4 + z3. (7)
We test H{0,1}0 , i.e., whether outcomes are equal between exposures fi = 0 and fi = 1. Figure 1 displaysthe null exposure graph for this scenario. For example, when z = (1, 0, 0, 0)—denoted as populationassignment “1”—unit 1 is treated, and so the exposures are as follows: f1 = 2, f2 = 1, f3 = 0, f4 = 0.Since only units 2, 3, and 4 are exposed to the exposure levels in the null hypothesis, we draw edgesfrom assignment “1” only to those units. This process repeats for all assignments to produce the nullexposure graph shown in Figure 1 (left). On the right side of Figure 1, the blue edges connecting units 1and 4 to assignments “2” and “3” highlight a complete bipartite subgraph (biclique) of the null exposuregraph. Importantly, within this biclique, all missing potential outcomes are imputable under the null.
3.2. Bicliques and biclique decompositions
The “completeness” of bicliques in the null exposure graph means that, within a biclique, all unitsare connected to all assignments. As we note above, this implies that for the units and assignments thatcomprise the biclique, we can impute all potential outcomes for exposures F underHF0 . Below, we givedefinitions for these important objects and discuss how to partition the set of treatment assignments inthe null exposure graph. The resulting decomposition will be essential for the proposed randomizationtest. We describe algorithms for finding bicliques in Section 5.2.
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12
34
1 (1,0,0,0)
2 (0,1,0,0)
3 (0,0,1,0)
4 (0,0,0,1)
Units Assignments
12
34
1 (1,0,0,0)
2 (0,1,0,0)
3 (0,0,1,0)
4 (0,0,0,1)
Units Assignments
Figure 1. Left: Depiction of the null exposure graph for a clustered interference setting with four units andfour assignments. The left nodes represent the experimental units, and the right nodes represent thepopulation treatment assignments. There are only four assignments since we consider treating exactly onecluster. The graph is bipartite because no units and assignments are connected with other like nodes.Right: One biclique in the null exposure graph is highlighted in blue.
Definition 3 (Biclique). A biclique in the null exposure graph, GFf = (V,E), where V = U∪Z and Eis defined in Equation (6), is a set-pair C = (U,Z), with U ⊆ U and Z ⊆ Z, such that (i, fi(z)) ∈ E,for every i ∈ U and every z ∈ Z.
As an example, C = ({1, 4}, {z2, z3}) is a biclique in Figure 1 since it is a complete bipartitesubgraph. Note that we use the same notation “C” for bicliques as we did for conditioning events inSection 2.3. This is intentional since our proposed test (in Section 4) will condition on a biclique ofthe null exposure graph.
We now formalize the intuition that bicliques in the null exposure graph allow imputation of themissing potential outcomes.
Proposition 1. Consider a null exposure graph, GFf , with some biclique C = (U,Z). If Zobs ∈ Z,then Yi(z) = Yi(Z
obs) under HF0 , for all i ∈ U and all z ∈ Z.
Proof. For any unit i ∈ U in the biclique, fi(Zobs) ∈ F since Zobs and i are both in the biclique. Fixany other assignment z ∈ Z in the biclique, then there is also an edge between i and z by definitionof the biclique. This implies that fi(z) ∈ F as well, by construction of the null exposure graph inDefinition 2. Hence, fi(z) ∈ F as well, and so Yi(z) = Yi(Z
obs) under HF0 of Equation (1).
Proposition 1 shows that we can condition our test on a biclique of GFf because we can impute allthe missing potential outcomes in a biclique that contains the observed treatment assignment Zobs.Since there exist many such bicliques in GFf , we need to decide how to condition on one through anappropriate conditioning mechanism, P (C|Z). Choosing this mechanism is not a trivial task, however,because in order to calculate the randomization distribution, P (z|C), we also need to calculate P (C|z),for any assignment z in C. This can lead to meaningful statistical and computational challenges, evenunder seemingly reasonable choices of the conditioning mechanism.
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In this paper, our approach is designed so that the conditioning mechanism operates on a restrictedset of bicliques, such that, for each assignment z ∈ Z, there is only one biclique that contains z. Thebenefit of this approach is that conditioning on the unique biclique containing Zobs yields a simplemechanism of the form P (C|Zobs) = 1{Zobs ∈ Z(C)}. This approach only needs to test membershipof Z in biclique C, and, consequently, also yields a simple randomization test in which we only needto randomize the assignments in C weighted by the design, P (Z|C) ∝ 1{Z ∈ Z(C)}P (Z).
Such special construction of the conditioning mechanism can be implemented through the conceptof a biclique decomposition, which is a set of bicliques in the null exposure graph that fully partitionsthe population set of assignments, Z. This concept is defined formally as follows.
Definition 4 (Biclique Decomposition). Let d : Z → N denote the degree of assignment z in thenull exposure graph. Define ZF = {z ∈ Z : d(z) > 0} as the set of all assignments connected to atleast one unit in the null exposure graph. A biclique decomposition, C = {C1, . . . , CK}, of the nullexposure graph in Definition 2 is a finite set of bicliques, Ck = (Uk,Zk), k = 1, . . . ,K, such that ZFis partitioned, i.e., ⋃
k
Zk = ZF , and Zk ∩ Zk′ = ∅, for any k 6= k′.
Notably, it is not necessary that the population of units, U, is partitioned in a biclique decom-position. This is crucial because partitioning both U and ZF may not be possible, or could lead tolow-powered tests.
4. Biclique-based randomization tests
4.1. Main method and test validity
We can now describe our proposed conditional randomization test for HF0 in Equation (1), whichis the key methodological contribution of this paper. Throughout, let C be some fixed biclique decom-position of the null exposure graph GFf . Consider the following procedure:
Procedure 1. For observed assignment Zobs ∼ P (Zobs):
1. Find the unique biclique, C = (U,Z) ∈ C, such that Zobs ∈ Z. Consider a test statistic t(z, y;C)
restricted to biclique C.
2. Calculate the observed value of the test statistic, T obs = t(Zobs, Y obs;C).
3. Define the randomization distribution as r(Z) ∝ 1{Z ∈ Z} · P (Z).
4. Define the randomization p-value as follows:
pval(Zobs, Y obs;C) = EZ∼r[1{t(Z, Y obs;C) > T obs}
]. (8)
We describe an algorithm for choosing a unique biclique (Step 1) in Section 5.2. The following theoremshows that this procedure is valid; the proof is in Appendix B.
10
Theorem 2. Consider the null hypothesis HF0 in Equation (1). Construct the corresponding nullexposure graph, GFf , and compute a biclique decomposition C. Let C ∈ C be the unique biclique suchthat Zobs ∈ C. Then, the randomization test described in Procedure 1 is valid conditionally at anylevel, i.e., the p-value defined in Equation (8) satisfies:
E(1{pval(Zobs, Y obs;C) ≤ α} | C, HF0
)= α,
where the expectation is with respect to the design, P (Zobs), and α ∈ (0, 1).
Theorem 2 follows from Theorem 1 by recognizing that Procedure 1 describes a conditional ran-domization test in which the conditioning event is a biclique C ∈ C, and the conditioning mechanismis defined as P{C = (U,Z)|Z} = 1{C ∈ C}1{Z ∈ Z}. The proof first verifies that any test statisticrestricted on C is imputable: this follows from the construction of the biclique and Proposition 1. Itthen shows that the randomization distribution r(Z) defined in Step 3 of the procedure is the correctconditional distribution P (Z|C) implied by the design and the conditioning mechanism.
The computational tractability of the randomization distribution, r(Z) ∝ 1{Z ∈ Z}P (Z), isimmediate provided that we can compute P (Z) and enumerate the assignments Z in any bicliqueC ∈ C. This last condition, however, may be prohibitive if the support of the design P (Z) is too largesince biclique enumeration is NP-hard. Fortunately, a small modification of our test can address thisissue. The idea is to add a preliminary step in Procedure 1 that subsamples assignments to limit thesize of Z. We describe this modification in Appendix B and show that the procedure is still valid.
Finally, while Procedure 1 automates the construction of a conditioning mechanism, it still allowsflexibility in the choice of the test statistic. For instance, to test H{a,b}0 , the simplest choice is for thetest statistic to denote the difference-in-means between outcomes of units in C exposed to a and b.However, we may improve power by using test statistics motivated by a model of interference, suchas a network regression models with spillovers, which may fit the data better than standard linearregression. See also Athey et al. (2018, Section 5.3) for an excellent related discussion. We turn topower in Section 5.
4.2. Comparison to related work
In Section 2.3, we discussed how our method, along with those of Aronow (2012) and Athey et al.(2018), could be described within the general framework of Basse et al. (2019b). We can also describethese approaches using the framework in this paper, in which each method corresponds to a differentapproach for selecting bicliques from the null exposure graph.
The method of Basse et al. (2019b), for instance, can be viewed as implicitly considering bicliquesof the null exposure graph with possibly overlapping assignments. In other words, assignments inthe bicliques form a covering—not a partition—of Z, such that an assignment may belong to morethan one biclique. The conditioning mechanism is then uniform on the set of all bicliques containingthe observed assignment. This approach works in their particular setting and results in powerfultests because the conditioning is guided by the observed assignment, Zobs. However, the drawback of
11
this approach is that there is no general way to construct good biclique coverings, instead requiringcase-by-case derivations. Specifically, the covering that is implied by the conditioning mechanismconstructed in Basse et al. (2019b) works only in two-stage experiments with clustered interference.
The approach of Athey et al. (2018) applies to more general settings, but it may lead to under-powered tests. To understand their approach, we introduce some additional notation. For any U ⊆ U,let C(U ; z) denote the largest biclique of the null exposure graph that contains only units from U andalso contains assignment z. Then, the conditioning mechanism implicitly considered by Athey et al.(2018) is of the form:
P (C|Z) = P (U)1{C = C(U ;Z)}, (9)
where P (U) is specified by the analyst. In other words, the approach of Athey et al. (2018) implicitlysuggests first to sample units from one side of the bipartite null exposure graph, and then to calculatethe induced biclique that contains the observed assignment. This approach is generally applicable,but the random choice of U may lead to underpowered or even ill-defined tests. This is the case when,for example, C(U ;Z) is empty due to a poor initial choice of U .
Our method combines the benefits of both approaches. Unlike the biclique covering strategyimplicit in Basse et al. (2019b), our approach is not problem-specific and automates the constructionof conditioning mechanisms. Also, in contrast to the method in Athey et al. (2018), our proposedapproach gives concrete guidance on how to properly condition the test to achieve higher power. Wediscuss these advantages in the context of clustered and spatial interference in Sections 6 and 7.
5. Power for biclique-based randomization tests
While the proposed randomization test is guaranteed to be exact, the power of the test dependson many factors, including the size of the study, the interference structure, and the test statistic. Inthis section, we explore this problem via a theoretical analysis, turning to simulations in later sections.We then propose a biclique decomposition algorithm that improves power by finding larger cliques. InSection 6.3, we show how we can further leverage knowledge of the design to improve power.
5.1. Theoretical results for statistical power
In this section, we analyze the power of the main biclique randomization test of Procedure 1.As mentioned earlier, power results for Fisher randomization tests is challenging because power ulti-mately depends on the full schedule of potential outcomes—very few results are available, even in thesimple, non-conditional case (Rosenbaum et al., 2010). A standard approach in power analyses forrandomization tests is therefore to consider a simplified model of the problem for which formal resultscan be obtained (Lehmann and Romano, 2006, Chapter 15). These results then serve as a foundationfor useful heuristics. In this spirit, we consider a simplified model solely for the purpose of assessingpower under certain conditions — if these conditions do not hold, the underlying randomization testis still guaranteed to be exact.
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Theorem 3. In Procedure 1, let C = (U,Z) ∈ C be the conditioning biclique, where C is a fixedbiclique decomposition. Let |C|= (n,m) denote the size of the biclique, with n = |U | and m = |Z|. Letthe randomization distribution and the null distribution be denoted, respectively, by
t(Z, Y (Z);C) ∼ F̂1,n,m, and t(Z, Y obs;C) ∼ F̂0,n,m, where Z ∼ P (Z|C). (10)
Suppose that for any fixed n > 0:
(A1) There exist continuous cdfs F1,n and F0,n such that F̂1,n,m and F̂0,n,m in (25) are the empiricaldistribution functions over m independent samples from F1,n and F0,n, respectively.
(A2) There exists σn > 0, and a continuous cdf F , such that F0,n(t) = F (t/σn), for all t ∈ R.
(A3) The treatment effect (e.g., spillover contrast) is additive, that is, there exists a fixed τ ∈ R suchthat F1,n(t) = F0,n(t− τ), for all t ∈ R.
Let φn,m be the power of the biclique randomization test conditional on a biclique of size (n,m), i.e.,
φn,m = E(1{pval(Z, Y obs;C) ≤ α} | |C|= (n,m)
).
Fix any small δ > 0. Then, for large enough m,
φn,m ≥ 1− F (F−1(1− α)− τ/σn)−O(m−0.5+δ).
If, in addition, supx∈R|F (x) − 1/(1 + e−bx)|≤ ε for fixed b, ε > 0, and σn = O(1/√n), then for some
fixed A, a > 0
φn,m ≥1
1 +Ae−aτ√n−O(m−0.5+δ)− ε. (11)
The proof of Theorem 3 is in Appendix E, and can be summarized as follows. We start withthe random variable t(Z, Y (Z);C) —i.e., the test statistic— as induced by the joint distribution of(Z, Y (Z), C) conditional on C. The distribution of the test statistic is denoted by F̂1,n,m. Under thenull, the distribution of t(Z, Y (Z);C) is the same as that of t(Z, Y obs;C), and is denoted by F̂0,n,m.
The first condition, (A1), connects these two distributions to marginal and continuous distribu-tions denoted, respectively, by F0,n and F1,n. In particular, this condition claims that the null (oralternative) distribution consists of m i.i.d. samples from the corresponding marginal. The mainrestriction underlying this condition is that the actual biclique C we condition on does not matterfor any biclique decomposition, C. It is perhaps the least realistic condition, as we expect the teststatistic values to be correlated within a biclique C, and across different bicliques within the samedecomposition C. Note, however, that (A1) is only useful to leverage results from empirical processtheory, and thus derive asymptotic rates for the testing power as in Equation (26). The other twoconditions are more typical in randomization test analysis. For instance, the second condition, (A2),posits that the number of units, n, affects the null distribution only as a scale parameter. Condition(A3) posits a constant spillover effect, and thus relates the null distribution of the test statistic with
13
its distribution under the alternative. As mentioned earlier, these conditions describe a simplifiedmodel — neither can be strictly true in our randomization-based framework, but they can be usefulapproximations for large m and will allow us to derive helpful power heuristics. Indeed, the empiricalstudies in Section 6.2 suggest that even under potential violations of these conditions, the theoreticalpredictions from Theorem 3 remain valid.
Under these conditions, Theorem 3 shows that power is increasing in the size of the biclique.Specifically, the number of focal units (n) in the biclique controls the “sensitivity” of the test, that is,how quickly the test achieves maximum power (as a function of the treatment effect). The number offocal assignments (m) in the biclique instead affects the maximum power of the test. See Section E.2in the Appendix for a simulation study that confirms these results. These insights provide usefulguidance for tuning and diagnosing clique decomposition algorithms.
Finally, although Equation (11) shows the relationship between the power of the biclique test andthe biclique size, it is not clear how this relates to properties the null exposure graph. In AppendixE.3, we leverage an important result from extremal graph theory to show that the density of the nullexposure graph relates to the existence of bicliques of certain size.
5.2. Biclique decomposition algorithm and power considerations
Given Theorem 3, an important step for improving power is to condition on large bicliques inStep 1 of Procedure 1. We now present an algorithm for decomposing the null exposure graph intobicliques, which in turn enables finding larger bicliques. For a given biclique, C, in the null exposuregraph, let E(C), U(C), and Z(C) denote the set of edges, the set of units, and the set of assignments inthe biclique, respectively. Thus, |E(C)|= |U(C)||Z(C)| since the biclique is bipartite. Also, let C ∈ Gindicate that C includes nodes and edges from null exposure graph G.
Our proposed biclique decomposition algorithm can be described as follows:
1. Start with an empty biclique set: C = {}, and the original null exposure graph G = GFf thatcorresponds to the null hypothesis, HF0 .
2. Solve the “largest biclique problem”:
C∗ = arg maxC∈G
|E(C)|. (12)
3. Remove biclique edges: E(G)← E(G) \ E(C∗).
4. Remove biclique assignments: Z(G)← Z(G) \ Z(C∗).
5. Update biclique set, C ← C ∪ {C∗}, and repeat from Step 2 if |E(G)|> 0.
The output of this procedure is a biclique decomposition, C.The main computational challenge is in Equation (12), where we calculate bicliques with the largest
possible number of edges in the remaining null exposure graph. This is a variant of the maximal edge
14
biclique problem, and is computationally challenging. In fact, Peeters (2003) show that finding suchbicliques is NP-hard. See Zhang et al. (2014) for a review. Fortunately, our randomization test remainsvalid even when the solution to (12) is approximate, so we do not need to solve Equation (12) exactly.In this paper, we use the “binary inclusion-maximal biclustering” (Bimax) method (Prelić et al., 2006)for such approximation of (12).2 In the following section, we show that using this algorithm withinthe proposed method results in a powerful test.
Finally, we note that these power considerations are also useful for optimal experimental design.Specifically, one can parameterize the design, simulate outcomes —according to an outcome model, asmentioned above— and then estimate the power of the biclique test for a treatment effect of interest.The design with highest power can then be chosen for the experiment. We illustrate this idea inAppendix E.4 in the context of the Medellín data. In particular, we consider a design space with twoparameters: one parameter controls the treatment probability at the “center” of the city; and the othercontrols the treatment probability at the outskirts. The resulting optimization problem appears to beconvex, suggesting that this approach may be more useful for experimental design under interference.We leave this for future work.
6. Application to clustered interference: A simulation study
In this section, we illustrate the proposed biclique test in settings with clustered interference,continuing our running Example 1. In these settings, we assume that interactions between units occurwithin, but not between, well-defined clusters of units, such as households or classrooms.
6.1. Problem setup and comparison of available methods
Following Example 1, we have N units divided equally into K clusters. In the experiment, wefirst treat K/2 clusters at random and then randomly treat one unit in each treated cluster. Here,the exposure function is defined as fi(z) = zi +
∑j∈[i] zj , where [i] denotes all units in i’s cluster.
Thus, each unit i has three possible exposure levels, namely 0, 1 and 2, which we describe as “control,”“spillover”, and “treated” exposures, respectively. Overloading notation for simplicity, we denote thethree potential outcomes for each unit as Yi(“control”), Yi(“spillover”), and Yi(“treated”), respectively.
We focus on the following spillover effect hypothesis,
H0 : Yi(“spillover”) = Yi(“control”) + τ, for all i, (13)
and vary τ . We assess validity when τ = 0 and power when τ 6= 0. The null hypothesis in Equation (13)is therefore an instance of H{a,b}0 in Equation (3), with a = 1 and b = 0. Following our discussion in
2This is a fast divide-and-conquer method to find sub-blocks of ones in a binary matrix, known as biclusters, mainlyused in the bioinformatics and gene expression literature. The algorithm is implemented in the R package biclust andcan incorporate constraints on the solution space. The constraints are placed on the number of units and assignmentnodes so that Bimax returns all bicliques C∗ where, for example, |U(C∗)|≥ n0, |Z(C∗)|≥ n1 for specified, integer-valuedparameters n0 and n1. In practice, to approximate the solution of Equation (12), and to produce balanced bicliques,we suggest choosing n0 and n1 so that they are similar to each other in magnitude, and n0 · n1 is large.
15
1*2*3
*4*56
12345678...
Units Assignments
1*23
*456
12345678...
Units Assignments
1*23
45*6
12345678...
Units Assignments
Figure 2. Example bicliques used by the three methods of Section 6.1. Interference is clustered with sixunits divided equally into two clusters. Zobs is Assignment 1 and unit 6 is treated (colored green). The unitsof each clique (focal units) are marked by an asterisk, “*”. An edge denotes that the unit is exposed to eitherfi = 1 or fi = 0, the exposures in H0 of Equation (13).Left: conditioning event of the biclique test: two focals per cluster; Middle: conditioning event of “conditionalfocals” (Basse et al., 2019b): one untreated focal per cluster is randomly chosen (units 2 and 4); Right:conditioning event of “random focals” (Athey et al., 2018): only one focal per cluster is randomlychosen (units 2 and 6). However, unit 6 is effectively removed since unit 6 is treated, and there are no edgesbetween this unit and any assignment.
Section 4.2, we compare three methods for testing this null:
(i) The biclique test proposed in this paper (Procedure 1);
(ii) The method of Basse et al. (2019b), which samples one focal unit per cluster among those whoare not treated (“conditional focals”);
(iii) The method of Athey et al. (2018), which samples one focal unit per cluster at random (“randomfocals”).
In their current form with one focal unit per cluster, methods (ii) and (iii) can be implemented aspermutation tests. We could select multiple focals per cluster for these methods, but the resulting testis difficult to implement via a permutation test. Notably, the biclique method avoids such implemen-tation issues (see also Section 6.2). Figure 2 illustrates how these tests differ in a hypothetical examplewith six units arranged in two clusters, namely, {1, 2, 3} and {4, 5, 6}, and where unit 6 is assignedto treatment (colored green). For unit 6, we observe the “treated” potential outcome, Y6(“treated”);thus, unit 6 is not connected to any other assignment in the null exposure graph, since we have noinformation about Y6(“control”) or Y6(“spillover”) under the null. For units 4 and 5, we observe the“spillover” potential outcomes; and for units 1, 2, and 3 in the first cluster, we observe their “control”outcomes. Under the null, we can impute the “control” potential outcomes for units 4 and 5 and the“spillover” potential outcomes for units 1, 2, and 3.
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Figure 2 also depicts the conditioning events for every method, with the selected focal units in eachbiclique marked with an asterisk. The leftmost subfigure shows that the biclique test in Procedure 1conditions on the biclique that includes {2, 3, 4, 5} as focal units. Unit 6 is not included since it doesnot add any edges in the objective function of Equation (12). This biclique is balanced and densein the sense that there are two units in each exposure of interest, and it contains many edges. Themiddle subfigure shows the “conditional focals” method of Basse et al. (2019b). For this test, we seethat, by construction, only one focal unit is selected per cluster. This leads to a biclique that is lessdense than the biclique from our proposed test. The rightmost subfigure shows the “random focals”method of Athey et al. (2018). This differs from “conditional focals” because it may select treatedunits as focal units. For illustration, if the method selects unit 6, which is assigned to treatment, thisunit is effectively dropped from the conditioning biclique, thus reducing power.
6.2. Simulation study: Two-stage experiment
We follow Basse and Feller (2018) to generate data for the setting with clustered interference:
yi,0 ∼ N(µ0, σ2µ),
τPi ∼ N(τP , σ2τ ),
τSi ∼ N(τS , σ2τ ),
yi,2 = yi,0 + τPi ,
yi,1 = yi,0 + τSi .
In the experiment, bK/2c clusters are assigned to treatment, and within treated clusters, oneunit is randomly assigned to treatment (as in Example 1). We sample Yi(“control”) ∼ N(yi,0, σ
2y),
Yi(“spillover”) ∼ N(yi,1, σ2y), and Yi(“treated”) ∼ N(yi,2, σ
2y), in order to generate the individual
potential outcomes. As such, τPi and τSi correspond to idiosyncratic primary and spillover effects,respectively. We consider the following specifications: N = 300, µ0 = 2, σµ = στ = 0.1, σy = 0.5,τS = 0.7, τP = 1.5, and K ∈ {20, 30, 75}. In the simulation we vary K to see how different methodsperform with small and large-sized clusters. The different cluster sizes, N/K, are therefore containedin {15, 10, 4}. For the simulations, we generate 5,000 different assignments and construct the nullexposure graph for the biclique method. For each cluster size and fixed τ , we generate 2,000 data setsfrom the DGP given above; τ is varied among 300 equally spaced values from 0 to 1.
The power plots are shown in Figure 3. We see that the biclique method performs substantiallybetter with larger cluster sizes. Recalling the conclusions from Figure 2, this is explained by notingthat the biclique method can select multiple focal units per cluster in the biclique decomposition. Incontrast, the methods of Basse et al. (2019b) and Athey et al. (2018), as defined here, both use onefocal unit per cluster. To confirm, we calculated the average number of focal units per cluster for eachmethod and data configuration with 15 units per cluster (N = 300,K = 20). The biclique methodhad 5.24 focal units per cluster compared to 1 for method (ii) and 0.97 for method (iii). In contrast,the biclique method underperforms in the right-most plot of Figure 3. This is because smaller cluster
17
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
τ
pow
er
bicliquedesign−assisted bicliqueconditional focalsrandom focals
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
τ
pow
er
bicliquedesign−assisted bicliqueconditional focalsrandom focals
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
τ
pow
er
bicliquedesign−assisted bicliqueconditional focalsrandom focals
Figure 3. Power plots for three data configurations; the design-assisted biclique procedure is described inSection 6.3. The results include the results shown in Figure 3; left: N = 300,K = 20; middle:N = 300,K = 30; and right: N = 300,K = 75.
sizes imply a sparser null exposure graph. In such settings, the biclique method has trouble findinggood biclique decompositions, and can drop entire clusters in its conditioning. By contrast, alternativemethods are fixed to sample exactly one focal from every cluster.
To further investigate the power of the biclique test, we gather data on the bicliques generatedby the biclique decomposition. We alter the optimization constraints in the biclique decompositionalgorithm we employ (see Section 5.2 for implementation details), and fix N = 300,K = 20 andτ = 0.3. Figure 4 displays the results. We see that the average number of units and assignments inthe decomposition are negatively correlated, which suggests a power trade-off. As the number of (focal)units per biclique increases, power also increases until some threshold. Beyond this threshold, moreunits in the biclique come at the expense of fewer treatment assignments, which on aggregate decreasespower. This highlights that the automated biclique test does not incorporate specific informationabout the clustered interference structure. In Section 6.3, we show that we can modify the standardprocedure to improve the test’s performance in this case.
Not evident in Figures 3 and 4 is the relative ease of implementation for each method. The bi-clique method is straightforward and essentially automated for a well-defined biclique decompositionalgorithm. On the other hand, methods (ii) and (iii) are generally hard to implement. For exam-ple, Basse et al. (2019b) show that their test can be implemented as a permutation test only underthe conditioning mechanism described in (ii). Selecting more units per cluster to increase power willgenerally break this property. Furthermore, the test described in (iii) is a permutation test only whenall clusters have equal size. It is not clear how to implement a valid permutation test based on themethod of Athey et al. (2018) with unequal cluster sizes.
Overall, these results suggest that our proposed biclique test can deliver reasonably powered ran-
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5 10 15 20 25 30 35 40
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average # assignments
pow
er
Figure 4. Power values for varying biclique characteristics. The data structure is fixed at{N = 300,K = 20} and τ = 0.3. Each individual gray dot corresponds to a biclique decomposition of the nullexposure graph which we condition on for the biclique method. The left graph shows power as a function ofthe average number of units (focals), and the right graph shows power as a function of the average number ofassignments. A figure combining this information into a single plot is shown in Appendix C.
domization tests that are comparable to methods specifically designed for the interference structureat hand. In settings where the null exposure graph is dense, the biclique test can even outperformsuch special-purpose methods, since it is able to discover a more flexible conditioning. Most im-portantly, the biclique test operates automatically under any interference structure, whereas othermethods require user input and specification.
6.3. Improving power: Design-assisted biclique tests
We now discuss how to leverage knowledge of the experimental design to improve the power ofthe biclique test. We describe the approach in detail and prove its validity in Appendix D. In thefollowing, we give a brief overview and illustrate the performance gains.
The main idea relies on understanding the structure of the null exposure graph in the clusteredinterference setting. Recall that, within each cluster, every unit i is connected to every cluster as-signment, except for the assignment where i is treated. Therefore, the subgraph of the null exposuregraph corresponding to cluster k (i.e., containing only the units in cluster k and all possible “clustersubvectors” of the population assignment vector) can be decomposed into two large bicliques. Thesetwo bicliques are constructed to each contain half of cluster k’s units (see Appendix D for details).This is useful because the biclique C that the test conditions on can then be constructed by joiningtogether the large bicliques from each individual cluster. As we will see, this modified approach largelyaddresses the power issues for large K in the simulation of the previous section.
Figure 3 shows the simulations with the “design-assisted” biclique test described above. We see
19
that the design-assisted test dominates all other tests. The main benefit of the design-assisted testis that it uses half of the units in the cluster as focals, whereas the tests of Basse et al. (2019b) andAthey et al. (2018) only use one unit. This also explains why the performance gap between these testsnarrows for larger K. For instance, when K = 75, the design-assisted biclique test uses roughly twounits per cluster, which still yields power improvement compared to using just one.
The caveat of this analysis is that it may not always be easy to leverage the design in constructinga powerful biclique test because it requires understanding the relation between the design and thestructure of the null exposure graph. However, the analysis and empirical results of this sectionsuggest that when such combination is possible, the design-assisted biclique test dominates alternativeapproaches.
7. Application to spatial interference: Crime in Medellín
In this section, we illustrate our method in a spatial interference setting in which interactions occurbetween “neighboring” units, but without the simpler structure of clustered interference. We focus onre-analyzing a large-scale experiment in Medellín, Colombia studying the impact of “hotspot policing”on crime (Collazos et al., 2019).
7.1. Problem setup and comparison of available methods
Following Collazos et al. (2019), the units are N = 37, 055 street segments, 967 of which wereidentified as hotspots using geo-located police data and further consultation with police. Of thesehotspots, 384 were randomly assigned to treatment, a six-month increase in daily police presence,via a completely randomized design over a domain Z of roughly 10, 000 possible assignments. Theoutcome of interest is a crime index, a weighted sum of the crime counts on each street segment.3
To define the spillover hypothesis, we need to define the exposure function, f . Following Collazoset al. (2019), we use geographic distance as a measure for spillover exposure. Let d(i, j) be the distance(in meters) between unit i and unit j. We then define:
fi(z) =
“pure control”, if zi = 0 and
∑j 6=i 1{d(i, j) ≤ 500}zj = 0;
“spilloverr”, if zi = 0 and∑
j 6=i 1{d(i, j) ≤ r}zj > 0;
“other”.
(14)
This defines a “pure control” as a control unit that has no treated units closer than 500 meters, ensuringsignificant spatial separation from treated streets. A control unit is assigned to “spillover” when thereare treated units closer than the specified distance.
3As discussed in Collazos et al. (2019), the index weights are chosen based on the length of sentence for a given crime.They are: 0.550 for homicides, 0.112 for assaults, 0.221 for car and motorbike theft, and 0.116 for personal robbery.Crime data is matched to street segment within 40-meter buffers. In other words, if a crime happened in an alley, itwill be matched to the closest street segment within a 40-meter radius. If there is overlap, it is matched to the streetsegment closest in terms of Euclidean distance.
20
Figure 5. Street segments, hotspots, and treated hotspots for the data set. The left figure is the observedassignment for the experiment. The right figure is an example randomization of the assignment vector. Thedots cover different hotspots, but they are still within the 967 segments representing the hotspots.Additionally, the light colored dots represent the 125m spillover units, i.e.: street segments that are within125m of the treated hotspots for a given randomization.
Our goal is to test contrast hypotheses of the form Ha,br0 , with a = “pure control” and br =
“spilloverr”, where r is a free variable in the set:
r = {75, 100, 125, 150, 175, 225, 275, 325, 375, 425}.
Each hypothesis for a given r will have its own null exposure graph and biclique decomposition.Following Collazos et al. (2019), we refer to “short-range spillovers” as the set of hypotheses Ha,br
0
with r ≤ 125m.To illustrate, Figure 5 shows the induced exposures under the randomization realized in the ex-
periment (left), and an alternative randomization that was not realized (right), for r = 125m. Thefigure highlights short-range spillover units (among all available units) in light blue and light green forthe observed treatment and randomization, respectively. Figure 5 also depicts important geographicfeatures of Medellín, with many hotspots near the city center, and where dark areas or holes cor-respond to physical barriers (e.g., mountains) or major infrastructure (e.g., airports). Even thoughonly 967 street segments can receive the active treatment, every street segment can potentially receivespillovers, and so we use the entire street network in our analysis. A key challenge is that streetsegments near the city center have a much higher probability of being exposed to crime spillovers thansegments on the outskirts of the city.
Finally, unlike our analysis of clustered interference, we will only apply the biclique test in this
21
case study. In principle, we could adapt the test of Athey et al. (2018) to this setting, but theimplementation of their procedure would be underpowered due to the spatial structure. Specifically,we will see that (under the short-range spillover hypothesis) focal units are concentrated either at thecenter or outskirts of the city; see Section 7.3 and Figure 8. It is unlikely that this “center-outskirts”pattern could be generated through a random selection of focals. Furthermore, the ε-net method offocal selection of Athey et al. (2018, Section 5.4.2) would not work well because it would generatepatterns of focal units that are spatially uniform. Similarly, it is not clear how to apply the approachof Basse et al. (2019b), which is more narrowly tailored to the clustered interference setting.
7.2. Spillover hypotheses: A simulation study
We now assess our proposed method via a simulation study calibrated to the actual Medellín streetnetwork. For these simulations, each biclique decomposition is constrained to include bicliques withat least 100 focal units and 1000 assignments. We explore the effect of radius r on test power usingthe following simple model for the outcomes, Y :
Yi(“pure control”) ∼ Gamma(α, β),
Yi(“spilloverr”) = Yi(“pure control”) + τr. (15)
The shape and rate parameters (α, β, respectively) are selected to match the mean and variance ofthe observed outcome in the actual experiment, the crime index. The parameter τr determines anadditive spillover effect at radius r. We set τr ∝ 1/r2 to allow for heterogeneity of the spillover effectwith respect to radius.4 In the simulation, we sample assignments according to the true design forevery value of r and outcomes according to Equation (15). We then test the null hypothesis Ha,br
0
on spillovers at distance r, defined at the beginning of this section. The test statistic is the simpledifference in means between focal units exposed to a and br.
The results are shown in Figure 6. In the left subfigure, we fix the additive treatment effect atzero (τr = 0) to assess validity. Our rejection level is 0.05, and so we confirm the validity of thebiclique method since all power values gather around the 5% rejection rate. In the right subfigure, weconsider nonzero additive spillovers effects (τr > 0) that are calibrated based on the spillover radius.We observe that the power curve is generally concave and nonmonotonic since it increases until someradius and then decreases. At first, this may be counterintuitive since as r increases the spillovereffect, τr, decreases (by definition), which should make it harder to detect. However, as shown inAppendix C, the number of focal units also increases sharply with respect to r. The net effect is anincrease of power. At the same time, as we discussed in Section 6.2, an increase in the number offocals per biclique generally results in a decrease in the number of assignments per clique (see alsoFigure 11 in Appendix C), and, eventually, in decreased testing power. Under our assumed outcomemodel, the maximum power is achieved approximately at a 275m spillover radius.
4This is in line with existing literature (Thomas, 2013; Barr and Pease, 1990; Verbitsky-Savitz and Raudenbush,2012), which has pointed out that “the likelihood that an offender will target an opportunity will be inversely relatedto the distance it is located from their routine activity spaces” (Eck, 1993; Johnson et al., 2014).
22
● ● ● ● ● ● ● ● ● ●
100 150 200 250 300 350 400
0.0
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er, τ
=0
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100 150 200 250 300 350 400
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spillover radius
pow
er, τ
r≠
0
Figure 6. The left figure shows the power analysis across 1000 simulations when τr = 0. We reject the nullat the 0.05 level. Therefore, the left figure confirms the validity of the biclique method. The right figuredisplays the power analysis for nonzero τr ∝ 1/r2.
This kind of analysis gives a useful estimate for the power profile (Figure 6, right) of our proposedbiclique test for a given biclique decomposition algorithm. In practice, we could compare between thepower profiles of different biclique decomposition algorithms, and choose the most favorable algorithmto apply on the real data. See Section 5.1 for additional discussion on testing power.
7.3. Spillover hypotheses: biclique test on real data
In this section, we demonstrate our proposed biclique test using the actual outcome data. We focuson the spillover hypothesis H{a,br}0 with radius r = 125m, following Collazos et al. (2019) who definethis type of exposure as “short-range spillover”. Results for all radius values in the previous simulationare included in Appendix F. We therefore test whether there is a difference between outcomes of purecontrol units and units who receive short-range spillovers:
H0 : Yi(z) = Yi(z′), for all z, z′ such that fi(z), fi(z′) ∈ {“pure control”, “spilloverr”}, (16)
where r = 125m, and the exposures are defined in Equation (14). As we discuss in Section 2.2,rejecting H{a,br}0 does not necessarily imply that the treatment exposures are different; instead it ispossible that any of the singleton hypotheses H{a}0 and H{br}0 does not hold.
The first step of the biclique test is to construct the null exposure graph. This graph has 37,055nodes on one side (number of streets/units), and 10,000 nodes on the other (number of assignments).Figure 7 visualizes this graph. The left subfigure shows a block of the null exposure graph that isreasonably discernible, where a dot corresponds to a unit-assignment pair, say (i, z). If the dot haswhite color, then there is no edge between unit i and assignment z in the graph. The light blue and
23
units
assi
gnm
ents
units
assi
gnm
ents
Figure 7. The left figure visually depicts the null exposure graph. The vertical axis corresponds to theassignments from the randomization procedure, and the horizontal axis displays the units. Light blue denotesan untreated unit that is a spillover and close to a treated hotspot, and navy denotes pure control. The rightfigure is a (zoomed-in) biclique of the null exposure graph containing the observed assignment. There is nowhite in the biclique since the biclique only contains units exposed to spillover or pure control. To conservespace, we only display the first 100 assignments and units for both.
navy blue colors indicate whether i receives short-range spillover or pure control exposure under z,respectively. We note that the null hypothesis is not sharp for units exposed to “white”, therefore weneed to condition on a biclique where the white components are effectively removed. This is analogousto choosing a biclique where all units are either exposed to “navy blue” or “light blue”. The right sideof Figure 7 shows such a clique (zoomed-in to have same size as the left side).5
We now apply our proposed test in Procedure 1 to the spillover hypothesis of Equation (16). Theleft side of Figure 8 displays hotspots, treated hotspots, and the focal units identified by the bicliquedecomposition. As mentioned earlier, most focal units are at the center or outskirts of the city dueto the particular spatial structure of spillovers. The right side of Figure 8 displays the randomizationdistribution of the test statistic measuring the difference in means between crime index values onshort-range spillover units and pure control units:
t(z, y;C) =1
Nb
∑i∈C
1{fi(z) = b}Yi −1
Na
∑i∈C
1{fi(z) = a}Yi, (17)
where C denotes the biclique we condition on; i ∈ C denotes that unit i is a node in the clique; a =
5Figure 7 displays additional important information about the test. For example, the colorings on the right side ofFigure 7 reveal that many units in the biclique are always “pure control” (navy blue columns), and a handful of unitsalways receive “short-range spillovers” (light blue columns). Conceptually, more variation in exposures across units (i.e.,more random colorings in right side of Figure 7) leads to more power. Currently, our biclique decomposition algorithmcannot guarantee such variation; we leave this problem open for future work.
24
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hotspotstreated hotspots125m spillover streetpure control streetfocal units
0.00 0.05 0.10 0.15 0.20 0.25
05
1015
20
Randomization distribution
test statisticD
ensi
ty
Figure 8. Left: Representation of unit exposures for Zobs. Also shown in green are the focal units from thebiclique represented on the right in Figure 7. Right: Test of the r = 125m spillover radius hypothesis, wherethe test statistic is a difference in average outcomes between “short-range spillover” units and pure controlunits, defined in Equation (17). Shown is the distribution of the test statistic under the null, and the blue lineis the observed test statistic. The p-value of the observed test statistic is approximately 0.077.
“pure control”, b = “short-range spillover”; Na =∑N
i=1 1{fi(z) = a} and Nb =∑N
i=1 1{fi(z) = b}are the exposure counts. Thus, positive values of the test statistic indicate that the average crimeoutcome for units exposed to “short-range spillovers” is larger than the average outcome for units thatare exposed to “pure control”.
The randomization values of the test statistic are computed conditional on the biclique depictedon the right of Figure 8. The observed test statistic is only larger than 7.7% of all the random-ization values, which is not significant at the 5% level. Assuming an additive short-range spillovereffect, inversion of the randomization test gives us [−0.51, 0.03] as the 95% confidence interval, indi-cating that negative values for the spillover effect are more plausible under the additivity assumption.This observation suggests a decrease in crime of street segments surrounding an area with increasedlaw enforcement/community policing, which is consistent with the literature (Collazos et al., 2019;Verbitsky-Savitz and Raudenbush, 2012).
Several caveats are in order, however. First, as we discuss above, it is possible that we reject thenull hypothesis because the underlying exposures themselves are not correctly specified; that is, anyof the singleton hypotheses H{a}0 and H{br}0 does not hold. Another caveat is that, even though ourrandomization test is always valid, the power of the test may be affected by the inherent differencesbetween the focal units. Specifically, due to the particular spatial arrangement in our application, the
25
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100 150 200 250 300 350 400
0.00
0.02
0.04
0.06
0.08
0.10
spillover radius
p−va
lue
p−values using raw outcome
Figure 9. P-values (left vertical axis) for biclique tests with varying spillover radii (horizontal axis). Theblue line shows p-values for tests using the raw crime index.
focal units that receive spillovers are mostly downtown streets, whereas units that are pure controlsare mostly on the outskirts of the city (see Figure 8, left). The observed differences in crime outcomesof these focal units could depend, say, on differences in demographics between these two city areas—failing to account for such differences could reduce the power of the randomization test. We discussthis issue in more detail, along with potential solutions using covariate adjustment, in Section 7.4.
Finally, we conduct biclique randomization tests, as described earlier, while varying the spilloverdistance, r. The results are shown in Figure 9, which also includes the results for the 125m-spilloverpresented in Figure 8. For outcomes, we consider the raw crime index. We see that the p-values forthe raw crime index are all small for varying radii; see the flat blue line in Figure 9. This suggeststhat some form of spillovers exists, where the distance does not seem to matter. Alternatively, theresults could indicate that the spillover effects may be heterogeneous with respect to distance.
7.4. Covariate adjustment for heterogeneous focals
The difference between short-range spillovers and pure control affects the power of the spilloverhypothesis tests. The concern becomes evident in Figure 8, where we see that the focal units thatreceive spillovers are mostly downtown streets, whereas units that are pure controls are mostly on theoutskirts of the city.
One straightforward way to address such possible heterogeneity in the focal units is to adjust forknown covariates. For example, we could regress outcomes on observed covariates, and then performour proposed biclique test on the residuals (Rosenbaum et al., 2010). To illustrate, we used thisapproach for the test of Section 7.3 and adjusted the outcome by distance from important societalcenter points, such as school, police station, courthouse, church, park, as well as “comuna”, which
26
0.00 0.02 0.04 0.06 0.08 0.10
010
2030
40
Randomization distribution
test statistic
Den
sity
Figure 10. Randomization test for 125m spillover radius hypothesis, applied on the residuals of a regressionof outcomes on known covariates. The p-value of the observed test statistic is approximately 0.13.
represent a neighborhood or district. The results are shown in Figure 10. The new p-value is 0.13,and suggests that outcomes under short-range spillovers are not statistically different from outcomesunder pure control. In Appendix F, however, we include a randomization analysis for many differentradii, which suggests that spillovers may exist at distances larger than 125m. This result hints atthe insufficiency of geographic distance to fully capture the intensity of spillovers. In future work, wecould incorporate additional information in the distance function, such as socioeconomic differencesbetween street segments. An advantage of the biclique-based testing methodology is the ability toarbitrarily define the distance function (and thus exposures) of interest.
More broadly, this analysis suggests that adjusting for heterogeneity may be important in practice.The regression-based approach could be extended to adapt related randomization-based approachesthat account for treatment effect heterogeneity (Ding et al., 2016). Another approach could be tobalance the focal units while incorporating covariates. We leave these ideas open for future work.
8. Extension to complex null hypotheses
Thus far, we have focused on testing null hypotheses that contrast two exposures, as in Equa-tion (1). We now turn to testing more complex null hypotheses that can be written as intersectionsof singleton hypotheses, such as testing null hypotheses that restrict interference between units. Themain difficulty in testing such hypotheses through our framework is that they are based on multiplesets of exposures, rather than just one as above. Thus, constructing the null exposure graph requiresmore care.
Here, we extend our biclique method to test composite intersection hypotheses. Let I = {Fj ⊂
27
F, j = 1, . . . , J} be a set of non-overlapping exposure sets, such that Fj ∩ Fj′ = ∅ when j 6= j′. Ourgoal is to develop a biclique test for the intersection hypothesis defined as follows:
HI0 =
J⋂j=1
HFj0 . (18)
As discussed in Section 2.2, when I is a partition of F into singletons, such that⋃j Fj = F and
|Fj |= 1, then the intersection hypothesis in (18) is equivalent to the exclusion-restriction hypothesis,Hex
0 , of Equation (4). While Hex0 is a special case of HI0 , we focus on it here because of its connection
to standard hypotheses on the “extent of interference” (see Example 3). We describe two methodsto test Hex
0 . The first method “patches together” the null exposure graphs that correspond to theindividual hypothesesHFj0 . The second method extends the approach of Athey et al. (2018) to leverageinformation from the multiple exposure graphs that correspond to Hex
0 . Finally, in Appendix G, weextend both of these methods to test HI0 in its more general form (18).
8.1. Multi-null exposure graph
Our first approach to test Hex0 relies on the key observation that, for any unit i, Hex
0 essentiallysplits Z into equivalence classes, such that any two assignments within the same equivalence classinduce the same potential outcome for unit i. In general, these equivalence classes are allowed todiffer across units. In practice, however, these classes typically correspond to the same exposure level,and thus have the same interpretation across units; e.g., the exposure levels “pure control”, “spilloverr”,and “other” in (14) for any unit i can also be viewed as equivalence classes of Z. This differs fromH{a,b}0 , which posits equal potential outcomes only within the two specified classes, a and b, rather
than within every class. To test Hex0 , we will therefore need to expand the definition of the null
exposure graph.In particular, we define the multi-null exposure graph of Hex
0 with respect to z ∈ Z,Z ⊆ Z asfollows:
G(z;Z) = (V,E), with V = U and E = {(i, z′) : i ∈ U, z′ ∈ Z, fi(z′) = fi(z)}. (19)
In words, G(z;Z) encodes which potential outcomes would be imputable if z is realized in the exper-iment, just like the simpler null exposure graph above. We need this new definition because Hex
0 iscomprised of many individual null hypotheses, HFj0 . As such, for any given pair (i, z′), the missingpotential outcome Yi(z′) could be imputed via a different HFj0 depending on which assignment Zobs
is realized in the experiment. The idea for a test of Hex0 is then to partition Z through a “patchwork”
of bicliques from multi-null exposure graphs. This idea is formalized below.
Procedure 2. To generate a biclique decomposition for testing Hex0 , we do the following:
1. Initialize C ← ∅, and Z0 ← Z.
2. While Z0 6= ∅:
28
(a) Sample Z uniformly from Z0.
(b) Obtain a non-empty biclique C = (U,Z) within G(Z;Z0); e.g., as in Equation (12).
(c) Update C ← C ∪ {C} and Z0 ← Z0 \ Z.
The main output of this procedure is a biclique decomposition, i.e., a collection C of bicliques thatfully partitions the treatment assignment space (see Definition 3). However, it is different from theearlier definition since the decomposition here is not applied to a single null exposure graph, but isinstead comprised of bicliques from various null exposure graphs, G(z;Z). To test Hex
0 we then simplycondition the test of Procedure 1 on the biclique of C that contains Zobs, as in the original test ofProcedure 1. We prove the validity of this test in the following theorem.
Theorem 4. Consider the intersection hypothesis HI0 defined in (18). Then, the biclique test inProcedure 1 operating on the biclique decomposition C from Procedure 2 is a conditionally valid testfor HI0 .
Theorem 4 shows that our proposed method can in fact test more complex hypotheses than thecontrast hypothesis of Equation (1). As with contrast hypotheses, the power of this test mainly de-pends on the sizes of the conditioning bicliques in Step 2(b) of Procedure 2; see also the power analysisin Section 5.1. However, Procedure 2 offers no guarantees for power because, like the decompositionalgorithm of Section 5.2, it also employs a greedy approach in constructing bicliques. We leave forfuture work a more sophisticated procedure that could overcome this.
8.2. Approach based on Athey et. al. (2018)
Hypothesis Hex0 can also be tested using the approach of Athey et al. (2018). As described in
Section 4.2, this approach is equivalent to conditioning on a biclique C = (U,Z), where we first sampleU using an arbitrary distribution g(U) on U, and then set Z = {z ∈ Z : fi(z) = fi(Z
obs), i ∈ U}.However, as also argued earlier, this approach may lack power because an arbitrary sampling of focalunits U may lead to small bicliques, which, as we discuss in Section 5.1, can be detrimental to power.
We therefore propose to adjust g(U) to give more weight to focal units that allow for largerbicliques. Specifically, we first estimate the average degree of every unit in U over a large sample ofmulti-null exposure graphs. We then sample units with larger weight to higher-degree units; theseunits are on average connected to more assignments, and so will lead to larger conditioning bicliques.The concrete procedure can be described as follows:
1. Initialize, di ← 0 for all i ∈ U.
2. For r = 1, . . . , R replications:
(a) Sample Z ∼ P (Z) according to the design.
(b) Set di ← di + degi(G(Z;Z)), where degi(G) denotes the degree of node i in graph G.
29
3. Define g(U) as sampling over U weighted by (d1, . . . , dn), and apply the approach of Athey et al.(2018).
This test is valid using the same justification as for the original test of Athey et al. (2018) since westill randomly sample the focal units independently of Zobs. However, our approach may increasepower as the sampled focal units will generally lead to larger bicliques. We leave to future work amore detailed power comparison with the original and “ε-nets” methods of Athey et al. (2018).
9. Concluding remarks
In this paper, we extend the classical Fisher randomization test to settings with general inter-ference. Our main contribution is the concept of the null exposure graph, which represents the nullhypothesis as a bipartite graph between units and assignments. Conditional on a biclique in thisgraph, the null hypothesis is sharp, and the corresponding (conditional) randomization test is valid.We showed the benefits of this approach in both clustered and spatial interference settings. Fur-thermore, our approach inherits the properties and theory of conditional randomization tests. Thisincludes covariate adjustment methods (Rosenbaum et al., 2002), as well as recent developments onstudentization of test statistics (Wu and Ding, 2018).
There are a number of promising directions for future work. First, we can explore how to combineinformation across biclique tests that condition on different bicliques. For instance, we could followGeyer and Meeden (2005) and leverage the distribution of p-values across tests to improve power, asdiscussed in Basse et al. (2019a). We could also adapt recent proposals on multiple randomizationtests from Zhang and Zhao (2021). Second, we could investigate how much data “is thrown away” byconditioning, and suggest biclique decompositions of the null exposure graph that minimize data loss.Third, building on our results for power, we could further develop the problem of optimal design fora given set of spillover hypotheses. Finally, it would be interesting to know under which conditionsour proposed tests can be implemented more efficiently.
A. Null hypothesis HF0
An alternative way to view the null hypothesis, HF0 , in Equation (1), is to think of it as a compositehypotheiss. In particular, HF0 is equivalent to testing a composite hypothesis, denoted as {H0,k},where H0,k is any simple hypothesis of the form H0,k : Y = Y0,k. Here, Y0,k is any N × 2N potentialoutcomes matrix with the following constraints: (i) it has Y obs at the column corresponding to Zobs,(ii) when unit i under z is exposed, i.e., fi(z) ∈ {a, b}, its outcome in Y0,k is fixed to Y obs
i . This revealsthat testing under interference entails a problem of identification. Specifically, the challenges here arethat the simple hypotheses are not testable since not all units are exposed under all assignments, andthe number of Y0,k is intractably large.
30
B. Proofs
B.1. Main results
Theorem 2. Consider the null hypothesis HF0 in Equation (1). Construct the corresponding nullexposure graph, GFf , and compute a biclique decomposition C. Let C ∈ C be the unique biclique suchthat Zobs ∈ C. Then, the randomization test described in Procedure 1 is valid conditionally at anylevel, i.e., the p-value defined in Equation (8) satisfies:
E(1{pval(Zobs, Y obs;C) ≤ α} | C, HF0
)= α,
where the expectation is with respect to the design, P (Zobs), and α ∈ (0, 1).
Proof. Let Zobs be the observed assignment and C be the biclique that contains Zobs. Let Z(C)
denote the set of assignments in C, In the formalism of Basse et al. (2019b), C is the conditioningevent of our test. We will use (Basse et al., 2019b, Theorem 1) to prove the validity of our proposedtest. This requires to show that the following two conditions hold. For notational simplicity, belowwe implicitly condition on C and also assume that 1{C ∈ C} = 1 whenever we use “C” to denote aconditioning biclique.
1. Imputability of test statistic: The potential outcomes are imputable within the biclique C underHF0 , since, by definition, the biclique units are exposed to some exposure in F for any assignment inthe biclique.Our test statistic is using only outcomes from units and assignments within the biclique,and so the condition in Equation (4) of Basse et al. (2019b, Theorem 1) holds.
2. Correct randomization distribution.: It remains to show that r(Z) = p(C|Z); i.e., that the random-ization distribution, r(Z), of our test coincides with the actual conditional distribution, P (Z|C) ∝P (C|Z)P (Z), induced by the conditioning mechanism p(C|Z) (Basse et al., 2019b, Section 3.2), andthe design P (Z). The conditioning mechanism of our procedure is equal to:
p(C|Zobs = z) = 1{z ∈ Z(C)}, (20)
since the test simply conditions on the biclique that contains the assignment. The marginal probabilityof conditioning on biclique C is therefore equal to:
p(C) =∑z′
p(C|z′)P (z′) =∑z′
1{z′ ∈ Z(C)}P (z′). (21)
31
The randomization distribution defined in Step 3 of the testing procedure of Section 4 is equal to:
r(z) = 1{z ∈ Z(C)} P (z)∑z′ 1{z′ ∈ Z(C)}P (z′)
[ By definition, in Step 3 of the biclique test ]
= 1{z ∈ Z(C)}P (z)
p(C)[ by Equation (21) ]
= P (C|Z)P (z)
p(C)[ by Equation (20) ]
= P (Z|C). (22)
The conditional validity of our test now follows from (Basse et al., 2019b, Theorem 1).
Theorem 4. Consider the intersection hypothesis HI0 defined in (18). Then, the biclique test inProcedure 1 operating on the biclique decomposition C from Procedure 2 is a conditionally valid testfor HI0 .
Proof. The main difference with Theorem 2 is that the test now also conditions on A = {Ai : i =
1, . . . , N}, the exposure sets of each unit. In this setting, there is no single biclique decomposition.Instead, for every possible value of A there corresponds one biclique decomposition, say, D(A). Thus,Equation (20) is updated to:
p(C|Zobs = z) = 1{C ∈ D(Az)}1{z ∈ Z(C)}, (23)
where Az denotes the value of A, which is uniquely determined by the observed assignment z. Wewill show that the equality
1{C ∈ D(Az)}1{z ∈ Z(C)} = 1{z ∈ Z(C)}
is guaranteed by construction of our null exposure graph. For that, it suffices to show that 1{C ∈D(Az)} = 0 implies that 1{z ∈ Z(C)} = 0. We prove by contradiction. Suppose that 1{C ∈D(Az)} = 0 and 1{z ∈ Z(C)} = 1 for some biclique C and observed assignment z. Note thatthe construction of the null exposure graph implies that all units in the null exposure graph receiveexposures contained in Az. Thus, 1{z ∈ Z(C)} = 1 implies that all units in C receive exposurescontained in Az. However, since the exposures in F[i] are all distinct, 1{C ∈ D(Az)} = 0 implies thatthere exists at least one unit in C that receives an exposure that is not in Az. This is a contradiction.We conclude that:
p(C|Zobs = z) = 1{z ∈ Z(C)},
and so the rest of the proof of Theorem 2 follows.
32
B.2. Implementation for arbitrary designs
We now address a practical implementation issue. Although our method works for arbitrarydesigns, it can become computationally intractable if the support, Z = {z : P (z) > 0}, is too large (onthe order of hundreds of thousands of nodes), since biclique enumeration is NP-hard. Fortunately,a small modification of our test can address this issue. The idea is to add a step at the beginningof Procedure 1 that subsamples assignments to limit the size of Z. We now show that the followingprocedure is still valid:
1. Draw Zobs ∼ P (Zobs).
2. Draw M − 1 assignments uniformly at random from Z \ {Zobs} and let ZM be the set of size Mformed as the union of {Zobs} and the set of size M − 1 just constructed.
3. Run our biclique test in Procedure 1, using the null exposure graph of the new support set, ZM .
The key for the proof is to consider the conditioning event as the pair (C,ZM ), where ZM is thesampled support set, and C is the biclique we condition on. The original test in Procedure 1 conditionsonly on a biclique, and ZM = Z. We obtain:
P (Z|C,ZM ) ∝ P (C,ZM |Z)P (Z)
∝ P (C|ZM , Z)P (ZM | Z)P (Z)
∝ 1{Z ∈ C}1{C ∈ ZM}P (Z), (24)
where we used the fact that P (ZM |Z) is independent of Z by construction of ZM ; also, P (C|ZM , Z) =
1{Z ∈ C}1{C ∈ ZM} because we simply condition on the biclique in ZM that assignment Z is con-tained in. Equation (24) ensures validity of this test if we simply make sure to make a bicliquedecomposition on ZM , so that 1{C ∈ ZM} = 1. The remainder terms in the randomization distribu-tion, 1{Z ∈ C}P (Z), correspond to the sampling distribution of Procedure 1 (Step 3).
C. More on clustered interference
Here, we continue our discussion on testing power in Section 6.2. In Figure 11, we reexamine howbiclique characteristics affect the testing power. The data shown are the same as in Figure 4, nowdisplayed on a single plot with color denoting power. Each dot corresponds to a different bicliquedecomposition of the null exposure graph, and we compute the average number of assignments andunits within each decomposition. Requiring more assignments in a biclique will make including moreunits a challenge. Intuitively, this results from the graphical nature of bicliques – they are completesubgraphs, and including more left nodes will dampen the size of the right node set. This inverse(nonlinear) relationship can be seen on the plot. Note also that there is a balancing of power fordifferent sized cliques. The highest powered tests come from bicliques with ∼ 150 units and ∼ 25
assignments. In practice, this is a tradeoff that can be navigated.
33
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100 150 200 250
510
1520
2530
3540
average # units
aver
age
# as
sign
men
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small powerlarge power
Figure 11. Average number of biclique assignments and units versus the power. Each dot corresponds tobiclique decomposition of the null exposure graph, and the color denotes the power value from the simulation.Red (blue) correspond to large (small) power values.
D. Design-assisted biclique test
We now define the “design-assisted” biclique test.
Procedure 3. For the two-stage experimental design in the cluster interference setting of Section 6.2,define the following procedure.
1. For each cluster k = 1, . . . ,K,
(a) Let Uk ⊆ U denote the set of units in cluster k, and Zk ⊆ Z the set of possible treatmentassignments of cluster k (N/K-length binary vector).
(b) Take Uk,1 to be a random half of Uk and Uk,2 = Uk \ Uk,1. Define Zk,j = {z ∈ Zk : zi =
1 & i ∈ Uk,j} for j = 1, 2, as the set of assignments for which the cluster is treated, and thesingle unit that is treated is from Uk,j.
(c) Define the following two bicliques in cluster k: Ck,1 = (Uk,1,Zk,2) and Ck,2 = (Uk,2,Zk,1).
(d) Let 0k denote the “all control” assignment in cluster k. We attach this assignment to oneof the two bicliques from step 1(c) at random. That is, with probability 1/2 we set Ck,1 =
(Uk,1,Zk,2 ∪ {0k}), otherwise we set Ck,2 = (Uk,2,Zk,1 ∪ {0k}).
2. Finally, let Z be the global assignment vector and let Z[k] denote the subvector corresponding tocluster k. Let Ck(Z) be the unique biclique from {Ck,1, Ck,2} such that Z[k] ∈ c(k, Z). We define
34
the following conditioning mechanism:
P (C|Z) ∝ 1{C =K⋃k=1
Ck(Z)}.
3. Run the randomization test of Procedure 1 conditional on C ∼ p(C|Zobs) as usual.
The above procedure is using the structure of the two-stage experimental design to define a bet-ter conditioning mechanism, P (C|Z). The structure of the design is leveraged mainly in step 1(c).Specifically, note that only one unit is being treated within each cluster, and that we are testing thenull hypothesis in (13), which allows imputation of potential outcomes for some unit i for any clusterassignment for which i is not treated. Thus, Ck,1, Ck,2, as defined, are indeed bicliques. The rest ofthe Procedure 3 makes sure to utilize these two large bicliques per cluster in the joint conditioningmechanism, which leads to conditioning the test on larger bicliques. Theorem 3 implies that this canyield more powerful tests. The following theorem shows that this procedure is valid.
Theorem 5. The randomization implied by Procedure 3 is valid at the nominal level for the nullhypothesis of Equation (13) under the two-stage experimental design of Section 6.2.
The proof of Theorem 5 is straightforward and relies on the fact that, by construction, the condi-tioning mechanism relies on a partitioning of the assignment space, such that P (C|Z) ∝ 1{Z ∈ C}as in Procedure 1.
Proof. By definition of the two-stage design, there is a constant proportion, say p, of clusters beingtreated out of K in total. As defined in the main text, let Ck(Z) denote the unique cluster from set{Ck,1, Ck,2} in cluster. By construction, when Z[k] = 0k — i.e., when the cluster is in control —thenP (Ck(Z) = Ck,j) = 1/2 for j = 1, 2. And when Z[k] 6= 0k — i.e., when the cluster is treated —thenP (Ck(Z) = Ck,j) = 1{Z[k] ∈ Ck,j}, for j = 1, 2.
Now, for some biclique C let C =⋃Kk=1Ck be its unique decomposition into cluster bicliques.
Following the definition of the conditioning mechanism in Procedure 3.
P (C|Z) =
K∏k=1:Z[k]=0k
P (0k ∈ Ck)K∏
k=1:Z[k] 6=0k
P (Z[k] ∈ Ck) [By definition of conditioning mechanism]
=K∏
k=1:Z[k]=0k
1{0k ∈ Ck}(1/2)K∏
k=1:Z[k] 6=0k
1{Z[k] ∈ Ck} [From analysis above]
= 2−(1−p)KK∏k=1
1{Z[k] ∈ Ck} = 2−(1−p)K · 1{Z ∈ C}. [Always (1− p)K clusters in control]
Recall that in our original test of Procedure 1, p(C|Z) ∝ 1{Z ∈ C}. We can now follow the proofof Theorem 2. The additional term 2−(1−p)K in the revised procedure does not affect the proof becauseit is fixed and drops from the calculations in the final derivations for r(Z) in Equation (22).
35
E. Power analysis
E.1. Proof of Theorem 3
Here, we prove the main theorem on power analysis of the biclique test.
Theorem 3. In Procedure 1, let C = (U,Z) ∈ C be the conditioning biclique, where C is a fixedbiclique decomposition. Let |C|= (n,m) denote the size of the biclique, with n = |U | and m = |Z|. Letthe randomization distribution and the null distribution be denoted, respectively, by
t(Z, Y (Z);C) ∼ F̂1,n,m, and t(Z, Y obs;C) ∼ F̂0,n,m, where Z ∼ P (Z|C). (25)
Suppose that for any fixed n > 0:
(A1) There exist continuous cdfs F1,n and F0,n such that F̂1,n,m and F̂0,n,m in (25) are the empiricaldistribution functions over m independent samples from F1,n and F0,n, respectively.
(A2) There exists σn > 0, and a continuous cdf F , such that F0,n(t) = F (t/σn), for all t ∈ R.
(A3) The treatment effect (e.g., spillover contrast) is additive, that is, there exists a fixed τ ∈ R suchthat F1,n(t) = F0,n(t− τ), for all t ∈ R.
Let φn,m be the power of the biclique randomization test conditional on a biclique of size (n,m), i.e.,
φn,m = E(1{pval(Z, Y obs;C) ≤ α} | |C|= (n,m)
).
Fix any small δ > 0. Then, for large enough m,
φn,m ≥ 1− F (F−1(1− α)− τ/σn)−O(m−0.5+δ).
If, in addition, supx∈R|F (x) − 1/(1 + e−bx)|≤ ε for fixed b, ε > 0, and σn = O(1/√n), then for some
fixed A, a > 0
φn,m ≥1
1 +Ae−aτ√n−O(m−0.5+δ)− ε.
Proof.
Lemma 1. Under Assumptions (A2) and (A3), it holds:
1− F1,nF−10,n(1− α) = 1− F (F−1(1− α)− τ/σn).
Proof. First, note that y = F0,n(x) implies that y = F (x/σn) from (A2) and so x = F−1(y)σn; thus,
36
F−10,n(y) = σnF−1(y). Now, for the first part:
1− F1,nF−10,n(1− α) = 1− F0,n(F−10,n(1− α)− τ) [From (A3)]
= 1− F (F−10,n(1− α)/σn − τ/σn) [From (A2)]
= 1− F (F−1(1− α)− τ/σn). [From result above]
Lemma 2. Let F−10,n(1 − α) , qα and F̂−10,n,m(1 − α) , q̂α,m. Then, for small enough ε there existsλ > 0 such that
P (|q̂α,m − qα|> ε) ≤ 2 exp(−λmε2).
Proof. Assumption (A1) implies that q̂α,m is the sample (1− α)-quantile of qα, and so
P (|q̂α,m − qα|> ε) ≤ 2 exp(−md2ε ),
where dε = min{F0,n(qα+ε)−F0,n(qα), F0,n(qα)−F0,n(qα−ε)}. See, for example, (Xia, 2019, Theorem2). Since F0,n is continuous and bounded, we can write F0,n(qα+ε) = F0,n(qα)+f0,n(qα)ε+O(ε2), andF0,n(qα− ε) = F0,n(qα)− f0,n(qα)ε+O(ε2), where f0,n is the pdf that corresponds to F0,n. Therefore,dε/ε = f0,n(qα) +O(ε). So, for small enough ε, there exists λ > 0 such that dε ≥ λε.
Lemma 3. Let fm(x) = λm−x + 2e−λm1−2x . Then, for large enough m the minimum of fm(x) is in
(1/2, 1 +O(log−1m)]. Specifically, for any large enough m,
minx∈R+
fm(x) = minx∈(1/2, 1+O(log−1m)]
fm(x).
Proof. Let gm(x) = λm−x = λe−x logm, then we can write fm(x) as fm(x) = gm(x) + 2e−(m/λ)g2m(x).
By differentiation, the minimum of fm(x) is at
g′m(x)− 4e−(m/λ)g2m(x)(m/λ)g′m(x)gm(x) = 0
e(m/λ)g2m(x) = 4(m/λ)gm(x)
(m/λ)g2m(x) = log (4(m/λ)gm(x))
(mλ)e−2x logm = log(4m) + log e−x logm
(mλ)e−2y = log(4m)− y [we set y = x logm]
y +mλe−2y = log(4m). (26)
It is straightforward to see from (26) that for large enough m the solution to this equation satisfiesy = θ log(4m) for some θ ∈ [1/2, 1]. Indeed, when θ = 0.5, the RHS in (26) is larger than theLHS for large enough m; and when θ = 1.0, the LHS is larger. Thus, x = θ log(4m)/logm =
θ(1 +O(1/logm)) ∈ (1/2, 1 +O(log−1m)).
37
Lemma 4. Suppose that (A1) holds, and fix any arbitrarily small constant δ > 0, with δ < 0.5. Then,for large enough m,
E(F̂1,n,m(z)− F1,n(z)) = O(m−0.5+δ), for any z ∈ R. (27)
Proof. Let ∆ = F̂1,n,m(z)− F1,n(z). Then,
F̂1,n,m(z)− F1,n(z) ≤ |∆|·I{|∆|≤ ε}+ |∆|·I{|∆|> ε} [ for any ε > 0 ]
≤ |∆|·I{|∆|≤ ε}+ I{|∆|> ε} [ since |∆|≤ 1, by definition ]
≤ ε+ I{|∆|> ε}
E(F̂1,n,m(z)− F1,n(z)) ≤ ε+ P (|∆|> ε)
≤ ε+ P (supz|F̂1,n,m(z)− F1,n(z)|> ε)
≤ ε+ 2e−2mε2
[ from (A1) and the DKW inequality ]
≤ m−0.5+δ + e−2m2δ
[ set ε = m−0.5+δ ]
= O(m−0.5+δ). [ for large enough m ].
Lemma 5. Suppose that Assumptions (A2) and (A3) hold. Then, for some r ∈ (0.5, 1+O(log−1m)),
E(F1,n(qα)− F1,n(q̂α,m)) ≥ −O(m−r).
Proof. Let ∆n,m = F1,n(qα)− F1,n(q̂α,m). Then,
∆n,m ≥ −I{|q̂α,m − qα|≤ εm}|F1,n(qα)− F1,n(q̂α,m)|−I{|q̂α,m − qα|> εm} [for any εm > 0 ]
≥ −µI{|q̂α,m − qα|≤ εm}|q̂α,m − qα|−I{|q̂α,m − qα|> εm} [F1,n is µ-Lipschitz]
≥ −µεm − I{|q̂α,m − qα|> εm}
E(∆n,m) ≥ −µεm − P (|q̂α,m − qα|> εm)
≥ −µεm − 2e−mµε2m [From Lemma 2]
≥ − minx∈R+
{µm−x + 2e−µm1−2x} [Tight bound along εm = m−x, x ∈ R+]
≥ −O(m−r). [For some r ∈ (1/2, 1) by Lemma 3].
Now, we put all the pieces together. Let
Φn,m =: E(1{pval(Z, Y obs;C) ≤ α} | |C|= (n,m), q̂α,m).
38
From the law of iterated expectation:
E(Φn,m | |C|= (n,m)) = φn,m. (28)
Moreover, by Assumption (A1) we have:
Φn,m = 1− F̂1,n,m(q̂α,m). (29)
It follows that
Φn,m − (1− F1,n(qα)) = (F1,n(qα)− F1,n(q̂α,m)) +(F1,n(q̂α,m)− F̂1,n,m(q̂α,m)
)[ by Eq. (29) ]
E(Φn,m) ≥ 1− F1,n(qα)−O(m−r)−O(m−0.5+δ) [ From Lemma 4 and Lemma 5 ]
φn,m ≥ 1− F1,n(qα)−O(m−0.5+δ) [ Since r > 0.5; and Eq. (28) ]
= 1− F (F−1(1− α)− τ/σn)−O(m−0.5+δ). [From Lemma 1]
If we also assume that supx∈R|F (x)− 1/(1 + e−bx)|≤ ε for fixed b, ε > 0, and σn = O(1/√n), then
φn,m ≥ 1− eb(F−1(1−α)−τc
√n)
1 + eb(F−1(1−α)−τc√n)−O(m−0.5+δ)− ε [For some c > 0]
φn,m ≥1
1 +Ae−aτ√n−O(m−0.5+δ)− ε. [Where A = ebF
−1(1−α) and a = bc]
(30)
E.2. Empirical confirmation of Theorem 3
In this section, we perform a simple simulation study to illustrate the result of Theorem 3. Inthe simulation we assume that the Assumptions (A1)-(A3) of Theorem 3. Another simulation studywhere the assumptions do not hold is presented in Section 6.2.
Thus, we only have to simulation the test statistic values. Let T denote our test statistic. Then,we define T |H0 ∼ N(0, 1/
√n) and T |H1 ∼ N(τ, 1/
√n), where τ is the treatment effect. This amounts
to F0,n(t) = Φ(t√n) and F1,n = Φ((t− τ)
√n). These definitions satisfy Assumptions (A1) and (A2)
of Theorem 3.In every simulation run, we repeat the following steps:
1. Sample tj ∼ F0,n, j = 1, . . . ,m, i.i.d.
2. For a given τ , sample T obs ∼ F1,n.
3. Define p-value: pval =∑m
j=1 1{tj ≥ T obs}.
4. Reject if pval ≤ 0.05.
39
In summary, under these assumptions, n — which is the number of focal units in the biclique test— controls the precision of the test statistic distributions. On the hand, m — which is the numberof focal assignments in the biclique test — controls the number of samples taken from H0 in order totest the hypothesis; in other words, m controls the support of the randomization distribution.
The results for various values of (n,m) are shown in Figure 12 below. We see that n controls howfast the power function increases to its maximum (sensitivity). On the other hand, m controls themaximum power of the test, but not its sensitivity. This shows that the power of the biclique test isaffected by the biclique size in different ways.
Figure 12. Power simulation study of the biclique test. The three panels correspond to n = 20, 50, 100.Different line types correspond to various m (number of assignments). The x-axis corresponds to the strengthof (spillover) treatment effect τ .
E.3. Implications for biclique-based randomization tests
The goal of this section is to connect graph-theoretic properties of the null exposure graph tobounds on the size of the biclique drawn from the conditioning mechanism. Since the structure of thenull exposure graph is fixed once the null hypothesis and design are specified, the final goal will be tounderstand how the test can be conditioned and relate that back to testing power (Theorem 3).
We will focus on density of the null exposure graph. There is classical work in extremal graphtheory that relates the number of graph edges to existence of certain-sized bicliques (Turán, 1941;Zarankiewicz, 1951). Others have built upon this work for bipartite graphs, including Kovári et al.(1954) and Hyltén-Cavallius (1958). We rely on analysis in the latter two references (especially exten-sions of what is known as the Kovári-Sós-Turán Theorem), formulated here in the context of the nullexposure graph. See Theorem 2.2 in Chapter 6, Section 2 of Bollobás (2004) for the precise bound
40
stated below in the Theorem.
Theorem (Bollobás (2004)). Consider the null exposure graph GFf = (V,E). Denote the total numberof units as N = |U| and assignments as H = |Z|. Define integers 0 < n < N and 0 < h < H. If thefollowing bound holds:
|E| ≥ (n− 1)H + (h− 1)1/nH1−1/n(N − n+ 1), (31)
then, there exists a biclique of GFf , C = (U,Z), with U ⊆ U, Z ⊆ Z and number of focal units n = |U |and focal assignments h = |Z|.
This result states that if there are “enough edges” in the null exposure graph (defined by thebound), then we are guaranteed existence of a certain-sized biclique. Moreover, we can translate thisbound to one on the density of the null exposure graph.
Definition 5 (null exposure graph density). Given the null exposure graph GFf = (V,E), its densityis defined as the number of edges divided by the total number of possible edges: d(GFf ) = |E|
NH .
Taking Definition 5 and the above theorem together, we conclude that if the null exposure graphis “dense enough,” there exists bicliques of a desired size. To give a concrete example of this bound,suppose we have an experiment like the clustered interference simulation of Section 6.2 with 300 units.We generate 100,000 treatment assignments and construct the null exposure graph as in Section B.2.In order to guarantee the existence of a biclique with 30 focal units and 500 focal assignments, wewould need the null exposure graph density to be at least 85% (calculated by the bound in (31)). Ifwe desired a biclique with 100 focal units and 500 focal assignments, we would need the null exposuregraph density to be at least 96%.
The example above illustrates the important connection between density and existence of suffi-ciently large bicliques for the randomization test. If the null exposure graph is not very dense, we arelikely to condition on bicliques with a small number of focal units and be underpowered (Theorem 3).On the other hand, if the null exposure graph has a large enough density, we will find bicliques withmany focal units and the test will be powerful. Indeed, for a fixed number of units N , assignments H,and focal assignments h, the bound in (31) is monotonically increasing in the number of focal unitsn. In words, we need large density to find large bicliques and have large power.
We now return to the clustered interference setting of Section 6.2. The power results presentedin Figure 3 are better understood in the context of the above theorem and the discussion presentedin Section 5.1. The cluster size defined by N/K in the experiment determines the density of the nullexposure graph. As before, density is defined as the number of edges in the graph – given by spilloverand control exposures – divided by the total number of possible edges. The experiment in Section 6.2fixed the number of units N and varied the cluster size K. The left panel of Figure 13 displays howdensity varies with cluster size. We can see that the density is increasing in the number of units percluster.
41
2 4 6 8 10 12 14
0.75
0.80
0.85
0.90
0.95
1.00
units per cluster (N/K)
null−
expo
sure
gra
ph d
ensi
ty
2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
units per cluster
pow
er
cliqueconditional focalsrandom focals
Figure 13. Left: The null exposure graph density as a function of cluster size for the clustered interferencesimulation study. Right:. The power of the biclique-based randomization test and competing methods for afixed τ = 0.3 and varying cluster size.
The right panel of Figure 13 shows how power varies with cluster size when τ = 0.3 for the bicliquemethod and the methods of Basse et al. (2019b) (conditional focals) and Athey et al. (2018) (randomfocals). Notice how the density of the null exposure graph follows the same trend as the bicliquemethod’s power. As explored theoretically above, increasing density will lead to increasing power.
Finally, in light of this discussion, we are able to better understand the power tradeoffs between thethree methods on the right in Figure 13. The objective of the graph decomposition algorithm is bicliquesize, and not explicitly power. Therefore, if the null exposure graph is too sparse, the biclique-basedrandomization will be underpowered relative to permutation-based alternatives specifically designedfor clustered interference. This is shown on the right panel of Figure 13 where the blue line is belowthe red and green lines. However, if the null exposure graph is dense enough, the biclique-based test iscomparable to and can easily exceed the power of alternative methods. For our clustered interferencesetting, the method starts outperforming at around 90% null exposure graph density (Figure 13 – left)and 6-8 units per cluster (Figure 13 – right).
E.4. Experimental design
In this section, we investigate our method’s use for experimental design. Since the biclique-basedrandomization test represents the first all-purpose approach for testing under general interference,including spatial intereference, we consider a spatial network like that of the Medellín example. Wegenerate 1000 points from a bivariate Gaussian with non-diagonal covariance to simulate the network(shown in Figure 14). Suppose we would like to design an experiment where the probability oftreatment in the city center p0 is different from the outskirts p1. As shown in Figure 14, the city
42
center is defined by the black circle. There are 362 units within the city center and 638 units in theoutskirts. The design question is: What are the optimal choices of p0, p1 for testing the spilloverhypothesis defined in (16) (with spillover radius equal to 0.1)?
−3 −2 −1 0 1 2 3 4
−3
−2
−1
01
23
4
p0
p1
Figure 14. The simulated network for the experimental design study. The black circle denotes the borderbetween the city center and outskirts. The treatment probabilities for the center and outskirts are given by p0and p1, respectively.
We answer this question by performing a power analysis for a grid of p0, p1. Like the previouspower analyses, we assume that the potential outcomes differ by an additive treatment effect τ = 0.3
and are normally distributed. For 40,000 different combinations of p0, p1, we compute the power ofthe test and display the results in the left of Figure 15. The power values range from 0.25 to 0.73, withthe largest values occurring for p1 ∼ 0.068 and p0 ∼ 0.061. The power surface is relatively convex,suggesting that there is a specific range of probabilities lead to high power, while other combinationsin the unit cube lead to underpowered tests.
The right side of Figure 15 shows the average number of focal units included in the biclique testsfor a given design. Interestingly, this surface is nearly identical to the power surface except for verysmall values of p0, p1. As mentioned in the power discussion of Section 5.1, the number of focal unitsis positively related to power. Therefore, the surfaces are close to each other. In the bottom leftcorner of the surface, the treatment probabilities are very low. Since the exposures in the hypothesisare defined as untreated spillover and pure control units, treating a very small number of units willleave the majority as pure control units. Moreover, this exposure status will rarely change amongthe different (sparse) treatment assignments. This part of the design space illustrates the tradeoffbetween the number of focal units and exposure balance. Very small treatment probabilities lead tolarge bicliques with focal units mostly exposed to pure control. In order to achieve optimal power,slightly smaller bicliques with a balance between spillover and pure control exposures are necessary.The biclique method is able to navigate this tradeoff to find a strictly interior solution.
43
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Power
p0
p 1
0.02 0.09 0.16 0.23 0.30
0.02
0.09
0.16
0.23
0.30
0.25
0.49
0.73
*
Average number of focal units
p0
p 1
0.02 0.09 0.16 0.23 0.30
0.02
0.09
0.16
0.23
0.30
36
431
827
Figure 15. Left: The power of the test for different combinations of p0, p1. Darker colors denote largerpower values, while lighter colors denote smaller power values. Right: null exposure graph density fordifferent combinations of p0, p1. Darker colors denote larger density values, while lighter colors denote smallerdensity values.
F. More on spatial interference
Here, we show more information on how properties of the bicliques (such as biclique size) affecttesting power in the context of spatial interference. Figure 16 displays the number of focals containedin the biclique for each hypothesis, Ha,br
0 as a function of r. We see that for larger radii, there are morefocals per biclique, on average, since more units are exposed to spillovers as the radius gets larger.
44
● ● ● ● ●●
●
●
●
●
100 150 200 250 300 350 400
4000
8000
1200
016
000
spillover radius
num
ber
of fo
cals
in c
lique
Figure 16. Number of focals versus spillover radius for bicliques containing the observed assignment. Theradii considered are in the set {75, 100, 125, 150, 175, 225, 275, 325, 375, 425}. Notice that the number of focalsincreases nonlinearly as the spillover radius increases.
Next, we show an extended version of the randomization analysis of Figure 9 shown in Figure 16.First, we show the p-values of the biclique randomization test (left vertical axis) with respect todistance radius r, as described earlier (“raw outcome" curve). Second, we show a version of thetest where we first regress the crime outcomes on known covariates, including information about theneighborhood and social center points, and then perform the biclique test on the residuals (“adjustedoutcome" curve). Finally, as a baseline, we also show regression coefficients from a simple OLS modelthat includes a binary variable indicating whether a unit receives spillovers at distance r or not, andknown covariates.
45
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●
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spillover radius
p−va
lue
● ● ●● ●
●● ● ●
●
0.0
0.1
0.2
0.3
0.4
0.5
−0.
0030
−0.
0020
−0.
0010
0.00
00
100 150 200 250 300 350 400
regr
essi
on c
oeffi
cien
t
p−values using raw outcomep−values using adjusted outcomeregression coefficient
Figure 17. P-values (left vertical axis) for biclique tests with varying spillover radii (horizontal axis). Theblue line shows p-values for tests using the raw crime index, and the black line shows p-values for tests usingthe the crime index adjusted for known covariates. The right vertical axis displays regression coefficients onthe binary variable defined by spillover or pure control statuses. For each radii, we restrict the OLSestimation to observations such that they are either exposed to spillover or pure control.
We see that the p-values for the raw outcome are all small for varying radii; see the flat blueline. This suggests that some form of spillovers exists, where the distance does not seem to matter.However, the biclique test on the adjusted outcomes (black curve in Figure 17) points to the otherdirection, as it does not indicate significance of spillover effects at any distance. This result suggeststhat the covariate distributions of “pure control" and “spillover" units are very different, such that thesignificance of the raw outcome FRTs may be attributed to that difference. The regression coefficient(green curve) agrees with the adjusted outcome results: no regression coefficient is significant at the0.05 level, and there is a similar, though concave, trend for increasing radii.
Finally, we show here additional information on the Medellín policing experiment. Figure 18 showsthe randomization distribution of the test statistics for various radii, r, and for both raw outcomesand adjusted outcomes. We see that the tests with the adjusted outcomes are sharper than the testswith the raw outcomes, indicating heterogeneity in the spillover effects.
46
0.00 0.10 0.20
020
40
r = 75, Crime index (adjusted)
0.00 0.10 0.20
010
20
r = 75, Crime index
0.00 0.10 0.20
020
40
r = 225, Crime index (adjusted)
0.00 0.10 0.20
010
20
r = 225, Crime index
0.00 0.10 0.20
010
2030
r = 100, Crime index (adjusted)
0.00 0.10 0.20
05
15
r = 100, Crime index
0.00 0.10 0.20
040
80
r = 275, Crime index (adjusted)
0.00 0.10 0.20
020
40
r = 275, Crime index
0.00 0.10 0.20
020
40
r = 125, Crime index (adjusted)
0.00 0.10 0.20
05
15
r = 125, Crime index
0.00 0.10 0.20
010
020
0
r = 325, Crime index (adjusted)
0.00 0.10 0.20
040
80
r = 325, Crime index
0.00 0.10 0.20
020
40
r = 150, Crime index (adjusted)
0.00 0.10 0.20
010
20
r = 150, Crime index
0.00 0.10 0.20
010
025
0
r = 375, Crime index (adjusted)
0.00 0.10 0.20
050
150
r = 375, Crime index
0.00 0.10 0.20
020
40
r = 175, Crime index (adjusted)
0.00 0.10 0.20
010
2030
r = 175, Crime index
0.00 0.10 0.20
010
020
0
r = 425, Crime index (adjusted)
0.00 0.10 0.20
010
020
0
r = 425, Crime index
Figure 18. Randomization distributions and observed test statistics (blue) for 20 biclique tests. The firstand third columns use the adjusted crime index as the outcome, while the second and fourth columns use theraw crime index. The radius defining spillover exposure status varies from 75 meters (top left) to 425 meters(bottom right). Note that the randomization distribution using the adjusted crime index have lower varianceand are centered at smaller positive values compared to their raw index counterparts. However, the p-valuesfrom the tests are largely same.
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G. Testing the general intersection hypothesis, HI0
G.1. Proof of Theorem 4
Theorem 4. Consider the intersection hypothesis HI0 defined in (18). Then, the biclique test inProcedure 1 operating on the biclique decomposition C from Procedure 2 is a conditionally valid testfor HI0 .
Proof. The main difference with Theorem 2 is that the test conditions on a biclique decomposition, C,that is comprised of multiple null exposure graphs, G(z;Z). Given C, as in Theorem 2, our conditioningmechanism satisfies:
P (C|Zobs) = 1{Zobs ∈ Z(C)}1{C ∈ C},
and so the derivations in Step 2 of Theorem 2 still hold. Thus, we only need to show that Step 1about imputability of potential outcomes is correct too.
To see this, let C be the conditioning biclique calculated from G(Zobs;Z0) for some Z0 ⊆ Z, asdescribed in Procedure 2. Take any unit i and assignment z′ in C. Since (i, z′) is an edge in C, then,by definition of G(z;Z), fi(z′) = fi(Z
obs). Under Hex0 , it follows that Yi(z′) = Yi(Z
obs).
G.2. Multi-null exposure graph
Here, we show how to extend the test for Hex0 in order to test HI0 in its more general form (18).
Below we define formally the concept of multi-null exposure graph.
Definition 6 (Multi-null exposure graph). Consider the intersection hypothesis HI0 of Equation (18).For any unit i ∈ U and z ∈ Z ⊆ Z, let A(i, z) be the unique set F ∈ I such that fi(z) ∈ F , if such setexists; otherwise, let A(i, z) = {}. Also, let G(z;Z) = (V,E) be the graph such that V = U ∪ Z andE = {(i, z′) ∈ U × Z : fi(z
′) ∈ A(i, z)}. Then, G(z;Z) is the multi-null exposure graph of HI0 withrespect to z ∈ Z,Z ⊆ Z.
This definition is a generalization with respect to the definition based on Hex0 . The main difference
is that Definition 6 may leave out some units-assignment pairs if the corresponding potential outcomescannot be imputed underHI0 . To testHI0 , Procedure 2 then needs to be slightly updated by initializingZ0 not as Z but as
Z0 ← Z \
z ∈ Z : fi(z) /∈J⋃j=1
Fj for all i ∈ U
.
Definition 6 of the multi-null exposure graphs G(z;Z) can also be used to test HI0 based on themethod of Athey et al. (2018) as described in Section 8.2.
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