appendixReferences

# Randomization-only Inference in Experiments with Interference

David Choi

###### Abstract

In experiments that study social phenomena, such as peer influence or herd
immunity, the treatment of one unit may influence the outcomes of others. Such
“interference between units” violates traditional approaches for causal
inference, so that additional assumptions are often imposed to model or limit
the underlying social mechanism. For binary outcomes, we propose an approach
that does not require such assumptions, allowing for interference that is both
unmodeled and strong, with confidence intervals derived using only the
randomization of treatment. However, the estimates will have wider confidence
intervals and weaker causal implications than those attainable under stronger
assumptions. The approach allows for the usage of regression, matching, or
weighting, as may best fit the application at hand. Inference is done by
bounding the distribution of the estimation error over all possible values of
the unknown counterfactual, using an integer program. Examples are shown using
using a vaccination trial and two experiments investigating social influence.

##  1 Introduction

In experiments where the assumption of “no interference between units” does
not apply, the effects of the treatment may extend to units receiving the
control. Examples include vaccination studies where units receiving the
placebo may be protected by the vaccine through herd immunity; economic
effects such as those occurring in a marketplace; or social networks in which
units may be influenced by information received from others. In such settings,
generally an assumption is made to limit the effects of interference; for
example, one might divide the units into groups that are assumed not to
interfere with each other. In settings where such restrictions are implausible
or have questionable validity, analyses relying on those assumptions may yield
misleading conclusions.

For such settings, we propose to estimate contrasts in attributable treatment
effects. The attributable effect is the difference between a unit’s observed
outcome and that which would have resulted under a uniformity trial, such as
the counterfactual in which all units had received the control (Rosenbaum,,
2001). In this paper, we show that such effects can be partially characterized
as a function of each unit’s direct treatment and indirect exposure to the
treatment of others, potentially with no assumptions on interference.

For example, to give a simple characterization we might regress effects
against treatment,

$\textit{Effect}_{i}\sim\beta_{0}+\beta_{1}\cdot\textit{Treatment}_{i}$ (1)

where $\textit{Effect}_{i}$ is the attributable effect for unit $i$, and
$\textit{Treatment}_{i}$ is the binary indicator of that unit’s treatment
assignment. This is equivalent to the least squares problem,

$\min_{\beta_{0},\beta_{1}}\sum_{i=1}^{N}\left(\textit{Effect}_{i}-\beta_{0}-\beta_{1}\cdot\textit{Treatment}_{i}\right)^{2}$
(2)

As a more complex example, given an observed social network we might regress
effects against each unit’s treatment, their number of treated friends, and an
interaction term,

$\displaystyle\textit{Effect}_{i}\sim\beta_{0}+\beta_{1}\cdot\textit{Treatment}_{i}+\beta_{2}\cdot\textit{\\#
Treated Friends}_{i}+$ (3)
$\displaystyle\beta_{3}\cdot(\textit{Treatment}_{i}\cdot\textit{\\# Treated
Friends}_{i})$

In both (1) and (3), the variable Effect is not observed, and hence
statistical inference is required to estimate the regression parameters.

The main result of this paper is that in regression specifications such as (1)
and (3) with binary outcomes, contrasts such as $\beta_{1}$, $\beta_{2}$, and
$\beta_{3}$ can be interval estimated with no assumptions beyond those
regarding randomization of treatment. Thus, these parameters can be reliably
inferred even when interference between individuals is difficult to accurately
model, including settings in which an observed network only crudely reflects
the true underlying social mechanisms. The regression may include
transformations of the treatment variable, or adjustments (i.e., “controls”)
if the indirect exposure to treatment is non-identically distributed. The
approach also generalizes beyond regression, to estimands corresponding to
weighted or matching-based comparisons of units. Inference requires novel
computations to bound the interval widths.

In the regression (1), the coefficient $\beta_{1}$ measures the relative
benefits of being treated versus not treated. For example, consider a
vaccination trial where those receiving the placebo might be indirectly
affected by the vaccines. In this setting, an appropriately signed $\beta_{1}$
would indicate that the vaccinated units were relatively more protected than
those who received the placebo, by an additional $\beta_{1}$ cases prevented
per capita. Similarly, in (3), the coefficients $\beta_{1}$, $\beta_{2}$, and
$\beta_{3}$ characterize the relative effects as a function of direct and
indirect exposure to treatment. In vaccination settings, this might imply that
units were relatively more protected when vaccinated, and additionally that
placebo units were relatively more protected when their friends were
vaccinated.

Relative effects are a partial characterization, and carry the caveat that
“more protected” may actually mean “less harmed”, if making the treatment
available causes everyone to become worse off. For example, a vaccine trial
might prevent an even more effective intervention from being deployed. Such
scenarios, when legitimately plausible, will have implications for all methods
of analysis.

How might such estimates be used? Decision-makers must often operate with
incomplete information, and a rigorous estimate of relative effects might be
more convincing than one that measures total effects, but hides uncertainty by
making overly strong assumptions. When communicating with policy makers,
Manski, (2013) and Nagin and Sampson, (2019) emphasize the importance of
properly conveying uncertainties that might otherwise be hidden by strong
assumptions, with Nagin and Sampson, (2019) specifically focusing on the
challenges of interference. Along these lines, doing an analysis both with and
without questionable assumptions may communicate the range of possible answers
to a question. Thus our approach might complement others.

In some settings, the research question may be of the form “is X contagious?”,
for behaviors and conditions such as product adoption, criminal behavior, and
obesity. In such studies, the treatment might serve primarily to trigger a
social influence mechanism so that it can be observed under randomization.
Then estimation of relative effects might advance knowledge by showing that a
social effect exists, and that it can be partially identified in a rigorous
way. This could fill an important role. For example, Christakis and Fowler,
(2007) studied contagion of obesity and received widespread attention, but
also strong criticism for lack of rigor (Lyons,, 2011).

###  1.1 Related Work

Our work is part of a growing literature on statistical inference in the
presence of interference between units. Approaches vary in their target
estimand and severity of modeling assumptions, with stronger inferences
generally requiring stronger assumptions.

Under the assumption of a known exposure model (Aronow and Samii,, 2017;
Eckles et al.,, 2017), the outcome of each unit depends on the treatments
through a set of statistics, termed the unit exposure; for example, a unit’s
exposure might equal the treatment of its neighbors in an ellicited social
network. For this setting, Aronow and Samii, (2017) gives propensity-weighted
estimates and confidence intervals for the average treatment effect, which
they define to be the difference in average outcome between two treatment
allocations (such as full treatment and full control).

The papers Hudgens and Halloran, (2008); Sävje et al., (2021); Leung, (2019);
Liu and Hudgens, (2014); Hu et al., (2021); Li and Wager, (2020) consider the
expected effects of intervening on the treatment of a single unit, assuming
the treatment of all other units to be randomized according to the experiment
design. Estimands of this type (sometimes termed the EATE) have been defined
for direct and indirect effects (Hu et al.,, 2021), and also generalized to
misspecified exposure models (Leung,, 2019). For this class of estimands,
unbiased point estimation requires no assumptions beyond randomization;
however, variance estimates and valid confidence intervals require assumptions
on interference, although not on its exact structure. For example, Leung,
(2019) requires a bandwidth parameter to be chosen according to the strength
of the assumed interference; Sävje et al., (2021) proposes variance inflation
factors based on the degrees of the underlying dependency graph; and Li and
Wager, (2020) consider random networks with distributional interactions
between units.

With the exception of Section 4.3, we will assume only that treatments are
randomized, so that interference may be unknown and unbounded in strength.
Previous work in this setting includes testing for null hypotheses related to
peer effects, such as the hypothesis of direct effects but no spillovers
(Athey et al.,, 2018; Aronow,, 2012; Basse et al.,, 2019); and estimation of
effects on the ranks of the treated and control units when ordering by outcome
(Rosenbaum,, 2007).

##### Other related works

Estimation under parametric modeling assumptions is studied in Rigdon and
Hudgens, (2015), and under an assumption that the effect is nonnegative in
Choi, (2017). The definition and estimation of vaccine-specific effects is an
active field, whose breadth is beyond our scope; we refer to Cai et al.,
(2021) for a recent discussion. Inference in observational studies with
interference is also an active area of research, and we refer to Forastiere et
al., (2021) and Ogburn et al., (2017). A technical report (Choi,, 2018)
considers the estimand $\tau_{1}$ given by (6), but without generalization to
other regressions, weighting, or matching.

##  2 Basic Approach

To introduce the main concepts, and to draw parallels with existing
approaches, we first describe our method for the simple regression (1) that
measures differences in attributable effects for the treated and control.

###  2.1 Defining the Attributable Effect

We assume a randomized experiment on $N$ units, and let $X\in\\{0,1\\}^{N}$
denote the treatment assignment (whose distribution is known),
$Y\in\\{0,1\\}^{N}$ the observed outcomes111For some estimands, we will allow
$Y$ and $\theta$ to be $[0,1]$-valued instead of binary., and
$\theta\in\\{0,1\\}^{N}$ the counterfactual outcomes that would result under a
“uniformity trial” (Rosenbaum,, 2007). Under the uniformity trial, the random
treatment assignment $X$ is generated but not actually administered, so that
the outcomes are unaffected by it. A common choice of uniformity trial is to
let $\theta$ denote the outcomes if all units were to receive the control
regardless of $X$. We do not assume or require $\theta$ to be observed (even
indirectly) for any unit.

We define the individual-level attributable treatment effect to be the vector

$Y-\theta.$

For example, in a vaccination trial the treatment $X_{i}$ might denote whether
unit $i$ received the vaccine or a placebo, $Y_{i}$ whether the unit had the
disease, and $\theta_{i}$ the counterfactual outcome that would have resulted
if all units were given the placebo. Letting an outcome of $1$ denote the
disease and $0$ otherwise, an effect of $Y_{i}-\theta_{i}=-1$ would mean that
unit $i$ did not have the disease, but would have if the placebo had been
administered to all participants. In such a case, it can be said that
administering the vaccine prevented this unit from having the disease. This
can occur even if the unit received the placebo in the trial, due to social
effects such as herd immunity.

The vector $Y-\theta$ clearly has a causal interpretation, and is the effect
of the treatment under the observed treatment assignment. It can be seen that
averaging $Y-\theta$ equals the attributable effect

$\frac{1}{N}\sum_{i=1}^{N}(Y_{i}-\theta_{i}),$ (4)

which is studied in Rosenbaum, (2001). Our focus will be to partially
characterize $Y-\theta$ by estimating a partial regression coefficient, or a
weighted or matching-based comparison of the unit outcomes. We will typically
make no assumptions on $\theta$ except on its range of allowable values. In
particular, we do not assume that $\theta$ is observed for any unit, and even
allow $Y$ to be uninformative for the values of $\theta$.

###  2.2 A Simple Example

Let $N_{1}$ and $N_{0}$ denote the number of treated and control units, and
let $\bar{X}=N_{1}/N$ denote the average treatment. We consider a slight
modification of the regression (1),

$Y_{i}-\theta_{i}\sim\tau_{0}+\tau_{1}(X_{i}-\bar{X}),$

which is equivalent to the least squares problem

$\min_{\tau_{0},\tau_{1}}\sum_{i=1}^{N}\left(Y_{i}-\theta_{i}-\tau_{0}-\tau_{1}(X_{i}-\bar{X})\right)^{2}.$

It can be seen that the intercept $\tau_{0}$ and slope $\tau_{1}$ are given by

$\displaystyle\tau_{0}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}(Y_{i}-\theta_{i})$ (5)
$\displaystyle\tau_{1}$
$\displaystyle=\underbrace{\frac{1}{N_{1}}\sum_{i:X_{i}=1}(Y_{i}-\theta_{i})}_{\text{attributable
effect on
treated}}-\underbrace{\frac{1}{N_{0}}\sum_{i:X_{i}=0}(Y_{i}-\theta_{i})}_{\text{attributable
effect on control}},$ (6)

so that $\tau_{0}$ is the average attributable effect, and $\tau_{1}$ is the
difference in attributable effects for the treated and control. The estimand
$\tau_{1}$ answers whether the units receiving the treatment were affected
differently than those receiving the control (and by how much), while
knowledge of both parameters $(\tau_{0},\tau_{1})$ specifies the attributable
effects for each group.

To give a possible usage for $\tau_{1}$, suppose that an intervention is
believed to have positive effects, perhaps due to previous experiments in
which units were effectively isolated, preventing interference. Then given a
more operational deployment in which the treatments are not isolated, one
might ask if the assignment was sufficiently dense to saturate the population.
For example, one might wish to know whether placebo units were nearly as well-
protected as those receiving a vaccination, or whether an assignment of
service providers provided evenly distributed spatial coverage. Estimating
$\tau_{1}$, the difference in effects that were experienced by treated and
untreated units, could be part of an analysis to answer such questions.

##### Estimation of $\tau_{1}$

The quantity $\tau_{1}$ is not a parameter, but rather an unobserved random
variable. Conservative prediction intervals (PIs) can be found for $\tau_{1}$
when $X$ is assigned by sampling without replacement. Letting $\hat{\tau}_{1}$
denote a point estimate equal to the difference in observed outcomes

$\hat{\tau}_{1}=\frac{1}{N_{1}}\sum_{i:X_{i}=1}Y_{i}-\frac{1}{N_{0}}\sum_{i:X_{i}=0}Y_{i},$
(7)

it can be seen that the estimation error $\tau_{1}-\hat{\tau}_{1}$ is given by

$\displaystyle\tau_{1}-\hat{\tau}_{1}$
$\displaystyle=\sum_{i:X_{i}=1}\frac{\theta_{i}}{N_{1}}-\sum_{i:X_{i}=0}\frac{\theta_{i}}{N_{0}},$
(8)

which is the difference between a random sample and its complement from the
unknown $\theta$. As a result, we can apply a finite population central limit
theorem to find a prediction interval for $(\tau_{1}-\hat{\tau}_{1})$, and
hence for the estimand $\tau_{1}$. The width of this interval will depend on
the unknown variance $\sigma_{\theta}^{2}$, given by

$\displaystyle\bar{\theta}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\theta_{i},$
$\displaystyle\sigma_{\theta}^{2}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{n}(\theta_{i}-\bar{\theta})^{2}.$ (9)

Proposition 1 gives the prediction interval, using the bound
$\sigma_{\theta}^{2}\leq 1/4$ which holds for binary outcomes. It is proven in
Section B.1 of the supplement.

######  Proposition 1.

Given $\rho\in(0,1)$, assume a sequence of experiments in which
$N\rightarrow\infty$, where treatment is assigned by sampling $N_{1}$ units
without replacement; $N_{1}/N$ converges to $\rho$; and
$\theta\in\\{0,1\\}^{N}$. Then $\tau_{1}$ is contained the interval

$\displaystyle\hat{\tau}_{1}\pm
z_{1-\alpha}\sqrt{\frac{N}{N-1}\frac{N}{N_{1}N_{0}}\frac{1}{4}}$

with probability asymptotically lower bounded by $1-2\alpha$.

##### Comparison with SATT

The point estimate $\hat{\tau}_{1}$ given by (7) is the difference in mean
outcomes, and also appears in estimation for the average treatment effect
(ATE), the average treatment effect on the treated (ATT), and the expected
average treatment effect (EATE).

The prediction interval for $\tau_{1}$ is most closely related to that of the
sample ATT discussed in Sekhon and Shem-Tov, (2021). To see this, assume an
experiment where treatment is assigned by sampling without replacement, and
let $\bar{Y}_{1}$ and $\bar{Y}_{0}$ respectively denote the average observed
outcomes for the treated and control groups, and let $\theta$ denote the
counterfactual outcomes that would occur if all units received the control,
with $\bar{\theta}_{1}$ and $\bar{\theta}_{0}$ defined analogously.

In the absence of interference, it holds for the control units that
$Y_{i}=\theta_{i}$. The sample ATT is defined as the difference between the
observed and counterfactual outcomes for the treated units,

$\tau^{\textup{SATT}}=\frac{1}{N_{1}}\sum_{i:X_{i}=1}(Y_{i}-\theta_{i})=\bar{Y}_{1}-\bar{\theta}_{1}.$

The estimator $\hat{\tau}_{1}$ equals $\bar{Y}_{1}-\bar{Y}_{0}$, and has error
$\hat{\tau}_{1}-\tau^{\textup{SATT}}=\bar{\theta}_{0}-\bar{\theta}_{1}$, where
we have used $\bar{Y}_{0}=\bar{\theta}_{0}$. This error is zero mean and has
variance

$\operatorname{Var}\left(\hat{\tau}_{1}-\tau^{\textup{SATT}}\right)=\frac{N}{N-1}\frac{N}{N_{1}N_{0}}\sigma_{\theta}^{2},$
(10)

with $\sigma_{\theta}^{2}$ given by (9). Since $Y_{i}=\theta_{i}$ for the
control units, the empirical variance of the control outcomes
$\\{Y_{i}:X_{i}=0\\}$ is consistent for $\sigma_{\theta}^{2}$ and can be
plugged into (10) to estimate
$\operatorname{Var}\left(\hat{\tau}_{1}-\tau^{\textup{SATT}}\right)$.

In the presence of interference, $\theta_{i}$ need not equal $Y_{i}$ for the
control units. For example, the effects of the treatment may be
redistributive, helping the treated by hurting the control. Then
$\bar{\theta}$ might approximately satisfy
$\bar{Y}_{0}\leq\bar{\theta}\leq\bar{Y}_{1}$. More generally,
$\bar{\theta}\in[0,1]$ if no assumptions are made.

In such settings, we cannot estimate $\tau^{\textup{SATT}}$, but we can
estimate $\tau_{1}$, the difference in effects for the treated and control. As
shown by (8), the estimation error $\hat{\tau}_{1}-\tau_{1}$ again equals
$\bar{\theta}_{1}-\bar{\theta}_{0}$, and hence its variance is the same as
(10). However, $\sigma_{\theta}^{2}$ can no longer be estimated from the
control outcomes, and must be bounded by 1/4 if no further assumptions are
made, resulting in a prediction interval that is wider than that of
$\tau^{\textup{SATT}}$ by a factor of $(2\hat{\sigma}_{Y_{0}})^{-1}$.

In some cases, it may be reasonable to reduce this factor by bounding
$\sigma_{\theta}^{2}=\bar{\theta}(1-\bar{\theta})$ more tightly. For example,
in Section 4.3 we consider a vaccination trial in which the disease rate
ranged from 0.002 to 0.005 in the preceding years. In such a case, it might
reasonable to bound $\bar{\theta}$ by a safety factor of the historical
values, as opposed to allowing $\bar{\theta}$ to equal $0.5$, which would be
implausibly high for this setting.

##### Comparison with EATE

The point estimate $\hat{\tau}_{1}$ is also unbiased for the expected average
treatment effect (Sävje et al.,, 2021), which can be written as

$\textup{EATE}=\frac{1}{N}\sum_{i=1}^{N}\left(\mathbb{E}[Y_{i}|X_{i}=1]-\mathbb{E}[Y_{i}|X_{i}=0]\right).$
(11)

If the units are independently assigned to treatment or control, the EATE is
the expected effect of intervening on a unit on that unit’s outcome, where the
expectation is taken over the randomly assigned treatment of the others.

While $\tau_{1}$ and the EATE have the same point estimate, they have quite
different sources of estimation error, which are not “nested” in any sense.
Unlike $\tau_{1}$, the estimation error for the EATE is the difference between
$\hat{\tau}_{1}$ and its expectation $\mathbb{E}[\hat{\tau}_{1}]$, and hence
depends on the observed outcome vector $Y$, whose values may have unknown
dependencies that must be controlled by additional assumptions. As a result,
the confidence interval width for the EATE will depend on the strength of the
assumption that is made. Under strong assumptions, the interval width for the
EATE may approach those attainable under an assumption of no interference;
under very weak assumptions, consistent estimation of the EATE may not be
possible.

To give a simple example of the latter case, suppose that the outcomes are
given by

$Y_{i}=X_{1}f_{i}(X_{i})+(1-X_{1})g_{i}(X_{i}),$

so that depending on the treatment of unit 1, the outcome of unit $i$ is given
either by $f_{i}(X_{i})$ or by $g_{i}(X_{i})$. In this case the point estimate
will be inconsistent for the EATE (while remaining $\sqrt{N}$-consistent for
$\tau_{1}$), as the EATE depends on both $f$ and $g$, but only one of the two
will be observed.

##  3 General Case

We generalize $\tau_{1}$ to allow the estimand to be a linear function of
$Y-\theta$,

$\text{estimand}=\sum_{i=1}^{N}w_{i}(X)(Y_{i}-\theta_{i}),$ (12)

where the weight $w$ should depend only on $X$ and be approximately zero-mean
in order for our interval estimates to be small. To implement this requirement
when the treatment (or the indirect exposure to treatment) is non-identically
distributed, we can use matching, weighting, or regression adjustment to draw
approximately unbiased comparisons between units.

##### Preliminaries

To measure each unit’s indirect exposure to the treatment of others, let
$A\in\\{0,1\\}^{N\times N}$ denote the adjacency matrix of an observed
network. Given a treatment assignment $X$, let $Z=(Z_{1},\ldots,Z_{N})$ denote
the number of treated neighbors for each unit,

$Z_{i}=\sum_{j:A_{ij}=1}X_{j},$ (13)

and let $W=(W_{1},\ldots,W_{N})$ the thresholding at a fixed level
$z_{\textup{min}}$:

$W_{i}=1\\{Z_{i}\geq z_{\textup{min}}\\}.$ (14)

We will assume that the units can be partitioned into $K$ propensity classes
$\Pi_{1},\ldots,\Pi_{K}$, where $n_{k}=|\Pi_{k}|$ denotes the size of class
$k$, and $k(i)$ equals the propensity class of unit $i$. Nominally, $Z_{i}$
and $Z_{j}$ should be identically distributed if $k(i)=k(j)$. For example, if
$X$ is assigned by sampling without replacement, then the distribution of
$Z_{i}$ varies only with $i$’s number of neighbors in the network $A$. In some
cases, the propensity classes may be misspecified, so that units belonging to
the same class may have non-identical exposure distributions. Then our
approach will produce wider intervals, so as to not be invalidated.

###  3.1 Examples

##### Regression

Let $\beta\in\mathbb{R}^{d}$ denote a general regression specification,
solving the least squares problem

$\min_{\beta}\sum_{i=1}^{N}\left(Y_{i}-\theta_{i}-\xi_{i}(X)^{T}\beta\right)^{2},$

for some choice of regressors $\xi_{i}(X)\in\mathbb{R}^{d}$ for $i\in[N]$.
Then $\beta$ is given by

$\beta=\left(\sum_{i=1}^{N}\xi_{i}(X)\xi_{i}(X)^{T}\right)^{-1}\sum_{i=1}^{N}\xi_{i}(X)(Y_{i}-\theta_{i})$
(15)

and it can be seen that the $\ell$th entry of $\beta$ can be written in the
form $w(X)^{T}(Y-\theta)$ for $w$ given by

$\displaystyle w_{i}(X)$
$\displaystyle=e_{\ell}^{T}\left(\sum_{i=1}^{N}\xi_{i}(X)\xi_{i}(X)^{T}\right)^{-1}\xi_{i}(X),$
(16)

where $e_{\ell}\in\mathbb{R}^{d}$ is the $\ell$th column of the $d\times d$
identity matrix.

We give two examples. Let $\beta^{\text{adj}}$ denote the partial regression
coefficient that arises when regressing the treatment effects $Y-\theta$ on
the exposures $Z$ and the set of $K$ indicator variables representing the
membership of each unit in the propensity classes $\Pi_{1},\ldots,\Pi_{K}$:

$\beta^{\text{adj}}=\left(\sum_{i=1}^{N}(Z_{i}-\zeta_{k(i)})^{2}\right)^{-1}\sum_{i=1}^{N}(Z_{i}-\zeta_{k(i)})(Y_{i}-\theta_{i}),$
(17)

where $\zeta_{k}$ is the average exposure for the $k$th propensity class

$\zeta_{k}=\frac{1}{n_{k}}\sum_{i\in\Pi_{k}}Z_{i}$

To interpret $\beta^{{\text{adj}}}$, we may say that for two individuals in
the same propensity class, a unit difference in the level of exposure is
associated with a $\beta^{\text{adj}}$ difference in their individual-level
attributable effects.

As a second example, regress $Y-\theta$ on a set of indicator variables for
the possible values of $Z$ and also on the $K$ propensity classes. This
results in the least squares problem

$\min_{\gamma\in\mathbb{R}^{D+1},\nu\in\mathbb{R}^{K}}\sum_{i=1}^{N}\left(\gamma_{Z_{i}}+\nu_{k(i)}-(Y_{i}-\theta_{i})\right)^{2},$
(18)

where $D$ denotes the maximum value of $Z$. Here the regression coefficients
$\gamma_{0},\ldots,\gamma_{D}$ correspond to the levels of $Z$, and $\nu$
corresponds to the propensity classes. We may then choose the contrasts
$\gamma_{d}-\gamma_{0}$, for $d=1,\ldots,D$, as the estimand of interest. To
interpret these contrasts, we may say that $\gamma_{d}-\gamma_{0}$ predicts
the difference in attributable effects for a unit with exposure level $d$
compared to one in the same propensity class who had zero exposure.

##### Weighting and Matching

Given thresholded exposures $W$ and propensity classes
$\Pi_{1},\ldots,\Pi_{K}$, let $n_{k1}$ and $n_{k0}$ denote the number of units
in $\Pi_{k}$ for which $W_{i}=1$ and $W_{i}=0$, respectively. Let
$\tau^{\text{weighted}}$ denote the weighted difference

$\tau^{\text{weighted}}=\sum_{k=1}^{K}\frac{n_{k}}{N}\sum_{i\in\Pi_{k}}\left(\frac{W_{i}(Y_{i}-\theta_{i})}{n_{k1}}-\frac{(1-W_{i})(Y_{i}-\theta_{i})}{n_{k0}}\right),$
(19)

which can be interpreted as the difference in attributable effects within each
propensity class, averaged together after weighting by class size. It can be
seen that $\tau^{{\text{weighted}}}$ can be written in the form
$w(X)^{T}(Y-\theta)$ for $w(X)$ given by

$w_{i}(X)=\frac{n_{k}}{N}\left(\frac{W_{i}}{n_{k(i)1}}-\frac{1-W_{i}}{n_{k(i)0}}\right)$
(20)

Alternatively, we can construct a matched subset of units by randomly pairing
units within the same propensity class (where one unit in each pair has
$W_{i}=1$ and the other has $W_{i}=0$), without replacement until no more
pairs can be formed. Let $\mathcal{M}$ denote the set of all units included in
the matching, and let $m=|\mathcal{M}|/2$ denote the number of matched pairs.
Let $\tau^{\text{matched}}$ denote the difference in attributable treatment
effects between the matched units,

$\tau^{\text{matched}}=\frac{1}{m}\left(\sum_{i\in\mathcal{M}:W_{i}=1}(Y_{i}-\theta_{i})-\sum_{i\in\mathcal{M}:W_{i}=0}(Y_{i}-\theta_{i})\right).$
(21)

The value $\tau^{\text{matched}}$ is analogous to $\tau$, measuring the
difference in attributable effects for the units in the matching. It can be
seen that $\tau^{\text{matched}}$ equals $w(X)^{T}(Y-\theta)$ for $w(X)$ given
by

$w_{i}(X)=\begin{cases}\frac{1}{m}&\text{ if }i\in\mathcal{M},W_{i}=1\\\
-\frac{1}{m}&\text{ if }i\in\mathcal{M},W_{i}=0\\\ 0&\text{
otherwise}\end{cases}$ (22)

As units are randomly paired within the same propensity class, the quantity
$\tau^{\text{matched}}$ is random given $X$. If this is undesireable, the
expectation of $\tau^{\text{matched}}$ may be used instead. This quantity
closely resembles $\tau^{\text{weighted}}$, and is given by

${\text{E}}[\tau^{\text{matched}}|X]=\sum_{k=1}^{K}\frac{m_{k}}{m}\sum_{i\in\Pi_{k}}\left(\frac{W_{i}(Y_{i}-\theta_{i})}{n_{k1}}-\frac{(1-W_{i})(Y_{i}-\theta_{i})}{n_{k0}}\right),$
(23)

where $m_{k}=\min(n_{k1},n_{k0})$ equals the number of matched pairs formed in
the $k$th strata, with $m=\sum_{k}m_{k}$.

###  3.2 Estimation

In principle, estimation for the general estimand $w(X)^{T}(Y-\theta)$
proceeds similarly to that of $\tau_{1}$, by defining the point estimate to
equal $w(X)^{T}Y$.

This point estimate will usually take intuitive form. For example, if the
estimand $w(X)^{T}(Y-\theta)$ is a regression on the attributable effects,
then $w(X)^{T}Y$ is simply the regression applied to the observed outcomes.
Likewise, if the estimand is weighted or matching-based comparison, then
$w(X)^{T}Y$ is simply that comparison applied to the observed outcomes. This
point estimate need not be unbiased, in the sense that the estimation error
$w(X)^{T}\theta$ may not be zero-mean; however, the expectation of the error
can be bounded by

$\min_{\vartheta\in\\{0,1\\}^{N}}\mathbb{E}w(X)^{T}\vartheta\
\leq\mathbb{E}w(X)^{T}\theta\leq\
\max_{\vartheta\in\\{0,1\\}^{N}}\mathbb{E}w(X)^{T}\vartheta,$ (24)

which can be easily evaluated when $\mathbb{E}w(X)$ is known, as is the case
for a randomized experiment.

To find a prediction interval, we will first show that a central limit theorem
holds for $w(X)^{T}\theta$,

$\frac{w(X)^{T}\theta-\bar{w}^{T}\theta}{\sqrt{\operatorname{Var}(w(X)^{T}\theta)}}\rightarrow
N(0,1),$

where $\bar{w}^{T}\theta$ is a location parameter for the asymptotic
distribution. This implies a normal-based interval for $w(X)^{T}\theta$ given
by

$\bar{w}^{T}\theta\pm z_{1-\alpha}\sqrt{\operatorname{Var}[w(X)^{T}\theta}],$

As $\theta$ is unobserved, the boundaries of this interval are unknown, but we
can find an outer bound by maximizing and minimizing over the unknown values
of $\theta$, resulting in the upper and lower bounds $U$ and $L$ given by

$\displaystyle U$ $\displaystyle=\max_{\vartheta\in\\{0,1\\}^{N}}\
\bar{w}^{T}\vartheta+z_{1-{\alpha}}\sqrt{\operatorname{Var}[w(X)^{T}\vartheta]}$
(25) $\displaystyle L$ $\displaystyle=\min_{\vartheta\in\\{0,1\\}^{N}}\
\bar{w}^{T}\vartheta-
z_{1-{\alpha}}\sqrt{\operatorname{Var}[w(X)^{T}\vartheta]},$ (26)

and a $(1-2\alpha)$ level prediction interval for the estimand is given by
$w(X)^{T}Y-[U,\,L]$.222To see this, observe that if $a\leq f(X)\leq b$ with
probability $1-\alpha$, then for any $g(X)$ it holds that $a+g(X)\leq
f(X)+g(X)\leq b+g(X)$ with the same probability.

Proposition 2 gives conditions under which prediction intervals for a variety
of estimands will asymptotically have correct coverage. It assumes a sequence
of experiments in which the number of units $N$ goes to infinity, where the
estimation error $w(X)^{T}\theta$ is a smooth function $h$ of an averaged
random vector $T(X)$ of sufficient statistics. The theorem applies the delta
method and uses a central limit theorem for sums of variables whose dependency
graph has bounded degree (Rinott,, 1994, Thm 2.2), whose application requires
Bernoulli randomization of the treatment vector $X$.

######  Proposition 2.

Given constants $d_{\max}$, $\rho$, $B$, $q$,
$\bar{\Sigma}\in\mathbb{R}^{q\times q}$, $\bar{T}\in\mathbb{R}^{q}$, and
mapping $h:\mathbb{R}^{q}\rightarrow\infty$ that is continuously
differentiable in a neighborhood of $\bar{T}$, assume a sequence of
experiments with estimands $w(X)^{T}(Y-\theta)$, in which $N\rightarrow\infty$
and the following hold

  1. A.1

Each unit is assigned to treatment independently with probability
$\rho\in(0,1)$

  2. A.2

The adjacency matrix $A\in\\{0,1\\}^{N\times N}$ has in-degree and out-degree
bounded by $d_{\max}$

  3. A.3

The quantity $w(X)^{T}\theta$ equals $h(T(X))$, where $T(X)$ equals

$T(X)=\frac{1}{N}\sum_{i=1}^{N}\phi_{i}(X),$

for mappings $\phi_{1},\ldots,\phi_{N}:\\{0,1\\}^{N}\mapsto\mathbb{R}^{q}$
satisfying

$\|\phi_{i}(X)\|_{\infty}\leq B,\qquad\forall X$

and

$\displaystyle\phi_{i}(X)=\phi_{i}(X^{\prime})\quad\textup{ if
}X_{i}=X_{i}^{\prime}\textup{ and }X_{j}=X_{j}^{\prime}$
$\displaystyle\textup{ for all }j:A_{ij}=1$

so that each $\phi_{i}$ depends only on the treatment of unit $i$ and $i$’s
neighbors in $A$.

  4. A.4

$\mathbb{E}T(X;\theta)$ converges to $\bar{T}$, and $\sqrt{N}\cdot
T(X;\theta)$ has covariance matrix converging to $\bar{\Sigma}$, which is non-
zero but may be non-invertible

  5. A.5

The variance of $\sqrt{N}\cdot w(X)^{T}\theta$ is asymptotically non-zero.

Under these conditions, let $\bar{w}$ satisfy
$\bar{w}^{T}\theta=h(\mathbb{E}T)$, and let $U$ and $L$ equal (25) and (26)
respectively. Then the quantity $w^{T}(Y-\theta)$ is contained by the interval
$w^{T}Y-[U,\,L]$ with probability asymptotically lower bounded by
$(1-2\alpha)$.

Corollary 1 gives conditions under which Proposition 2 can be applied to the
regression weights $\beta$ given by (15).

######  Corollary 1.

Assume a sequence of experiments with $N\rightarrow\infty$ which satisfy
conditions A.1 and A.2 of Theorem 2, where $w(X)$ as given by (16) for any
$\ell$ satisfies condition A.5, with $\xi_{i}$ satisfying
$\|\xi_{i}(X)\|_{\infty}\leq B$ for all $X$, and

$\displaystyle\xi_{i}(X)=\xi_{i}(X^{\prime})\quad\textup{ if
}X_{i}=X_{i}^{\prime}\textup{ and }X_{j}=X_{j}^{\prime}$
$\displaystyle\textup{ for all }j:A_{ij}=1$

so that $\xi_{i}(X)$ depends only on the treatment of unit $i$ and $i$’s
neighbors in $A$. Let $\phi_{i}(X)$ be given by

$\phi_{i}(X)=\left[\begin{array}[]{c}\xi_{i}(X)\otimes\xi_{i}(X)\\\
\theta_{i}\xi_{i}(X)\end{array}\right],$

where $\otimes$ denotes kronecker product, and let
$T(X)=N^{-1}\sum_{i}\phi_{i}(X)$ satisfy condition A.4. Then for the estimand
$\beta$ given by (15), the conclusions of Proposition 2 apply to the interval
estimation of $\beta_{\ell}$ with $\bar{w}$ given by

$\displaystyle\bar{w}_{i}$
$\displaystyle=e_{l}^{T}\left(\sum_{i=1}^{N}\mathbb{E}[\xi_{i}(X)\xi_{i}(X)^{T}]\right)^{-1}\mathbb{E}\xi_{i}(X).$
(27)

Corollary 2 gives conditions under which Proposition 2 can be applied to the
weighted contrast $\tau^{\text{weighted}}$, as given by (19).

######  Corollary 2.

Assume a sequence of experiments with $N\rightarrow\infty$ which satisfy
conditions A.1 and A.2 of Theorem 2, with $w(X)$ given by (20) for fixed $K$
satisfying condition A.5. Let $\phi_{i}(X)\in\mathbb{R}^{4K}$ be given by

$\phi_{i}(X)=\left(\left[1,\ \frac{NW_{i}}{n_{k(i)}},\
\frac{N\theta_{i}}{n_{k(i)}},\
\frac{NW_{i}\theta_{i}}{n_{k(i)}}\right]^{T}\right)\otimes e_{k(i)}$

where $\otimes$ denotes kronecker product and $e_{k}$ denotes the $k$th column
of the $K\times K$ identity matrix, with $n_{k}/N\geq 1/B$ for all $k\in[K]$,
and $T(X)=N^{-1}\sum_{i}\phi_{i}(X)$ satisfying condition A.4. Then the
conclusions of Proposition 2 apply to the interval estimation of
$\tau^{\text{weighted}}$, with $\bar{w}$ given by

$\bar{w}_{i}=\frac{n_{ki}}{N}\left(\frac{\mathbb{E}W_{i}}{\mathbb{E}n_{k(i)1}}-\frac{1-\mathbb{E}W_{i}}{\mathbb{E}n_{k(i)0}}\right)$
(28)

##### Remarks

  1. 1.

Conditions A.1 and A.2 of Proposition 2 are required to apply the central
limit theorem of (Rinott,, 1994, Thm 2.2), which applies to sums of dependent
variables, such as the entries of $W$. Condition A.4 of the proposition
implies $\sqrt{N}$-asymptotics, under which A.5 is required for the delta
method to yield intervals with the desired coverage. Conditions A.1 and A.5
can potentially be removed when delta method and central limit theorem of
Rinott, (1994) are not used, as shown for $\tau_{1}$ in Proposition 1 and for
$\beta^{\text{adj}}$ in Proposition 3 contained in the appendix.

  2. 2.

To enforce the degree constraint given by A.2, we might sparsify the observed
network $A$, for example by considering only same-neighborhood connections.
This changes the regression specification or estimand that is being
considered. However, it does not assume interference to be limited to the
sparsity pattern of $A$.

  3. 3.

While Corollary 1 implies that prediction intervals can be found for all
entries of $\beta$ and will have correct coverage, the intervals may be
vacuously wide for entries corresponding to non-contrasts (such as the
intercept term), or if the regression does not include adjustment for non-
identically distributed $X$ or $Z$. In such cases the entries of $\bar{w}$ may
be large, resulting in large bounds for $U$ and $L$.

###  3.3 Computation

As $\operatorname{Var}(w^{T}\theta)$ equals $\theta^{T}Q\theta$ for
$Q=\mathbb{E}ww^{T}-\mathbb{E}w\mathbb{E}w^{T}$, solving for $U$ given by (25)
requires solving the optimization problem

$\max_{\vartheta\in\\{0,1\\}^{N}}\bar{w}^{T}\vartheta+z_{1-\alpha}\sqrt{\vartheta^{T}Q\vartheta},$
(29)

To ensure coverage of the prediction intervals, a local optimum to (29) is not
sufficient – either the global solution or an upper bound must be found. As
$Q$ is positive semi-definite, this requires maximization of a convex
objective, which is computationally difficult even if the integer constraint
on $\vartheta$ is relaxed to the convex set $[0,1]^{N}$.

To solve or upper bound (29), we proceed in two steps. First, we find a
diagonal matrix $D$ such that $Q-D$ is negative semidefinite. This can be done
by solving the semidefinite program

$\displaystyle\min_{D\in\mathbb{R}^{n\times
n}}\sum_{i=1}^{n}D_{ii}\qquad\textup{subject to }D\textup{ diagonal, $D\geq
0,$ and }Q-D\preceq 0$

where $\geq 0$ denotes nonnegativity and $\preceq 0$ denotes negative
semidefinite. Then, we can rewrite (29) as

$\max_{\vartheta\in\\{0,1\\}^{N}}\bar{w}^{T}\vartheta+z_{1-\alpha}\sqrt{\vartheta^{T}(Q-D)\vartheta+\sum_{i}D_{ii}\vartheta_{i}}$
(30)

It can be seen that the objective of (30) is concave, but equals that of (29)
for $\vartheta\in\\{0,1\\}^{N}$, since $x=x^{2}$ if $x$ is binary valued. (30)
can be solved or bounded by Gurobi using branch-and-bound methods. For faster
computation, an upper bound can be found by relaxing the integrality
constraint of (30) to $\vartheta\in[0,1]^{N}$, resulting in a concave
maximization problem whose global optimum can be efficiently found. Solving
for $L$ given by (26) proceeds analogously.

In principle, assumptions on $\theta$ may be incorporated in the optimization
of (25) and (26), in the form of additional constraints. Doing so may result
in smaller confidence intervals. Such assumptions may be easier to consider
than those that directly bound interference; for example, in Section 4.3 we
consider a bound on $\theta$ based on historical rates of the outcome in the
years prior to treatment.

##  4 Examples

###  4.1 Social Networks and the Decision to Insure

In the experiment described in Cai et al., (2015), rural farmers in China were
randomly assigned to information sessions where they would be given the
opportunity to purchase weather insurance333data available at
https://www.aeaweb.org/articles?id=10.1257/app.20130442. First round sessions
were held three days before second round sessions, so that first round
attendees could have informal conversations with their second round friends,
in which they might share their opinions about the insurance product. A major
goal of the experiment was to demonstrate the effects of information sharing
through this channel.

The farmers were instructed to list 5 close friends with whom they
specifically discussed rice production or financial issues. One of the
conclusions of Cai et al., (2015) was that the decision to purchase insurance
was affected not only by a farmer’s own treatment assignment, but also by that
of their friends; specifically, farmers assigned to a second round session
were more likely to purchase insurance if more of their listed friends in the
first round were assigned to a high-information session.

##### Setup

We let $M$ and $N$ denote the number of units in the first and second rounds,
and define $X$, $Y$, $Z$, and $\theta$ as follows. For the first round units,
let $X\in\\{0,1\\}^{M}$ indicate whether each unit was assigned to a high or
low information session. For the second round units, let $Y\in\\{0,1\\}^{N}$
indicate whether or not they purchased insurance; $Z$ their number of same-
village first round friends assigned to a high information session; and
$\theta\in\\{0,1\\}^{N}$ their outcomes that would have occurred under a
counterfactual in which the first round sessions were not held. For example,
if $Y_{i}=1$ and $\theta_{i}=0$, then unit $i$ purchased insurance, but would
not have done so if the first round sessions were not held. Thus $Y-\theta$
equals the effect of the first round sessions on the purchasing decisions of
the second round.

We will assume that $X$ is randomized by simple random sampling within each
village, inducing propensity classes that were the interaction of village and
number of first round friends. In fact, Cai et al., (2015) reports that
treatments were stratified by family size and amount of land used for rice
farming; however, further details are not included, and additionally these
variables are missing for a significant fraction of the first round units. For
this reason, our results for this example should be viewed primarily as a
illustration of the proposed method.

##### Results

Table 1 gives estimates for various parameters described in Section 3.1. The
point estimate for $\beta^{\text{adj}}$ suggests that for two farmers in the
same propensity class, a unit difference in their number of high information
first round friends was associated with a 0.08 difference in the effects of
the first round sessions. However, the interval is weakly non-significant (95%
PI: [0.0, 0.16]).

The contrasts $\gamma_{d}-\gamma_{0}$ suggest an effect threshold at 2 high
information friends. Units with 1 high information friend experienced effects
that were indistinguishable from those with zero such friends. In comparison,
units with 2 such friends were more strongly encouraged to purchase insurance,
with effects that were 22% larger (95% PI: [2%, 41%]) than the effects on
those with zero high info friends. Contrasts $\gamma_{d}-\gamma_{0}$ for
$d\geq 3$ were non-significant due to small numbers of units with $Z\geq 3$.
Similar differences in effect were estimated by the weighted and matching-
based estimands $\tau^{\text{weighted}}$, $\tau^{\text{matched}}$, and
$E[\tau^{\text{matched}}|X]$, using the thresholded exposures
$W_{i}=1\\{Z_{i}\geq 2\\}$.

The interpretation of our estimates is that the second round units were
affected by the first round sessions, with effects that correlated with the
information content given to their first round friends, and in particular with
whether at least two such friends received high information content.

| Point Est. | Bias | 95% PI
---|---|---|---
$\beta^{\text{adj}}$ | 8% |  $\pm$ 0.4% | [0%, 17%]
$\gamma_{1}-\gamma_{0}$ | 2% |  $\pm$ 1% | [-11%, 14%]
$\gamma_{2}-\gamma_{0}$ | 21% |  $\pm$ 1% | [2%, 40%]
$\gamma_{3}-\gamma_{0}$ | 16% |  $\pm$ 2% | [-17%, 49%]
$\gamma_{4}-\gamma_{0}$ | 69% |  $\pm$ 2% | [-60%, 195%]
$\tau^{\text{weighted}}$ | 19% |  $\pm$ 2% | [2%, 36%]
$\tau^{\text{matched}}$ | 20% |  $\pm$ 1% | [3%, 37%]
$\operatorname{E}[\tau^{\text{matched}}|X]$ | 19% |  $\pm$ 1% | [3%, 35%]
Table 1: Estimation Results for Insurance Experiment

###  4.2 Simulated Vaccine Trial

We analyze the vaccinesim dataset, which simulates a vaccine trial using
models that were partially fit to an actual cholera vaccine trial, as
described in Perez-Heydrich et al., (2014)444available in the R package
inferference described in Saul and Hudgens, (2017).

##### Setup

We let $X_{i}$ denote whether unit $i$ received the vaccine or placebo;
$Y_{i}=1$ that the unit had cholera; $\theta_{i}$ their outcome under the
counterfactual in which all units had received the placebo; and $V_{i}$ their
neighborhood vaccination rate which equals the fraction of individuals in
$i$’s neighborhood (including non-participants) who were assigned to
vaccination. Participants were assigned independently with probability 2/3 to
receive the vaccine, otherwise receiving the placebo. Supplement C contains
additional simulation details.

##### Results

We estimate $\tau_{1}$ given by (6) and $\beta_{1}$, $\beta_{2}$, and
$\beta_{3}$ in the regression of effects on direct treatment, neighborhood
vaccination rates, and their interaction,

$(Y_{i}-\theta_{i})\sim\beta_{0}+\beta_{1}X_{i}+\beta_{2}V_{i}+\beta_{3}(X_{i}\cdot
V_{i})+\text{controls},$

where the control terms are the expectations $\mathbb{E}[V_{i}]$ and
$\mathbb{E}[X_{i}V_{i}]$, which depend on (and control for) the number of
participants and non-participants in each neighborhood.

Table 2 shows the resulting estimates. They suggest that vaccinated units were
relatively more protected than placebo units, and that placebo units were
relatively more protected when their neighbors were vaccinated. Estimates for
$\beta_{1}$, $\beta_{2}$ and $\beta_{3}$ had high uncertainty, and 90%
prediction intervals are shown. $\tau_{1}$ indicates that vaccinated units
were relatively more protected than placebo, with $0.12$ additional cases
prevented per capita. However, this may underrepresent the effects of the
vaccine, as placebo units may have been protected by the vaccines. To see
this, observe that the point estimates for $\beta_{1}$ and $\beta_{3}$ suggest
that the relative effects varied with neighborhood vaccination rates, with

$0.22-0.29\cdot\text{vaccination rate}$

additional cases prevented per capita for vaccinated units versus placebo.
Thus in neighborhoods with low vaccination rates, the difference in effects
for vaccinated and placebo units may have been significantly larger than the
overall difference given by $\tau_{1}$. The point estimate for $\beta_{2}$
indicates that placebo units were relatively more protected in neighborhoods
with high vaccination rates, with a 0.01 difference in rates associated with
0.006 additional cases per capita prevented by the trial.

| Point Est. | Bias | PI | Coverage
---|---|---|---|---
$\tau_{1}$ | -0.12 | 0 | [-0.17, -0.07] | 95%
$\beta_{1}$ | -0.21 | $\pm 0.005$ | [-0.38, -0.05] | 90%
$\beta_{2}$ | -0.59 | $\pm 0.02$ | [-1.16, -0.02] | 90%
$\beta_{3}$ | 0.29 | $\pm 0.01$ | [-0.03, 0.61] | 90%
Table 2: Estimates for vaccinesim data

##### Comparison with Other Approaches

For this application, the EATE can be interval estimated using the approach of
Leung, (2019). This method requires a choice of bandwidth, beyond which
interference is assumed to be negligible; we chose this bandwidth to imply
negligible interference between neighborhoods. Comparing the EATE estimate to
that of $\tau_{1}$, we find that the boundaries of their 95% intervals
differed by less than $0.01$.

Similarly, the estimand $\beta$ may be compared to $DE(\alpha)$ and
$IE_{0}(\alpha,\alpha^{\prime})$, which are discussed in Perez-Heydrich et
al., (2014); Tchetgen and VanderWeele, (2012), and are given by

$\displaystyle DE(\alpha)$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\bigg{(}\mathbb{E}[Y_{i}|X_{i}=1;\alpha]-\mathbb{E}[Y_{i}|X_{i}=0;\alpha]\bigg{)}$
$\displaystyle IE_{0}(\alpha,\alpha^{\prime})$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\bigg{(}\mathbb{E}[Y_{i}|X_{i}=0;\alpha]-\mathbb{E}[Y_{i}|X_{i}=0;\alpha^{\prime}]\bigg{)}$

where $\mathbb{E}[\,\cdot\,|\,\cdot\,;\alpha]$ denotes expectation under
i.i.d. Bernoulli($\alpha$) randomization of treatment for all individuals,
including non-participants. It can be seen that $DE(\alpha)$ is the expected
effect of intervening on a single random unit, if all individuals (including
non-participants) were randomized to receive the vaccine with probability
$\alpha$. Similarly, $IE_{0}(\alpha,\alpha^{\prime})$ is the expected effect
on a placebo unit of changing the probability of vaccination for all
individuals from $\alpha^{\prime}$ to $\alpha$.

(a) $\beta_{1}+\beta_{3}\cdot V_{i}$

(b) $DE(\alpha)$

(c) $\beta_{2}(V_{i}-0.4)$

(d) $I\\!E_{0}(\alpha,0.4)$

Figure 1: Estimates of effect heterogeneity/herd immunity. Shown are point
estimates, 90% PI, and true values (dashed red line). Fig. (c) and (d) show
effects relative to that of 40% vaccination.

Figure 1 shows estimates and true values for $IE_{0}(\alpha,0.4)$ and
$DE(\alpha)$ for a range of $\alpha$ values. In comparison to $\beta$, have
similar values, but show less uncertainty and describe the overall effects of
possible interventions, rather than relative effects of the observed
treatment. However, they assume no interference between neighborhoods, and
also assume that the decision to participate followed a known functional form
dependent only on observed covariates.

##### Coverage of Confidence Intervals

Using the simulation parameters given in Perez-Heydrich et al., (2014), we may
generate the unknown counterfactual outcomes $\theta$ and simulate $X$ to
measure the coverage rates of our confidence intervals. In 10K simulations,
the intervals for $\beta_{1}$, $\beta_{2}$, and $\beta_{3}$ achieved coverage
but were as much as 2.2 times wider than necessary. However, for other values
of $\theta$, found by solving (25), the coverage of the 90% interval was as
low as 91%. This suggests that the interval widths were necessary in order to
have coverage for all possible values of $\theta$, as required without further
assumptions.

###  4.3 Real Vaccine Trial (Aggregate Outcomes)

The paper Ali et al., (2005) analyzes the actual results of the cholera
vaccine trial, in order to investigate the possibility of indirect (herd)
protection under high levels of vaccination. A table of aggregate outcomes for
the actual cholera vaccine trial is given, which we aggregate further in Table
3 into two groups of roughly equal size, by dividing the units according to
whether their neighborhood vaccination rate was $\leq 40\%$ or $>40\%$. We
consider the regression

$(Y_{i}-\theta_{i})\sim\beta_{1}X_{i}+\beta_{2}(X_{i}\cdot i\in\textup{group
1})+\textup{controls},$

where the control terms are the membership indicators for groups 1 and 2.

With no further assumption on $\theta$, we allow for the possibility that in
the absence of the vaccinations, the cholera rate could have been hundreds of
times larger than the observed rate of 2.8 cases per thousand. As this may be
unrealistic, we will consider an additional assumption based on past cholera
rates. The trial covered the Matlab region of Bangladesh in 1985, and cholera
rates for this region are given in (Zaman et al.,, 2020, Fig. 1). We will
assume that under the counterfactual $\theta$ in which all units receive the
placebo, the cholera rate for the participants would be upper bounded by 7
cases per thousand,

$\frac{1}{N}\sum_{i=1}^{N}\theta_{i}\leq 0.007,$ (31)

which is substantially higher than the highest observed rate of 5 per thousand
in the ten years before the trial. As the participants ($\approx 30\%$ of the
population) are not a random sample, some caution is required; however, (31)
may easier to consider than a bound on interference between neighborhoods.

The coefficients $\beta_{1}$ and $\beta_{2}$ characterize the relative effects
of receiving the vaccine versus placebo, in each group. Table 4 shows the
results. The estimates suggest that vaccinated units were relatively more
protected that placebo ones, by a difference of

$1.3+2.2\cdot 1\\{\text{unit is in group 1}\\}$

additional cases prevented per thousand. Prediction intervals (weakly) exclude
zero for the interaction term $\beta_{2}$, indicating that the relative
effects were different for the two groups, and also for $\beta_{1}+\beta_{2}$,
which gives the relative effects for group 1 which had lower vaccination
rates. These results suggest that the overall difference in outcomes between
treatment and control, which equaled 2.4 cases per thousand, may under-
represent the effects of the vaccine, as larger relative effects occurred when
vaccination rates were low.

| Vaccination | Cholera Rate (per 1000)
---|---|---
Group | Rate | Treated | Control
1 | $\leq 40\%$ | 2.1 (54/25K) | 5.6 (72/13K)
2 | $>40\%$ | 1.7 (42/25K) | 3.0 (36/12K)
Total |  | 1.94 (96/49K) | 4.4 (108/25K)
Table 3: Cholera cases in treated and control units grouped by neighborhood vaccination rates. (Taken from Hudgens and Halloran, (2008); Ali et al., (2005).) | Point Estimate | Bias | 90% PI
---|---|---|---
$\beta_{1}$ | -1.3 | 0 | [-3.5, 0.9]
$\beta_{2}$ | -2.2 | 0 | [-4.4, 0.0]
$\beta_{1}+\beta_{2}$ | -3.5 | 0 | [-5.6, -1.4]
Table 4: Estimates (in cases per thousand) for aggregate-level cholera data

##  5 Conclusions

We have presented an approach for partial estimation of unit-level
attributable effects given a randomized trial on a single network with strong
unmodeled interference. The approach goes beyond testing to give a partial
characterization, returning interval estimates for the relative effects of
treatment.

##### Supplemental Materials

The supplement contains analysis of a social experiment (Paluck et al.,,
2016); proofs; and data summaries.

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## Supplemental Material for “Randomization-only Inference in Experiments with
Interference”

##  Appendix A Support for Conflict Reduction in Schools

The paper Paluck et al., (2016) describes an experiment555data available at
https://www.icpsr.umich.edu/web/civicleads/studies/37070 in which public
middle schools were randomized to receive a treatment designed to encourage
anti-conflict norms and behavior in students. Within each treated schools, a
small subset of eligible students (between 40-64) were randomized between
treatment and control. Treated students were invited to participate in bi-
monthly meetings in which they designed and enacted strategies to reduce
schoolwide conflict. Outcomes for all students in each school included the
self-reported wearing of an orange wristband representing support for anti-
conflict norms. Network information (“spent time with”) was recorded for all
students. Estimation of effects on the wristband outcome has been used in
Aronow, (2012) and Leung, (2019) to illustrate their proposed methods, as we
do here.

(a) $\mathbb{E}[\tau^{\text{matched}}]$ Figure 2: Estimates (point estimate,
bias bound, and 95% PI) for experiment described in Section A.

##### Setup

Let $X_{i}$ denote if student $i$ was invited to participate in the anti-
conflict meetings (which can only equal 1 for eligible students), and $Y_{i}$
their outcome of whether they self-reported wearing the wristband representing
support for anti-conflict norms. Breaking slightly from the previous examples,
we define the counterfactual $\theta_{i}$ to denote their outcome if all
eligible units were invited to participate in the meetings (“full treatment”).
With $\theta$ so defined, the difference $Y-\theta$ is interpretable as the
effect of not inviting the control units to participate.

##### Results

Following Aronow, (2012) and Leung, (2019), we estimate differences in
treatment effect on four groups of students, corresponding to whether or not
they were invited to participate in the meetings, and also whether or not they
had at least one network connection who was invited to participate. Figure
2(a) shows estimates of $\mathbb{E}[\tau^{\text{matched}}]$ for units that
were matched based on school and number of eligible friends. Similar results
were found using weighting or matching. A significant difference in effects
was found between two groups; as a result of not inviting the control units to
participate, units in control whose friends were also in control were affected
more adversely than those who were treated and had at least one treated
friend.

##  Appendix B Proofs

We prove Propositions 1 and 2, Corollaries 1 and 2, and also state and prove
Proposition 3, which gives an alternate interval estimate for
$\beta^{\text{adj}}$.

###  B.1 Proof of Proposition 1

Our proof of Propositions 1 and 3 will use the following finite population
central limit theorem.

######  Theorem 1.

[Thm. 1]li2017general Let $\bar{u}_{S}$ denote the average of $N_{1}$ units
sampled without replacement from the finite population $u_{1},\ldots,u_{N}$,
with $\bar{u}=N^{-1}\sum_{i}u_{i}$. As $N\rightarrow\infty$, if

$\frac{1}{\min(N_{1},N-N_{1})}\cdot\frac{m_{N}}{v_{N}}\rightarrow 0,$ (S.1)

where $m_{N}$ and $v_{N}$ are given by

$m_{N}=\max_{i}(u_{i}-\bar{u})^{2}\qquad\text{and}\qquad
v_{N}=\frac{1}{N-1}\sum_{i}(u_{i}-\bar{u})^{2},$

then $(\bar{u}_{S}-\bar{u})/\sqrt{\operatorname{Var}(\bar{u}_{S})}\rightarrow
N(0,1)$ in distribution.

Lemma 1 gives a simpler expression for the difference in means of $\theta$,
from which its variance can be derived.

######  Lemma 1.

Let $X$ be assigned by sampling $N_{1}$ units without replacement, and let
$\bar{\theta}$ and $\sigma_{\theta}^{2}$ denote the mean and variance of
$\theta$. Then it holds that

$\sum_{X_{i}=1}\frac{\theta_{i}}{N_{1}}-\sum_{X_{i}=0}\frac{\theta_{i}}{N_{0}}=\frac{1}{N_{1}}\sum_{i=1}^{N}X_{i}\frac{(\theta_{i}-\bar{\theta})N}{N_{0}},$

which is the average of $N_{1}$ samples without replacement from the finite
population $u_{1},\ldots,u_{N}$, where
$u_{i}=(\theta_{i}-\bar{\theta})N/N_{0}$, and which has variance

$\displaystyle\frac{1}{N_{1}}\frac{N_{0}}{N-1}\frac{N^{2}}{N_{0}^{2}}\sigma_{\theta}^{2}.$
(S.2)

###### Proof of Lemma 1.

$\displaystyle\sum_{X_{i}=1}\frac{\theta_{i}-\bar{\theta}}{N_{1}}-\sum_{X_{i}=0}\frac{\theta_{i}-\bar{\theta}}{N_{0}}$
$\displaystyle\hskip
8.5359pt=\sum_{i=1}^{N}\frac{X_{i}(\theta_{i}-\bar{\theta})}{N_{1}}-\sum_{i=1}^{N}\frac{(1-X_{i})(\theta_{i}-\bar{\theta})}{N_{0}}$
$\displaystyle\hskip
8.5359pt=\sum_{i=1}^{N}X_{i}(\theta_{i}-\bar{\theta})\frac{N}{N_{1}N_{0}}-\frac{1}{N_{0}}\underbrace{\sum_{i=1}^{N}(\theta_{i}-\bar{\theta})}_{=0}$
$\displaystyle\hskip
8.5359pt=\frac{1}{N_{1}}\sum_{i=1}^{N}X_{i}\frac{(\theta_{i}-\bar{\theta})N}{N_{0}}.$

This equals the average of $N_{1}$ samples without replacement from the finite
population $u_{1},\ldots,u_{N}$, where
$u_{i}=(\theta_{i}-\bar{\theta})N/N_{0}$, which is known to have variance
equal to $\frac{1}{N_{1}}\frac{N_{0}}{N-1}$ multiplied by the variance of
$\\{u_{i}\\}$, which equals (S.2). ∎

###### Proof of Proposition 1.

It can be seen that $\tau_{1}-\hat{\tau}_{1}$ equals

$\tau_{1}-\hat{\tau}_{1}=\sum_{X_{i}=1}\frac{\theta_{i}}{N_{1}}-\sum_{X_{i}=0}\frac{\theta_{i}}{N_{0}}.$

By Lemma 1, this equals the average of $N_{1}$ samples without replacement
from the finite population $u_{1},\ldots,u_{N}$, where
$u_{i}=(\theta_{i}-\bar{\theta})N/N_{0}$, with variance given by (S.2). Hence
the terms $m_{N}$ and $v_{N}$ as defined in Theorem 1 can be bounded by

$m_{N}\leq(N/N_{0})^{2}\qquad\text{and}\qquad
v_{N}=\sigma_{\theta}^{2}(N/N_{0})^{2}.$ (S.3)

To prove the proposition, we will divide the sequence of experiments into two
subsequences: (i) those for which $\sigma_{\theta}^{2}\geq(\log N)/N$, and
(ii) those for which $\sigma_{\theta}^{2}<(\log N)/N$, and then show coverage
of the proposed interval for each subsequence separately.

For subsequence (i), we observe that by (S.3), along with
$N_{1}\rightarrow\rho N$ and $\sigma_{\theta}^{2}\geq(\log N)/N$, it can be
seen that the CLT condition (S.1) holds, i.e.,

$\frac{1}{\min(N_{1},N-N_{1})}\cdot\frac{m_{N}}{v_{N}}\rightarrow 0.$

Thus by Theorem 1, it holds that

$\mathbb{P}\left(|\tau_{1}-\hat{\tau}_{1}|\geq
z_{1-\alpha}\sqrt{\operatorname{Var}(\tau_{1}-\hat{\tau}_{1})}\right)\rightarrow
1-2\alpha.$ (S.4)

Since $\theta\in\\{0,1\\}^{N}$, it holds that $\sigma_{\theta}^{2}\leq 1/4$,
and plugging this bound into (S.2) yields an upper bound on the variance,

$\displaystyle\operatorname{Var}(\tau_{1}-\hat{\tau}_{1})$
$\displaystyle\leq\frac{1}{4}\frac{N}{N-1}\frac{N}{N_{1}N_{0}},$ (S.5)

and combining (S.5) and (S.4) implies for subsequence (i) that

$\mathbb{P}\left(|\tau_{1}-\hat{\tau}_{1}|\geq
z_{1-\alpha}\sqrt{\frac{1}{4}\frac{N}{N-1}\frac{N}{N_{1}N_{0}}}\right)\rightarrow
2\alpha,$ (S.6)

and hence that $\tau_{1}$ is in the interval $\hat{\tau}_{1}\pm
z_{1-\alpha}\sqrt{\frac{1}{4}\frac{N}{N-1}\frac{N}{N_{1}N_{0}}}$ with the same
probability.

For subsequence (ii), plugging $\sigma_{\theta}^{2}<(\log N)/N$ into (S.2)
implies that $\sqrt{\operatorname{Var}(\tau_{1}-\hat{\tau}_{1})}$ is a
vanishing fraction of
$z_{1-\alpha}\sqrt{\frac{1}{4}\frac{N}{N-1}\frac{N}{N_{1}N_{0}}}$, the width
of the proposed confidence interval. As a result, for this subsequence it
holds by Chebychev’s inequality that

$\mathbb{P}\left(|\tau_{1}-\hat{\tau}_{1}|\geq
z_{1-\alpha}\sqrt{\frac{1}{4}\frac{N}{N-1}\frac{N}{N_{1}N_{0}}}\right)\rightarrow
0,$ (S.7)

and hence that $\tau_{1}$ is in the interval $\hat{\tau}_{1}\pm
z_{1-\alpha}\sqrt{\frac{1}{4}\frac{N}{N-1}\frac{N}{N_{1}N_{0}}}$ with the same
probability.

Combining (S.6) for subsequence (i) and (S.7) for subsequence (ii) implies
that for combined sequence of experiments, $\tau_{1}$ is in the interval
$\hat{\tau}_{1}\pm
z_{1-\alpha}\sqrt{\frac{1}{4}\frac{N}{N-1}\frac{N}{N_{1}N_{0}}}$ with
probability asymptotically lower bounded by $1-2\alpha$, proving the
proposition.

∎

###  B.2 Proof of Theorem 2

The proof of Theorem 2 will use several tools.

The Cramer-Wold device [Sec 2.3]van2000asymptotic states that a sequence of
random vectors $X_{n}\in\mathbb{R}^{d}$ converges in distribution to $X$ if
and only if $\varpi^{T}X_{n}$ converges in distribution to $\varpi^{T}X$ for
all vectors $t\in\mathbb{R}^{d}$.

Theorem 2 describes a uniform delta method [Thm 3.8]van2000asymptotic, which
implies that if $\sqrt{n}X$ converges to a multivariate normal distribution,
then $f(X)$ does so as well after centering and rescaling, if $f$ is
continuously differentiable around the limit point of $X$.

######  Theorem 2.

[Thm 3.8]van2000asymptotic Let $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a
continuously differentiable mapping in a neighborhood of $\bar{T}$, and let
random vectors $T_{n}\in\mathbb{R}^{d}$ and non-random
$(\bar{T}_{n},r_{n})\in\mathbb{R}^{d}\times\mathbb{R}$ satisfy that
$\bar{T}_{n}\rightarrow\bar{T}$, $r_{n}\rightarrow\infty$, and
$r_{n}(T_{n}-\bar{T}_{n})$ converges in distribution to a random vector $T$.
Then $r_{n}(f(T_{n})-f(\bar{T}_{n}))$ converges in distribution to $\nabla
f(\bar{T})^{T}T$, the inner product of $T$ and the gradient of $f$ at
$\bar{T}$. Moreover, the difference between $r_{n}(f(T_{n})-f(\bar{T}_{n}))$
and $\nabla f(\bar{T})^{T}r_{n}(T_{n}-\bar{T}_{n})$ converges in probability
to $0$.

Theorem 3 is a central limit theorem (Rinott,, 1994, Thm 2.2) that requires
the notion of a dependency graph: Given a collection of dependent random
variables $X_{1},\ldots,X_{N}$, a graph with $N$ nodes is said to be its
dependency graph if for any two disjoint subsets $S_{1}$ and $S_{2}$, if there
is no edge in the graph connecting $S_{1}$ to $S_{2}$ then the sets of random
variables $\\{X_{i}:i\in S_{1}\\}$ and $\\{X_{i}:i\in S_{2}\\}$ are
independent.

######  Theorem 3.

(Rinott,, 1994, Thm 2.2) Let $X_{1},\ldots,X_{N}$ be zero-mean random
variables having a dependency graph whose maximal degree is strictly less than
$D$, satisfying $|X_{i}-\mathbb{E}X_{i}|\leq B$ a.s., $i=1,\ldots,N$, and
$\operatorname{Var}\sum_{i=1}^{N}X_{i}=\sigma^{2}>0$. Let $\Phi$ denote the
CDF of a standard normal. Then for all $t\in\mathbb{R}$ it holds that

$\displaystyle\left|\mathbb{P}\left(\frac{\sum_{i=1}^{N}X_{i}}{\sigma}\leq
t\right)-\Phi(t)\right|$ $\displaystyle\hskip
8.5359pt{}\leq\frac{1}{\sigma}\left\\{\sqrt{\frac{1}{2\pi}}DB+16\left(\frac{n}{\sigma^{2}}\right)^{1/2}\
D^{3/2}B^{2}+10\left(\frac{n}{\sigma^{2}}\right)D^{2}B^{3}\right\\}$

Lemma 2 bounds the degree of the dependency graph for the thresholded
exposures.

######  Lemma 2.

Let the assumptions and definitions of Theorem 2 hold. Then the random vectors
$\phi_{1}(X),\ldots,\phi_{N}(X)$ have dependency graph with maximum degree
bounded by $d_{\max}^{2}$.

###### Proof of Lemma 2.

Let $G$ denote the graph with $N$ nodes which has an edge between $i$ and $j$
if $A_{i\ell}=A_{j\ell}=1$ for any node $\ell$, so that $\ell$ is a common
neighbor of both $i$ and $j$. Since $A$ has maximum degree $d_{\max}$, it can
be seen that $G$ has maximum degree $d_{\max}^{2}$.

To see that $G$ is the dependency graph for $\phi_{1}(X),\ldots,\phi_{N}(X)$,
we observe that since treatments are assigned independently, if nodes $i$ and
$j$ are not connected in $G$, they have no common neighbors in $A$, and hence
$\phi_{i}(X)$ and $\phi_{j}(X)$ depend on disjoint treatments and are
independent. Similarly, given disjoint subsets $S_{1}$ and $S_{2}$ with no
edges between them in $G$, it follows that the sets $\\{\phi_{i}(X):i\in
S_{1}\\}$ and $\\{\phi_{i}(X):i\in S_{2}\\}$ also depend on disjoint
treatments and are are independent, making $G$ the dependency graph of
$\phi_{1}(X),\ldots,\phi_{N}(X)$. ∎

###### Proof of Theorem 2.

Let $\Sigma$ equal the covariance matrix of $\sqrt{N}T(X)$, which may be non-
invertible. We will show that $\varpi^{T}T(X)$ is asymptotically normal for
every non-zero vector $\varpi\in\mathbb{R}^{L}$, from which multivariate
normality of $T(X)$ will follow by Cramer-Wold device666while $\Sigma$ may be
non-invertible, $T(X)$ is normal in the sense that its characteristic function
exists and equals that of a normal distribution in a subspace of
$\mathbb{R}^{K}$. In turn, the delta method will imply that $h(T(X))$ is
asymptotically normal after centering and rescaling.

Given $\varpi\in\mathbb{R}^{L}$, it can be seen from condition A.3 that

$\varpi^{T}(T(X)-\mathbb{E}T)=\frac{1}{N}\sum_{i=1}^{N}\varpi^{T}(\phi_{i}(X;\theta_{i})-\mathbb{E}\phi_{i}(X;\theta_{i}))$

Dividing by
$\sqrt{\operatorname{Var}(\varpi^{T}T(X))}=\sqrt{\varpi^{T}\Sigma\varpi/N}$
yields

$\displaystyle\frac{\varpi^{T}(T-\mathbb{E}T)}{\sqrt{\operatorname{Var}(\varpi^{T}T)}}$
$\displaystyle=\frac{\varpi^{T}(T-\mathbb{E}T)}{\sqrt{\varpi^{T}\Sigma\varpi}}\cdot\sqrt{N}$
$\displaystyle=\sum_{i=1}^{N}\frac{\varpi^{T}(\phi_{i}(X;\theta_{i})-\mathbb{E}\phi_{i}(X;\theta_{i}))}{N}\cdot\sqrt{\frac{N}{\varpi^{T}\Sigma\varpi}}$
(S.8)

For any fixed $\varpi$ not in the nullspace of $\bar{\Sigma}$, (S.8) can be
seen to equal a sum of $N$ zero-mean random variables that are bounded by
$O(1/\sqrt{N})$, where we have used that $\phi_{i}$ is bounded, and that
$\varpi^{T}\Sigma\varpi\rightarrow\varpi^{T}\bar{\Sigma}\varpi$, which is
bounded away from zero for $\varpi$ not in the nullspace. As the variables in
the sum are scaled and centered functions of $\\{\phi_{i}(X;\theta_{i})\\}$,
their dependency graph has maximum degree $d_{\max}^{2}$ by Lemma 2. As a
result, for $\varpi\notin\operatorname{nullspace}(\bar{\Sigma})$, Theorem 3
can be applied to show that

$\sqrt{N}\frac{\varpi^{T}(T-\mathbb{E}T)}{\sqrt{\varpi^{T}\Sigma\varpi}}\rightarrow_{d}N(0,1)$
(S.9)

To apply Cramer-Wold, we must show that
$\sqrt{N}\cdot\varpi^{T}(T-\mathbb{E}T)$ converges in distribution to
$N(0,\varpi^{T}\bar{\Sigma}\varpi)$ for all vectors $\varpi$. To show this
result for $\varpi\notin\operatorname{nullspace}(\bar{\Sigma})$, (S.9) implies
that

$\sqrt{N}\cdot\varpi^{T}(T-\mathbb{E}T)\rightarrow_{d}N(0,\varpi^{T}\Sigma\varpi)\rightarrow_{d}N(0,\varpi^{T}\bar{\Sigma}\varpi),$

as $\Sigma\rightarrow\bar{\Sigma}$. To show it for
$\varpi\in\operatorname{nullspace}(\bar{\Sigma})$, observe that

$\displaystyle N\operatorname{Var}(\varpi^{T}T)$
$\displaystyle=\varpi^{T}\Sigma\varpi$
$\displaystyle\rightarrow\varpi^{T}\bar{\Sigma}\varpi=0,$

and hence that
$\sqrt{N}\varpi^{T}(T-\mathbb{E}T)\rightarrow_{d}N(0,\varpi^{T}\bar{\Sigma}\varpi)$.

As a result, by Cramer-Wold it follows that $\sqrt{N}(T-\mathbb{E}T)$ is
asymptotically $N(0,\bar{\Sigma})$. By the delta method (Theorem 2), this
implies that $\sqrt{N}(h(T)-h(\mathbb{E}T))$ is asymptotically normal, with
non-vanishing variance by assumption A.5. Letting
$U=\max_{\vartheta}\left[\bar{w}^{T}\vartheta+z_{1-\alpha}\sqrt{\operatorname{Var}(w(X)^{T}\vartheta)}\right]$,
it follows that

$\displaystyle\mathbb{P}\left(w(X)^{T}\theta\geq U\right)$
$\displaystyle\hskip
14.22636pt=\mathbb{P}\left(w(X)^{T}\theta\geq\max_{\vartheta}\left[\bar{w}^{T}\vartheta+z_{1-\alpha}\sqrt{\operatorname{Var}(w(X)^{T}\vartheta)}\right]\right)$
$\displaystyle\hskip
14.22636pt\leq\mathbb{P}\left(w(X)^{T}\theta\geq\bar{w}^{T}\theta+z_{1-\alpha}\sqrt{\operatorname{Var}(w(X)^{T}\theta)}\right)$
$\displaystyle\hskip
14.22636pt=\mathbb{P}\left(\frac{w(X)^{T}\theta-\bar{w}^{T}\theta}{\sqrt{\operatorname{Var}(w(X)^{T}\theta)}}\geq
z_{1-\alpha}\right)$ $\displaystyle\hskip
14.22636pt=\mathbb{P}\left(\frac{h(T(X))-h(\mathbb{E}T)}{\sqrt{\operatorname{Var}[h(T(X))]}}\geq
z_{1-\alpha}\right)$ (S.10) $\displaystyle\hskip
14.22636pt\rightarrow\alpha+o(1),$ (S.11)

where (S.10) holds because $h(T(X))=w(X)^{T}\theta$,
$h(\mathbb{E}T)=\bar{w}^{T}\theta$, and
$N\operatorname{Var}(w(X)^{T}\theta)\rightarrow
N\operatorname{Var}[h(T(X))]>0$, and (S.11) follows from the asymptotic non-
degenerate normality of $\sqrt{N}h(T(X))$.

As this shows that $w^{T}\theta\leq U$ with probability asymptotically at
least $1-\alpha$, it follows that

$w^{T}Y-w^{T}\theta\geq w^{T}Y-U$

holds with the same probability. This proves the lower bound of the prediction
interval for $w^{T}(Y-\theta)$. The proof of the upper bound proceeds
analogously.

∎

###  B.3 Proof of Corollaries 1 and 2

To prove Corollaries 1 and 2, we must show that $w(X)^{T}\theta=h(T(X))$ for
some smooth function $h$, and verify that $\bar{w}^{T}\theta=h(\mathbb{E}T)$,
as required to apply Proposition 2.

###### Proof of Corollary 1.

Let $M\in\mathbb{R}^{d\times d}$ and $v\in\mathbb{R}$ equal

$\displaystyle M$ $\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\xi(X)\xi(X)^{T},$
$\displaystyle v$ $\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\theta_{i}\xi(X),$

It can be seen that $T(X)=N^{-1}\sum_{i}\phi_{i}(X)$ equals

$T(X)=\left[\begin{array}[]{c}\operatorname{vec}(M)\\\ v\end{array}\right],$

where $\operatorname{vec}(M)$ denotes the vector in which the columns of $M$
are stacked on top of each other. Since $w(X)^{T}\theta$ satisfies

$w(X)^{T}\theta=M^{-1}v,$

it follows that $w(X)^{T}\theta$ can be written as $h(T(X))$ for a smooth
function $h$. From this and the assumptions of Corollary 1, condition A.3 of
Proposition 2 is fulfilled. It can also be seen that
$\bar{w}^{T}\theta=h(\mathbb{E}T)$. As the remaining conditions of Proposition
2 are assumed by the corollary, the proposition follows.

∎

###### Proof of Corollary 2.

Let $v^{1}$, $v^{2}$, $v^{3}$, $v^{4}\in\mathbb{R}^{K}$ be given by

$\displaystyle v^{(1)}_{k}$ $\displaystyle=n_{k}/N,$ $\displaystyle
k=1,\ldots,K$ $\displaystyle v^{(2)}_{k}$
$\displaystyle=\frac{1}{n_{k}}\sum_{i\in\Pi_{k}}W_{i},$ $\displaystyle
k=1,\ldots,K$ $\displaystyle v^{(3)}_{k}$
$\displaystyle=\frac{1}{n_{k}}\sum_{i\in\Pi_{k}}\theta_{i},$ $\displaystyle
k=1,\ldots,K$ $\displaystyle v^{(4)}_{k}$
$\displaystyle=\frac{1}{n_{k}}\sum_{i\in\Pi_{k}}W_{i}\theta_{i},$
$\displaystyle k=1,\ldots,K$

It can be seen that $T(X)=N^{-1}\sum_{i}\phi_{i}(X)$ equals

$T(X)=\left[\begin{array}[]{c}v^{(1)}\\\ v^{(2)}\\\ v^{(3)}\\\
v^{(4)}\end{array}\right]$

It can be seen that $w(X)^{T}\theta$ can be written as $h(T(X))$ for a smooth
function $h$, by the following steps

$\displaystyle w(X)^{T}\theta$
$\displaystyle=\sum_{k=1}^{K}\frac{n_{k}}{N}\left[\sum_{i\in\Pi_{k}}\frac{W_{i}\theta_{i}}{n_{k1}}-\sum_{i\in\Pi_{k}}\frac{(1-W_{i})\theta_{i}}{n_{k0}}\right]$
$\displaystyle=\sum_{k=1}^{K}\frac{n_{k}}{N}\left[\sum_{i\in\Pi_{k}}W_{i}\theta_{i}\left(\frac{1}{n_{k1}}+\frac{1}{n_{k0}}\right)-\sum_{i\in\Pi_{k}}\frac{\theta_{i}}{n_{k0}}\right]$
$\displaystyle=\sum_{k=1}^{K}v_{k}^{(1)}\left[v_{k}^{(4)}\left(\frac{1}{v_{k}^{(2)}}+\frac{1}{1-v_{k}^{(2)}}\right)-\frac{v_{k}^{(3)}}{1-v_{k}^{(2)}}\right]$
$\displaystyle=h(T(X))$

From this and the assumptions of Corollary 2, condition A.3 of Proposition 2
is fulfilled. It can also be seen that $\bar{w}^{T}\theta=h(\mathbb{E}T)$. As
the remaining conditions of Proposition 2 are assumed by the corollary, the
proposition follows.

∎

###  B.4 An Alternate Prediction Interval for $\beta^{\text{adj}}$

Proposition 3 gives an alternative prediction interval for
$\beta^{\text{adj}}$, whose validity holds under different conditions than
those assumed by Proposition 2. In particular, here we assume that treatments
are assigned by sampling without replacement, and we do not require the
variance of $\sqrt{N}\cdot w(X)^{T}\theta$ to be non-vanishing.

######  Proposition 3.

Given constants $d_{\max}$, $b_{\min}>0$, and $\rho\in(0,1)$, assume a
sequence of experiments in which $N\rightarrow\infty$, where treatment is
assigned by sampling $N_{1}$ units without replacement, with
$N_{1}/N\rightarrow\rho$; the adjacency matrix $A$ has in-degree and out-
degree bounded by $d_{\max}$; $\mathbb{E}[(Z_{i}-\zeta_{k(i)})^{2}]\geq
b_{\min}$ for a non-vanishing fraction of units; and $\theta\in\\{0,1\\}^{N}$.

Let $\hat{\beta}^{{\text{adj}}}$ denote the estimate

$\hat{\beta}^{\text{adj}}=\left(\sum_{i=1}^{N}(Z_{i}-\zeta_{k(i)})^{2}\right)^{-1}\sum_{i=1}^{N}(Z_{i}-\zeta_{k(i)})Y_{i},$
(S.12)

and let $a(X)\in\mathbb{R}$ and $v(X)\in\mathbb{R}^{N}$ be given by

$a(X)=\left(\frac{1}{N}\sum_{i=1}^{N}(Z_{i}-\zeta_{k(i)})^{2}\right)^{-1}\text{
and }v_{i}(x)=\frac{Z_{i}-\zeta_{k(i)}}{N},i=1,\ldots,N,$

and let
$\bar{a}=\left(\mathbb{E}\left[\frac{1}{N}\sum_{i}(Z_{i}-\zeta_{k(i)})^{2}\right]\right)^{-1}$.
Let $U$ and $L$ equal

$\displaystyle U$ $\displaystyle=\ \max_{\theta}\ \
a(X)\mathbb{E}v(X)^{T}\theta+\bar{a}z_{1-\frac{\alpha}{2}}\max\left\\{\sqrt{\operatorname{Var}[v(X)^{T}\theta]},\frac{\log
N}{N}\right\\}$ (S.13) $\displaystyle L$ $\displaystyle=\ \min_{\theta}\ \
a(X)\mathbb{E}v(X)^{T}\theta-\bar{a}z_{1-\frac{\alpha}{2}}\max\left\\{\sqrt{\operatorname{Var}[v(X)^{T}\theta]},\frac{\log
N}{N}\right\\}.$ (S.14)

Then $\beta^{{\text{adj}}}$ is contained by the interval
$\hat{\beta}^{\text{adj}}-[U,L]$, with probability asymptotically lower
bounded by $(1-\alpha)$.

Proposition 3 contains two technicalities. First, it requires that
$\mathbb{E}[(Z_{i}-Z_{k(i)})^{2}]\geq b_{\min}$ for a non-vanishing fraction
of units. This limits degenerate cases, such as when a propensity class
contains only one unit. Second, the interval width is lower bounded by
$\frac{\log N}{N}$ (which is much smaller than the usual $N^{-1/2}$ rate).
This ensures coverage in the event that asymptotic normality does not hold for
$v(X)^{T}\theta$, such as when $\sum_{i}\theta_{i}=O(1)$.

The equations (S.13)-(S.14) can be solved by the methods discussed in Section
3.3.

####  B.4.1 Proof of Proposition 3

Our proof of Proposition 3 will use the following helper lemmas: Lemma 3,
which states that $a(X)\rightarrow_{p}\bar{a}$; and Lemma 4, which gives a
central limit theorem for $(v(X)-\bar{v})^{T}\theta$, and also gives an
expression for its variance.

######  Lemma 3.

Let the assumptions and definitions of Proposition 3 hold. Then $a(X)$
converges in probability to $\bar{a}$.

######  Lemma 4.

Let the assumptions and definitions of Proposition 3 hold. Let
$\bar{\theta}_{\Pi_{k(j)}}$ denote the average of $\theta$ over the propensity
class of unit $j$, and let $u_{1},\ldots,u_{N}$ be given by

$u_{i}=\frac{N_{1}}{N}\sum_{j=1}^{N}(\theta_{j}-\bar{\theta}_{\Pi_{k(j)}})A_{ji},$

and let $\sigma_{u}^{2}$ be given by

$\frac{1}{N-1}\sum_{i}(u_{i}-\bar{u})^{2},$

where $\bar{u}=\frac{1}{N}\sum_{i}u_{i}$. Then the variance of
$v(X)^{T}\theta$ is given by

$\operatorname{Var}(v(X)^{T}\theta)=\frac{1}{N_{1}}\frac{N_{0}}{N}\sigma_{u}^{2},$

and if $\sigma_{u}^{2}\geq N_{1}^{-1}\log N$, it holds that

$\frac{(v(X)-\bar{v})^{T}\theta}{\sqrt{\operatorname{Var}(v(X)^{T}\theta)}}$

is asymptotically normal distributed.

The proof of Lemma 3 will use the Azuma-Hoeffding (or McDiarmid’s) inequality
[Thm 6.2]boucheron2013concentration, which states that given independent
variables $X=(X_{1},\ldots,X_{N})$, and a function satisfying the bounded
difference property

$|f(X)-f(X^{\prime})|\leq c_{i}\text{ if }X_{j}-X_{j}^{\prime}\text{ for all
}j\neq i,$

it holds that $\mathbb{P}\left(|f(X)-\mathbb{E}f|>t\right)\leq
2\exp\left(-2t^{2}/\sum_{i}c_{i}^{2}\right)$.

We first prove Proposition 3, and then give proofs for Lemmas 3, and 4.

###### Proof of Proposition 3.

Let $\sigma_{u}^{2}$ be defined as in the statement of Lemma 4, and divide the
sequence of experiments assumed by Proposition 3 into two subsequences: (i)
those for which $\sigma_{u}^{2}>N_{1}^{-1}\log N$, and (ii) those for which
$\sigma_{u}^{2}\leq N_{1}^{-1}\log N$. Let $a$ and $v$ abbreviate $a(X)$ and
$v(X)$.

To prove the proposition, we will show that the probability that
$av^{T}\theta$ exceeds the upper bound of the prediction interval is
asymptotically bounded by $\alpha$ for both subsequences (i) and (ii). For
subsequence (i), this probability satisfies

$\displaystyle\mathbb{P}\left(av^{T}\theta\geq\max_{\theta}\left[a\bar{v}^{T}\theta+z_{1-\alpha}\bar{a}\sqrt{\operatorname{Var}(v^{T}\theta)}\right]\right)$
$\displaystyle\leq\mathbb{P}\left(av^{T}\theta\geq
a\bar{v}^{T}\theta+z_{1-\alpha}\bar{a}\sqrt{\operatorname{Var}(v^{T}\theta)}\right)$
$\displaystyle=\mathbb{P}\left(\frac{a(v-\bar{v})^{T}\theta}{\bar{a}\sqrt{\operatorname{Var}(v^{T}\theta)}}\geq
z_{1-\alpha}\right)$ $\displaystyle=\alpha+o(1)$ (S.15)

where (S.15) follows from Slutsky’s theorem as $a/\bar{a}\rightarrow_{p}1$ by
Lemma 3, and

$\frac{(v-\bar{v})^{T}\theta}{\sqrt{\operatorname{Var}(v^{T}\theta)}}$

converges in distribution to a standard normal by Lemma 4, and hence their
product is asymptotically normal as well.

We now show the same for subsequence (ii). Let $r_{n}=(\log N)^{2}/N^{2}$. It
holds that

$\displaystyle\mathbb{P}\left(av^{T}\theta\geq\max_{\theta}\left[a\bar{v}^{T}\theta+z_{1-\alpha}\bar{a}\sqrt{r_{n}}\right]\right)$
$\displaystyle\leq\mathbb{P}\left(av^{T}\theta\geq
a\bar{v}^{T}\theta+z_{1-\alpha}\bar{a}\sqrt{r_{n}}\right)$
$\displaystyle=\mathbb{P}\left(\frac{a(v-\bar{v})^{T}\theta}{\bar{a}\sqrt{r_{n}}}\geq
z_{1-\alpha}\right)$ $\displaystyle=o(1)$ (S.16)

where (S.16) follows from Slutsky’s theorem as $a/\bar{a}\rightarrow_{p}1$ by
Lemma 3, and

$\frac{(v-\bar{v})^{T}\theta}{\sqrt{r_{n}}}\rightarrow_{p}0$

by Chebychev’s inequality as $r_{n}\gg\operatorname{Var}(v(X)^{T}\theta)$ for
subsequence (ii), implying that their product converges to zero as well.

Combining (S.15) and (S.16) implies that

$\mathbb{P}\left(av^{T}\theta\geq\max_{\theta}\left[a\bar{v}^{T}\theta+z_{1-\alpha}\bar{a}\sqrt{\max\\{r_{n},\operatorname{Var}(v^{T}\theta)\\}}\right]\right)\leq\alpha+o(1)$
(S.17)

for both subsequences (i) and (ii). Substituting
$av^{T}\theta=\hat{\beta}^{\text{adj}}-\beta^{\text{adj}}$, and
$U=\max_{\theta}\left[a\bar{v}^{T}\theta+z_{1-\alpha}\bar{a}\sqrt{\max\\{r_{n},\operatorname{Var}(v^{T}\theta)\\}}\right]$
into (S.17) implies

$\mathbb{P}(\hat{\beta}^{\text{adj}}-\beta^{\text{adj}}\leq U)\geq
1-\alpha+o(1),$

and hence that $\beta^{\text{adj}}\geq\hat{\beta}^{\text{adj}}-U$ with
probability at least $1-\alpha+o(1)$ as well. This is the lower bound of the
PI for $\beta^{\text{adj}}$. The proof of the upper bound proceeds
analogously.

∎

###### Proof of Lemma 3.

We will couple $X$ with a vector $\tilde{X}\in\\{0,1\\}^{N}$ whose entries are
i.i.d. Bernoulli($N_{1}/N$). To do this, let $\tilde{N}_{1}$ denote the number
of selected units in $\tilde{X}$,

$\tilde{N}_{1}=\sum_{i}\tilde{X}_{i},$

and let $X$ be generated from $\tilde{X}$ in the following manner. If
$\tilde{N}_{1}>N_{1}$, randomly sample $\tilde{N}_{1}-N_{1}$ entries without
replacement from $\\{i:\tilde{X}_{i}=1\\}$, and change their values to 0; if
$\tilde{N}_{1}\leq N_{1}$, randomly sample and alter $N_{1}-\tilde{N}_{1}$
entries from $\\{i:\tilde{X}_{i}=0\\}$ in likewise fashion. The resulting
vector has exactly $N_{1}$ entries equal to 1, and is uniformly drawn from the
set of all such vectors; hence, the resulting vector has the required
distribution for $X$.

Let $b(x)=a^{-1}(x)$, so that $b(X)$ equals

$b(X)=\sum_{i=1}^{N}(Z_{i}-\zeta_{k(i)})^{2}.$

To prove the proposition, we will show 3 steps. First, we show that $b$
satisfies a bounded difference property, so that if $x$ and $x^{\prime}$ are
vectors in $\\{0,1\\}^{N}$ differing in a single entry, then

$|b(x)-b(x^{\prime})|\leq\frac{d_{\max}^{2}}{N}.$ (S.18)

Second, we will use (S.18) to show that

$a(\tilde{X})=\bar{a}(1+o_{P}(1)).$ (S.19)

Third, we will show that

$a(X)=a(\tilde{X})(1+o_{P}(1)).$ (S.20)

Together (S.19) and (S.20) imply that $a(X)=\bar{a}(1+o_{P}(1))$, proving the
claim.

We first show (S.18). It can be seen that $Z=AX$, and that the vector $v(X)$
can be written as

$v(X)=\frac{1}{N}(I-P)AX,$ (S.21)

where $P$ is a projection matrix given by

$P_{ij}=\frac{1\\{k(i)=k(j)\\}}{|\Pi_{k(i)}|}.$ (S.22)

Similarly, $b(X)$ can be written as

$b(X)=\frac{1}{N}X^{T}A^{T}(I-P)AX,$

where we have used the identity that $(I-P)^{T}(I-P)=I-P$ when $P$ is a
projection matrix. Letting $X^{\prime}=X+\delta$, it thus follows that

$\displaystyle b(X)-b(X^{\prime})$
$\displaystyle=\frac{1}{N}(X+\delta)^{T}A^{T}(I-P)A(X+\delta)-\frac{1}{N}X^{T}A^{T}(I-P)AX$
$\displaystyle=\frac{1}{N}2\delta^{T}A^{T}(I-P)AX+\frac{1}{N}\delta^{T}A^{T}(I-P)A\delta$
$\displaystyle=2\delta^{T}A^{T}v(X)+\frac{1}{N}\delta^{T}A^{T}(I-P)A\delta.$

For $X$ and $X^{\prime}$ differing in only one entry, $\delta$ is equal to the
vector of all zeros except one entry equal in $[-1,\,1]$, and it follows that

$\displaystyle|b(X)-b(X^{\prime})|$
$\displaystyle\leq|2\delta^{T}A^{T}v(X)|+\left|\frac{1}{N}\delta^{T}A^{T}(I-P)A\delta\right|$
$\displaystyle\leq\frac{d_{\max}^{2}}{N}+\frac{d_{\max}}{N},$

which proves (S.18). Here we have used that $A$ has maximum in-degree and out-
degree $d_{\max}$, and that the entries of $v(X)$ are bounded by $d_{\max}/N$.

We next show (S.19). Using the bounded difference property (S.18), the Azuma-
Hoeffding inequality implies that

$\mathbb{E}\left(|b(\tilde{X})-\mathbb{E}b|\geq
t\right)\leq\exp\left(\frac{-t^{2}N}{d_{\max}^{2}}\right),$ (S.23)

which implies that $b(\tilde{X})\rightarrow_{p}\mathbb{E}b(\tilde{X})$. Using
this with $\mathbb{E}b(\tilde{X})=\Omega(1)$ and
$a(\tilde{X})=1/b(\tilde{X})$, it follows that
$a(\tilde{X})=\bar{a}(1+o_{P}(1))$. This shows (S.19).

We finally show (S.20). We observe that since $\mathbb{E}\tilde{N}_{1}=N_{1}$,
Heoffding’s inequality implies that

$\mathbb{P}(|\tilde{N}_{1}-N_{1}|\geq\sqrt{N}\log N)\leq\exp(-\frac{1}{2}\log
N),$

which implies that $\|\tilde{X}-X\|_{1}=o_{p}(\sqrt{N}\log N)$. Hence by the
bounded difference property (S.18) it holds that

$|b(\tilde{X})-b(X)|=o_{P}(1),$

and hence by continuity that $a(\tilde{X})-a(X)=o_{P}(1)=o_{P}(a(X))$ as well,
proving (S.20). This completes the proof of the lemma.

∎

###### Proof of Lemma 4.

We begin with the observation that

$v(X)^{T}\theta=\frac{1}{N}\sum_{i=1}^{N}(Z_{i}-\zeta_{k(i)})\theta_{i},$

and hence can be written using projection matrix $P$ defined by (S.22) as

$\displaystyle v(X)^{T}\theta$ $\displaystyle=\frac{1}{N}\theta^{T}(I-P)AX$
$\displaystyle=\frac{1}{N_{1}}\frac{N_{1}}{N}\sum_{i=1}^{N}\left(\sum_{j=1}^{N}(\theta_{j}-\bar{\theta}_{\Pi_{k(j)}})A_{ji}\right)X_{i}$
$\displaystyle=\frac{1}{N_{1}}\sum_{i=1}^{N}u_{i}X_{i},$ (S.24)

This can be seen to equal the average of $N_{1}$ units sampled without
replacement from $u_{1},\ldots,u_{N}$, and thus has variance

$\operatorname{Var}[v(X)^{T}\theta]=\frac{1}{N_{1}}\frac{N_{0}}{N}\sigma_{u}^{2}.$

It can be seen that

$\max_{i}(u_{i}-\bar{u})^{2}=O\left(d_{\max}^{2}\right),$

and as $\sigma_{u}^{2}>N_{1}^{-1}\log N$, it holds that

$\frac{1}{\min(N_{1},N-N_{1})}\frac{\max_{i}(u_{i}-\bar{u})^{2}}{\sigma_{u}^{2}}\rightarrow
0,$

and hence the lemma follows from the CLT given by Theorem 1 [Thm.
1]li2017general. ∎

##  Appendix C Summary Information for Data Sets

Here we give tables to summarize the simulated cholera trial outcomes released
by Perez-Heydrich et al., (2014), insurance takeup by farmers in the
experiment of Cai et al., (2015), and wristband-wearing decisions made by
students in the school interventions described in Paluck et al., (2016). An
aggregate table of the actual outcomes for the cholera trial is given by Table
3.

Vaccination Rate | Cases, placebo | Cases, vaccine
---|---|---
0-25% | 50/111 (45%) | 37/144 (26%)
25-50% | 73/282 (25%) | 81/464 (17%)
50-75% | 24/183 (13%) | 42/480 (9%)
75-100% | 2/13 (15%) | 4/110 (4%)
Total | 149/589 (25%) | 164/1198 (14%)
Table S5: Outcomes for simulated cholera trial dataset

A comparison of Tables S5 and 3 shows that the actual population is much
larger than the simulated data set ($N$=74K vs 1.7K), and has much lower rates
of cholera incidence (0.003 vs 0.17). As the simulated outcomes were fit to
the actual data as a function of treatment (including neighborhood
vaccination) and observed covariates (age and distance to water), we speculate
that the covariate distributions were shifted to a higher-risk subpopulation.

Perez-Heydrich et al., (2014) gives some information regarding the simulation
parameters, stating that the outcome for unit $j$ in neighborhood $i$ was
determined by a logistic regression model

$\text{log
odds}P(Y_{ij}=1)=0.5-0.788a-2.953\alpha_{i}-0.098X_{ij1}-0.145X_{ij2}+0.35a\alpha_{i},$

where $a$ is the unit treatment; $\alpha_{i}$ was the neighborhood vaccination
rate if $j$’s treatment equals $a$; $X_{ij1}$ and $X_{ij2}$ denote $j$’s age
(in decades) and distance to the nearest river (in km), which were randomly
generated. Treatments were assigned i.i.d. Bernoulli(2/3). Units were
generated by simulating the covariate $X$, and then selecting units to
participate in the trial by a logistic regression model

$\text{log
odds}P(\text{participate}_{ij})=0.2727-0.0387X_{ij1}+0.2179X_{ij2}+b_{i},$

where $b_{i}$ is a random neighborhood-level effect.

Table S6 gives summary statistics for the experiment of Cai et al., (2015). In
this experiment, units were randomized not only by first or second round and
by high or low information, but also whether or not the second round units
were shown a list of names corresponding to people who had purchased the
insurance product in the first round. When studying the effects of informal
communication between rounds, the analysis of Cai et al., (2015) is restricted
to the “no info” group which excludes those units who were shown such a list,
and we follow their example in our analysis as well.

Exposure | Insurance Takeup
---|---
(# of high info friends) |
0 | 116/293 (40%)
1 | 196/454 (43%)
2 | 96/171 (56%)
3 | 21/46 (46%)
4 | 5/8 (63%)
5 | 1/1 (100%)
Total | 435/973 (45%)
Table S6: Outcomes for insurance purchases of farmers, for “no info” group
used by Cai et al., (2015)

Table S7 gives summary statistics for the experiment of Paluck et al., (2016).
For this experiment, the analysis of the wristband outcomes in Paluck et al.,
(2016) and Aronow and Samii, (2017) restricts to units who had positive
probability of being assigned to one of the four treatment outcomes that were
considered, meaning that they were eligible units with at least one eligible
friend, and we follow their example in our analysis as well.

Treated? |  $\geq 1$ Treated Friend? | Outcome
---|---|---
No | No | 12/158 (7%)
No | Yes | 61/374 (16%)
Yes | No | 53/201 (26%)
Yes | Yes | 103/333 (31%)
Total | 229/1066 (21%)
Table S7: Outcomes for school intervention

apalike bibfile