# Estimation of Causal Effects Under K-Nearest Neighbors Interference

Samirah Alzubaidi
Department of Mathematics, Al-Qunfudah University College,
Umm Al-Qura University
and
Michael J. Higgins
Department of Statistics, Kansas State University

###### Abstract

Considerable recent work has focused on methods for analyzing experiments
which exhibit treatment interference—that is, when the treatment status of one
unit may affect the response of another unit. Such settings are common in
experiments on social networks. We consider a model of treatment
interference—the $K$-nearest neighbors interference model (KNNIM)—for which
the response of one unit depends not only on the treatment status given to
that unit, but also the treatment status of its $K$ “closest” neighbors. We
derive causal estimands under KNNIM in a way that allows us to identify how
each of the $K$-nearest neighbors contributes to the indirect effect of
treatment. We propose unbiased estimators for these estimands and derive
conservative variance estimates for these unbiased estimators. We then
consider extensions of these estimators under an assumption of no weak
interaction between direct and indirect effects. We perform a simulation study
to determine the efficacy of these estimators under different treatment
interference scenarios. We apply our methodology to an experiment designed to
assess the impact of a conflict-reducing program in middle schools in New
Jersey, and we give evidence that the effect of treatment propagates primarily
through a unit’s closest connection.

Keywords: Causal inference under interference; Network effects; Randomization
inference; Peer effects; Spillover

##  1 Introduction

In randomized experiments, assessing causal effects requires special care when
the treatment condition assigned to one unit is allowed to affect the response
of other units. Under this setting, a unit’s outcome is not only influenced by
its own treatment status—a direct effect of treatment—but may also be
influenced by other units’ treatments—an indirect effect (Sobel, 2006;
Rosenbaum, 2007; Hudgens and Halloran, 2008). In causal inference terminology,
these experiments exhibit treatment interference, treatment spillover, network
effects, or peer effects. This interference is especially common in settings
with a social factor where units are allowed to interact with each other, for
example, in studies on social media networks.

Experiments exhibiting treatment interference violate the stable unit
treatment value assumption (SUTVA)—a foundational assumption of traditional
causal inference methods (Splawa-Neyman et al., 1990; Rubin, 1980; Holland,
1986; Imbens and Rubin, 2015). In particular, SUTVA requires that the
treatment assigned to one unit affects only the outcome of that unit and does
not affect the outcomes of other units. The presence of interference may
complicate statistical analysis and lead to inaccurate inference if not
carefully taken into account (Sobel, 2006).

Traditionally, the effect of social influence and interaction between units
has been viewed as a nuisance prohibiting accurate estimation of the direct
effect of treatment. Considerable work has focused on designing experiments to
mitigate the effect of treatment interference, for example, through clustering
units that are likely to interact with each other and assigning treatment to
these clusters instead of individual units (Ugander et al., 2013; Gui et al.,
2015; Eckles et al., 2016). However, recent applications—for example, studies
conducted on social media platforms and those evaluating the efficacy of
vaccination strategies—have giving rise to studies in which quantifying and
estimating interference effects is of primary interest (Hudgens and Halloran,
2008; Aronow and Samii, 2017; Forastiere et al., 2020; Sussman and Airoldi,
2017; Toulis and Kao, 2013; Alzubaidi and Higgins, 2022).

Methods for estimating indirect effects often begin by classifying the
interaction through defining an exposure mapping on the units under study—a
network where nodes represent units under study and edges between vertices
indicate that the corresponding units may interact with each other. There may
additionally be an interaction measure computed between each pair of units in
the exposure mapping indicating the strength of that interaction. Common
settings include partial interference, in which units are partitioned into
groups and treatment interference can persist within groups but not across
groups (Sobel, 2006; Rosenbaum, 2007; Hudgens and Halloran, 2008; Tchetgen and
VanderWeele, 2012; Basse and Feller, 2018), and neighborhood interference, in
which interference is not allowed to persist outside of a small neighborhood
of a unit (Sussman and Airoldi, 2017). Once the nature of the treatment
interference is specified, estimators can then be defined to estimate both
direct and indirect effects (Aronow and Samii, 2017).

We build on this literature by developing estimators for these treatment
effects under the $K$-nearest neighbors interaction model (KNNIM) (Alzubaidi
and Higgins, 2022). In this model, the treatment given to one unit may
interfere with response another unit if the first unit is one of the $K$
“closest” individuals to the second unit with respect to a given interaction
measure. Quantifying $K$-nearest neighbors effects may help researchers tease
out peer effects induced by, for example, interactions between best friends,
spouses, siblings, and/or close colleagues. Additionally, this model has
several appealing properties. First, this model allows for users with stronger
interactions to produce larger indirect effects than users with weaker
interactions, and will ignore potential indirect effects due to dilapidated,
but technically present, connections (e.g. Facebook friends that no longer
interact with each other). Second, the marginal and joint probabilities for
possible treatment exposures have closed-form expressions under common
experimental settings, allowing for unbiased estimation of treatment effects
and precise estimation of standard errors. Finally, KNNIM may be effective in
estimating indirect effects in the presence of non-transitive influencer
effects—effects induced by users that have influence over a large number of
individuals, but may themselves only directly interact with a handful of
individuals.

In this paper, using a potential outcomes approach, we define causal estimands
for direct and $K$-nearest-neighbor indirect effects. We then derive Horvitz-
Thompson estimators (Horvitz and Thompson, 1952) for these estimands that are
unbiased given exact marginal and joint probabilities for possible treatment
exposures, and derive conservative standard errors for these estimators. We
provide a closed-form solution to compute these exposure probabilities under
completely-randomized and Bernoulli-randomized experimental designs. We then
demonstrate how these estimators may have significantly stronger precision
under an assumption of no interaction between direct and indirect effects. We
conclude by showcasing the efficacy of these methods via simulation and
application to a field experiment designed to reduce conflict among middle
school students in New Jersey (Paluck et al., 2016; Aronow and Samii, 2017).

The paper is organized as follows. Section 2 sets up the notation and
preliminaries. The $K$-nearest neighbors interference model is provided in
Section 3. Section 4 defines causal effects under KNNIM. Proposed unbiased
estimators with the derived properties are given in Section 5. Section 6
provides estimators with stronger precision under an assumption of no
interaction between direct and indirect effects. Section 7 presents variance
estimation. Simulation studies and real data analysis to demonstrate the
proposed methods are provided in Sections 8, 9, and 10. We conclude in Section
11.

###  1.1 Motivating Example: Reducing Conflict in Schools

To motivate our approach, we refer to a recent randomized field experiment to
assess the efficacy of an anti-conflict program on middle schools in New
Jersey (Paluck et al., 2016). The experiment was explicitly designed to
determine whether benefits of the program can be propagated through social
interactions between students. In particular, the program was implemented
through randomly selecting “seed students” from a pre-selected list of
eligible seeds within all participating schools. Seed students were asked to
actively participate in and advocate for the anti-conflict program. School-
wide benefits of the program were then propagated through interactions with
these seed students. Analysis was performed only on students that were
eligible to be seeds ($N=$ 2,451). Overall, the program showed a statistically
significant improvement in anti-conflict behaviors among students. For details
of the randomization and analysis, please refer to Paluck et al. (2016) and
Aronow and Samii (2017).

Notably, to assess potential pathways for treatment interference, students
were asked to identify, in order, the 10 other students that they spent the
most time with during the previous few weeks. This yields a unique dataset in
which the strength of the interaction between two individuals under study is
explicitly recorded. Hence, an interference model that allows for direct
incorporation of these relative strengths of the interactions, such as KNNIM,
can yield additional insights into the structure of the treatment
interference. For example, in Section 10 , we give evidence that the indirect
effect of the treatment was propagated primarily through the student with the
“strongest connection” to the seed student.

##  2 Notation and Preliminaries

Consider an experiment on $N$ units where each unit is assigned either a
treatment status or a control status. The Neyman-Rubin Causal Model (NRCM)
(Splawa-Neyman et al., 1990; Rubin, 1974; Holland, 1986) is a commonly-assumed
model of response for causal inference. Under this model, the observed
response of a unit is determined by the treatment status given to that unit
and the potential outcomes for that unit—the hypothetical responses of that
unit under the possible treatment statuses. A fundamental assumption of this
model is the stable-unit treatment value assumption (SUTVA), which requires
that there is only a single version of each treatment status and the response
of a unit is unaffected by the treatment status of any other unit (Splawa-
Neyman et al., 1990; Rubin, 1980; Holland, 1986; Imbens and Rubin, 2015). Of
note, SUTVA is violated for experiments that exhibit treatment interference
(Cox, 1958; Rubin, 1980). Failing to account for violations of SUTVA can lead
to inaccurate treatment effect estimates (Sobel, 2006).

Under interference, the effect of a treatment on a unit may occur through
direct application of the treatment to that unit, indirectly through
application of treatment to units that affect the response of the original
unit, or both (Hudgens and Halloran, 2008). When interference is allowed to
take completely arbitrary forms, treatment effect estimates are often
estimated with very low power, or may be unidentifiable (Aronow and Samii,
2017). Thus, to make progress on this problem, researchers often make
assumptions restricting how interference can propagate across units (Toulis
and Kao, 2013; Aronow and Samii, 2017; Ugander et al., 2013; Sussman and
Airoldi, 2017).

To elucidate these ideas, we extend the potential outcomes framework to
account for both direct and indirect treatment components. Let $W_{i}$ be a
treatment indicator for unit $i$—that is, $W_{i}=1$ if unit $i$ is given
treatment and $W_{i}=0$ otherwise. Let
$y_{i}(\mathbf{W})=y_{i}(W_{i},\mathbf{W}_{-i})$ denote the potential outcome
of unit $i$ under treatment allocation
$\mathbf{W}=(W_{1},W_{2},\ldots,W_{N})\in\\{0,1\\}^{N}$, where unit $i$ is
given treatment $W_{i}$, and the remaining treatment statuses are allocated
according to $\mathbf{W}_{-i}$. Responses $Y_{i}$ satisfy

$Y_{i}=\sum_{\mathbf{W}\in\\{0,1\\}^{N}}y_{i}(\mathbf{W})\mathbf{1}(\mathbf{W}^{\prime}=\mathbf{W}),$
(1)

where $\mathbf{1}(\mathbf{W}^{\prime}=\mathbf{W})$ is an indicator variable
that is equal to 1 if and only if the observed treatment status
$\mathbf{W}^{\prime}=\mathbf{W}$. That is, the response of unit $i$ only
depends on the potential outcomes of unit $i$ and the observed treatment
allocation across all $N$ units.

Without making assumptions on the amount of interference allowed in a study,
it may be impossible to estimate common causal quantities of interest in any
practical way—for example, each unit may have up to $2^{N}$ potential outcomes
when interference is unconstrained. Thus, researchers often place strong
restrictions on the nature of the treatment interference (Toulis and Kao,
2013; Aronow and Samii, 2017; Ugander et al., 2013; Sussman and Airoldi, 2017;
Alzubaidi and Higgins, 2022). This often begins by constructing an exposure
mapping $G=(V,E)$—a directed graph where each vertex $i\in V$ represents a
unit under study and an edge $\vec{ij}\in E$ indicates that the treatment
status of unit $i$ may potentially interfere with the response of unit $j$.
Each edge $\vec{ij}\in E$ may also have a weight $d(i,j)$ denoting the
strength of the the potential interference which may be observed through
studying interactions between $i$ and $j$. Stronger interference between $i$
and $j$ may either correspond to a larger or smaller value of $d(i,j)$
depending on the context of the problem—for this paper, stronger interactions
are associated with smaller values of $d(i,j)$. Under an assumed exposure
mapping $G$, for any two treatment allocations $\mathbf{W}$,
$\mathbf{W}^{\prime}$, we have that
$y_{i}(\mathbf{W})=y_{i}(\mathbf{W}^{\prime})$ if $W_{i}=W_{i}^{\prime}$ and
$W_{j}=W_{j}^{\prime}$ for all $j\in V$ such that $\vec{ji}\in E$; if
$\vec{ki}\notin E$, then treatment statuses $W_{k}$, $W^{\prime}_{k}$ may
differ without affecting equality of the potential outcomes.

Once the exposure mapping is specified, models may further restrict the nature
of interference. For example, a common assumption is that interference can
only occur if a certain number or fraction of neighbors within the exposure
mapping are given the treatment condition. However, few existing models
specify exactly which neighbors in the exposure mapping are allowed to
interfere with a unit’s response, or allow for indirect effects to differ
across neighbors. As an alternative, we consider a model where interference of
treatment on a unit $i$ is restricted to its $K$–nearest neighbors (Alzubaidi
and Higgins, 2022). This model allows neighbors with stronger interactions to
contribute larger indirect effects, and limits the ability of weakly-
interacting units to affect response.

##  3 $K$-Nearest Neighbors Interference Model

The $K$-nearest neighbors interference model (KNNIM) is a recently-proposed
model in which a unit $j$ is only allowed to interfere with the response of
unit $i$ if $j$ is within $i$’s $K$-neighborhood (Alzubaidi and Higgins,
2022). The $K$-neighborhood of unit $i$, denoted $\mathcal{N}_{iK}$, is the
set of the $K$ “closest” units to unit $i$:

$\displaystyle\mathbf{\mathcal{N}}_{iK}=\\{j:d(i,j)\leq
d(i,(K)),j=1,2,\ldots,K,~{}j\neq i\\}.$

where $d(i,(K))$ denotes the $K^{\text{th}}$ smallest value of
$(d(i,j))^{N}_{j=1}$. Ties between $d(i,j)$ may be broken arbitrarily to
ensure that $|\mathbf{\mathcal{N}}_{iK}|=K$. Define
$\mathcal{N}_{-iK}=V\setminus(i\cup N_{iK})$ as all units in $V$ that are
outside of $i$’s $K$-neighborhood. Note that the sets
$\\{i\\},\\{\mathcal{N}_{ik}\\},\\{\mathcal{N}_{-ik}\\}$ form a partition of
$V$.

Let
$\mathbf{W}=(W_{i},\mathbf{W}_{\mathcal{N}_{iK}},\mathbf{W}_{\mathcal{N}_{-iK}})$
denote treatment assignment vector for all units $N$, partitioned into the
treatment given to unit $i$, the treatments given to $i$’s $K$-nearest
neighbors, and the treatments given to all other units. For clarity, treatment
statuses in $\mathbf{W}_{\mathcal{N}_{iK}}$ and
$\mathbf{W}_{\mathcal{N}_{-iK}}$ are given in increasing order with respect to
$d(i,j)$ (e.g. the first entry of $\mathbf{W}_{\mathcal{N}_{iK}}$ is the
treatment assignment given to the nearest neighbor of unit $i$). The defining
assumption of KNNIM is as follows:

######  Assumption 1.

($K$-Neighborhood Interference Assumption (K-NIA)). The potential outcomes
$y_{i}(\mathbf{W})$ for all units $i$ satisfy

$y_{i}(W_{i},\mathbf{W}_{\mathcal{N}_{iK}},\mathbf{W}_{\mathcal{N}_{-iK}})=y_{i}(W_{i},\mathbf{W}_{\mathcal{N}_{iK}},\mathbf{W}^{\prime}_{\mathcal{N}_{-iK}}).$
(2)

In words, K-NIA ensures that the potential outcome of unit $i$ is only
affected by its own treatment status and the treatment statuses of its
$K$-nearest neighbors. The treatment statuses of units outside the
$K$-neighborhood will not affect the potential outcome of unit $i$. Thus, this
model restricts the number of potential outcomes to be $2^{K+1}$ for each
unit. For brevity, we will suppress the treatment assignment outside of the
$K$-nearest neighbors when denoting potential outcomes under KNNIM:
$y_{i}(W_{i},\mathbf{W}_{\mathcal{N}_{iK}})=y_{i}(W_{i},\mathbf{W}_{\mathcal{N}_{iK}},\mathbf{W}_{\mathcal{N}_{-iK}})$.

The choice of $K$ is ultimately left to the researcher, though large values of
$K$ may not be sufficiently restrictive to allow for reliable estimates and
inferences. A detailed description on how to choose $K$ is given in Alzubaidi
and Higgins (2022). As a crude, but useful, rule-of-thumb, we find it
practical to choose $K$ such that each exposure used in the estimation of
treatment effects is observed at least 30 times.

##  4 Causal Estimands under KNNIM

Using the potential outcomes framework and following Hudgens and Halloran
(2008), we now define causal estimands under KNNIM. We start with general
definitions of direct and indirect effects and conclude with KNNIM-specific
nearest neighbors effects.

###  4.1 Direct, Indirect, and Total Effects

The average direct effect (ADE) $\delta_{dir}$ is the average difference in a
unit’s potential outcomes when changing that unit’s treatment status and
holding all other units’ treatment status fixed. It may be defined as

$\delta_{dir}=\frac{1}{N}\sum_{i=1}^{N}(y_{i}(1,\mathbf{1})-y_{i}(0,\mathbf{1})),$
(3)

where $\mathbf{1}$ denotes a vector of 1’s of length $K$. In contrast to
direct effect, the average indirect effect (AIE) $\delta_{ind}$ is defined as
the average difference in a unit’s potential outcome when changing all other
treatment statuses from control to treated, holding its own treatment fixed.
It may be defined as

$\delta_{ind}=\frac{1}{N}\sum_{i=1}^{N}(y_{i}(0,\mathbf{1})-y_{i}(0,\mathbf{0})),$
(4)

where $\mathbf{0}$ denotes a vector of 0’s of length $K$. The average total
effect (ATOT) $\delta_{tot}$ measures the average difference in potential
outcomes between all units receiving treatment and all units receiving
control:

$\delta_{tot}=\frac{1}{N}\sum_{i=1}^{N}(y_{i}(1,\mathbf{1})-y_{i}(0,\mathbf{0})).$
(5)

Note that these quantities are defined to satisfy

$\delta_{tot}=\delta_{dir}+\delta_{ind}.$ (6)

When SUTVA holds, $\delta_{tot}=\delta_{dir}$ and $\delta_{ind}=0$.

Note, we may define
$\delta_{dir}=N^{-1}\sum_{i=1}^{N}(y_{i}(1,\mathbf{0})-y_{i}(0,\mathbf{0}))$
and
$\delta_{ind}=N^{-1}\sum_{i=1}^{N}(y_{i}(1,\mathbf{1})-y_{i}(1,\mathbf{0}))$
while still ensuring that (6) holds. These quantities may differ from (3) and
(4) if there is interaction between direct effects and indirect effects—that
is, if the differences $y_{i}(1,\mathbf{W}_{-i})-y_{i}(0,\mathbf{W}_{-i})$
differ depending on the allocation of treatment given to $\mathbf{W}_{-i}$
(Alzubaidi and Higgins, 2022).

###  4.2 The $\ell^{th}$–Nearest Neighbor Indirect Effect

We now define nearest-neighbor average treatment effects. Under KNNIM, these
estimands are of primary researcher interest. Let
$\mathbf{W}^{*}_{\ell}=(W^{*}_{\ell,1},W^{*}_{\ell,2},\ldots,W^{*}_{\ell,K})\in\\{0,1\\}^{K}$
denote the treatment assignment vector of length $K$ where the first $\ell$
nearest neighbors are given treatment and the rest are given control:

$\mathbf{W}^{*}_{\ell,j}=\left\\{\begin{array}[]{ll}1,&j\leq\ell,\\\
0,&\text{otherwise}.\end{array}\right.$ (7)

Note that $\mathbf{W}^{*}_{K}=\mathbf{1}$ and $\mathbf{W}^{*}_{0}=\mathbf{0}$.
Following our definitions from Section 4.1, we define the average
$\ell^{th}$–nearest neighbor indirect effect (A$\ell$NNIE) as

$\delta_{\ell}=\frac{1}{N}\sum_{i=1}^{N}(y_{i}(0,\mathbf{W}^{*}_{\ell})-y_{i}(0,\mathbf{W}^{*}_{\ell-1})).$
(8)

Note that $\mathbf{W}^{*}_{\ell}$ and $\mathbf{W}^{*}_{\ell-1}$ are identical
except that $W^{*}_{\ell,\ell}=1$ and $W^{*}_{\ell-1,\ell}=0$. Hence,
$\delta_{\ell}$ may be interpreted as the average difference in response due
to the treatment status of the $\ell^{th}$–nearest-neighbor. Additionally,
under KNNIM, the AIE is the sum of the A$\ell$NNIEs.

$\delta_{ind}=\sum_{\ell=1}^{K}\delta_{\ell}.$ (9)

To see this, note that $\sum_{\ell=1}^{K}\delta_{\ell}$ is a telescoping sum
that simplifies to

$\sum_{\ell=1}^{K}\delta_{\ell}=\sum_{\ell=1}^{K}\frac{1}{N}\left(\sum_{i=1}^{N}y_{i}(0,\mathbf{W}^{*}_{\ell})-\sum_{i=1}^{N}y_{i}(0,\mathbf{W}^{*}_{\ell-1})\right)=\frac{1}{N}\sum_{i=1}^{N}(y_{i}(0,\mathbf{1})-y_{i}(0,\mathbf{0}))=\delta_{ind}.$
(10)

##  5 Horvitz–Thompson Estimators

We now derive Horvitz–Thompson (HT) estimators for the estimands described in
Section 4. Our approach closely follows that in Aronow and Samii (2017). Of
particular note, these HT estimators require computing, for each unit $i$,
various marginal and joint probabilities related to the treatment allocation
on $i\cup\mathcal{N}_{iK}$. Thankfully, for many common designs, these
probabilities can be computed exactly under KNNIM, and we give closed-form
solutions for these probabilities under completely-randomized and Bernoulli-
randomized designs. When these probabilities cannot be computed exactly, they
may still be estimated, for example, using the approach in Aronow and Samii
(2017).

Let $\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})$ denote the marginal probability
that unit $i$ is given exposure $(W,\mathbf{W}_{\mathcal{N}_{K}})$—that is,
the overall treatment allocation assigns treatment $W$ to unit $i$ and assigns
treatment conditions $\mathbf{W}_{\mathcal{N}_{K}}$ to $i$’s $K$-neighborhood.
Define $\pi_{ij}(W,\mathbf{W}_{\mathcal{N}_{K}})$ as the joint probability
that units $i$ and $j$ are both given exposure
$(W,\mathbf{W}_{\mathcal{N}_{K}})$ and define
$\pi_{ij}((W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))$
as the joint probability that unit $i$ receives exposure
$(W,\mathbf{W}_{\mathcal{N}_{K}})$ and unit $j$ receives exposure
$(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})$. Define indicator
variables $I_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})$ that are equal to 1 if unit
$i$ is given exposure $(W,\mathbf{W}_{\mathcal{N}_{K}})$ and are 0 otherwise.
Finally, let

$\bar{y}(W,\mathbf{W}_{\mathcal{N}_{K}})=\frac{1}{N}\sum_{i=1}^{N}y_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})$
(11)

denote the average potential outcome across all $N$ units for treatment
allocation $(W,\mathbf{W}_{\mathcal{N}_{K}})$. The Horvitz-Thompson (HT)
estimator (Horvitz and Thompson, 1952) for
$\bar{y}(W,\mathbf{W}_{\mathcal{N}_{K}})$ is

$\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}})=\frac{1}{N}\sum_{i=1}^{N}I_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})\frac{Y_{i}}{\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}.$
(12)

This estimator is unbiased for $\bar{y}(W,\mathbf{W}_{\mathcal{N}_{K}})$,

$\mathbf{E}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))=\bar{y}(W,\mathbf{W}_{\mathcal{N}_{K}}),$
(13)

and has variance

$\mathbf{Var}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))=\frac{1}{N^{2}}\sum_{i=1}^{N}\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})[1-\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})]\left[\frac{y_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}{\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}\right]^{2}\\\
+\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j\neq
i}[\pi_{ij}(W,\mathbf{W}_{\mathcal{N}_{K}})-\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})\pi_{j}(W,\mathbf{W}_{\mathcal{N}_{K}})]\frac{y_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}{\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}\frac{y_{j}(W,\mathbf{W}_{\mathcal{N}_{K}})}{\pi_{j}(W,\mathbf{W}_{\mathcal{N}_{K}})}.$
(14)

The covariance of the HT estimators under any two exposures
$(W,\mathbf{W}_{\mathcal{N}_{K}})$ and
$(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})$ is

$\mathbf{Cov}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))=\\\
\frac{1}{N^{2}}\left(\sum_{i=1}^{N}\sum_{j\neq
i}\left[\pi_{ij}((W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))-\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})\pi_{j}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})\right]\right.\\\
\times\left.\frac{y_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}{\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}\frac{y_{j}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})}{\pi_{j}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})}\right)-\frac{1}{N^{2}}\sum_{i=1}^{N}y_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})y_{i}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}).$
(15)

From (13), (14), and (15), the expectation and variance for the difference in
HT estimators for the average response under any two unique exposures
$(W,\mathbf{W}_{\mathcal{N}_{K}})$,
$(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})$ can be computed as
follows:

######  Theorem 1.

$\mathbf{E}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}})-\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}}))=\bar{y}(W,\mathbf{W}_{\mathcal{N}_{K}})-\bar{y}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}}),$
(16)
$\mathbf{Var}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}})-\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}}))=\\\
\mathbf{Var}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))+\mathbf{Var}(\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))-2\mathbf{Cov}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})).$
(17)

We can then obtain unbiased estimators for the ADE, AIE, ATOT, and the
A$\ell$NNIE as follows:

$\displaystyle\widehat{\delta}_{{HT,dir}}$
$\displaystyle=\bar{Y}_{HT}^{obs}(1,\mathbf{1})-\bar{Y}_{HT}^{obs}(0,\mathbf{1}),$
(18) $\displaystyle\widehat{\delta}_{{HT,ind}}$
$\displaystyle=\bar{Y}_{HT}^{obs}(0,\mathbf{1})-\bar{Y}_{HT}^{obs}(0,\mathbf{0}),$
(19) $\displaystyle\widehat{\delta}_{{HT,tot}}$
$\displaystyle=\bar{Y}_{HT}^{obs}(1,\mathbf{1})-\bar{Y}_{HT}^{obs}(0,\mathbf{0}),$
(20) $\displaystyle\widehat{\delta}_{HT,\ell}$
$\displaystyle=\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell})-\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell-1}).$
(21)

Variances for these estimators are derived as in (17). As with the estimands,
we have the following relationships between these estimators:

$\displaystyle\widehat{\delta}_{HT,tot}$
$\displaystyle=\widehat{\delta}_{HT,dir}+\widehat{\delta}_{HT,ind},$ (22)
$\displaystyle\widehat{\delta}_{HT,ind}$
$\displaystyle=\sum_{\ell=1}^{K}\widehat{\delta}_{HT,\ell}.$ (23)

###  5.1 Marginal and Joint Exposure Probabilities

One significant benefit of KNNIM is that this model allows for closed-form
expressions of the marginal and joint exposure probabilities for many common
experimental designs. We now provide these exposure probabilities under
completely-randomized and Bernoulli-randomized designs.

We first introduce some notation to facilitate this discussion. For any two
units $i$ and $j$, let $b_{ij}$ denote the number of units shared by both $i$
and $j$’s closed $K$-neighborhoods:
$b_{ij}=|(i\cup\mathcal{N}_{iK})\cap(j\cup\mathcal{N}_{jK})|$. Consider two
(not necessarily unique) treatment exposures
$(W,\mathbf{W}_{\mathcal{N}_{K}})$,
$(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})$ given to the closed
$K$-neighborhood of units $i$ and $j$ respectively. These exposures are called
compatible if they can co-occur within a given treatment assignment. For
example, if $j$ is unit $i$’s nearest neighbor, the exposures $(1,\mathbf{1})$
for unit $i$ and $(0,\mathbf{0})$ for unit $j$ are incompatible since unit $j$
is given treatment in the first exposure and control in the second
exposure—the probability that $i$ and $j$ can jointly observe two incompatible
treatment exposures is 0. For the exposure $(W,\mathbf{W}_{\mathcal{N}_{K}})$,
let $N_{itK}$ and $N_{icK}$ denote the number of treated and control units
respectively within $i\cup\mathcal{N}_{iK}$. For exposure
$(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})$, let $N_{jitK}$ and
$N_{jicK}$ denote the number of treated and control units in
$(j\cup\mathcal{N}_{jK})\setminus(i\cup\mathcal{N}_{iK}$)—that is, the number
of treated and control units for units not already included in $i$’s closed
neighborhood. The specific units given treatment or control do not factor into
the probability computations.

####  5.1.1 Exposure Probabilities Under Complete Randomization

In a completely-randomized design, the number of treated units $N_{t}$ in the
study is selected prior to randomization. Each possible treatment has the same
$\binom{N}{N_{t}}^{-1}$ probability of occurring.

The marginal probability that $i\cup\mathcal{N}_{iK}$ receives exposure
$(W,\mathbf{W}_{\mathcal{N}_{K}})$ is

$\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})=\frac{\binom{N-K-1}{N_{t}-N_{itK}}}{\binom{N}{N_{t}}}.$
(24)

The joint probability that unit $i$ receives exposure
$(W,\mathbf{W}_{\mathcal{N}_{K}})$ and unit $j$ receives exposure
$(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})$ is

$\pi_{ij}((W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))=\\\
\begin{cases}\frac{\binom{N-K-1}{N_{t}-N_{tK}}}{\binom{N}{N_{t}}}\frac{\binom{N-2K-2+b_{ij}}{N_{t}-N_{itK}-N_{jitK}}}{\binom{N-K-1}{N_{t}-N_{itK}}},&(W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})~{}\text{are
compatible for }i\text{ and }j,\\\ 0,&\text{otherwise}.\end{cases}$ (25)

####  5.1.2 Exposure Probabilities Under Bernoulli Randomization

In a Bernoulli-randomized design, each unit has a pre-specified probability
$p$ of being assigned treatment, and treatments are assigned independently
across units (e.g. treatment assignment is determined for each unit by
flipping a coin that has probability $p$ of landing heads). Under Bernoulli-
randomization, the marginal probability that $i\cup\mathcal{N}_{iK}$ receives
exposure $(W,\mathbf{W}_{\mathcal{N}_{K}})$ is

$\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})=p^{N_{itK}}(1-p)^{N_{icK}},$ (26)

and joint probability of units $i$ and $j$ being exposed to
$(W,\mathbf{W}_{\mathcal{N}_{K}})$ and
$(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})$ respectively is

$\pi_{ij}((W_{i},W_{\mathcal{N}_{ik}}),(W^{\prime}_{j},W^{\prime}_{\mathcal{N}_{jk}}))=\\\
\begin{cases}p^{N_{itK}}(1-p)^{N_{icK}}p^{N_{jitK}}(1-p)^{N_{jicK}},&(W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})~{}\text{are
compatible for }i\text{ and }j,\\\ 0,&\text{otherwise}.\end{cases}$ (27)

##  6 Estimation Under No Weak Interaction Between Direct and Indirect Effects

Next, we consider an additional assumption on the responses of potential
outcomes—no weak interaction between direct and indirect effects. When it
holds, this assumption allows for significantly more powerful estimators of
direct and indirect effects and makes the two competing definitions of direct
and indirect effects in Section 4.1 equivalent. Note, this assumption is
similar to the additivity of main effects (AME) assumption considered by
Sussman and Airoldi (2017), however, we strictly weaken this assumption by
only requiring it to hold for the average of the potential outcomes.

######  Assumption 2.

(No Weak Interaction Between Direct and Indirect Effects) There is no weak
interaction between direct and indirect effects if, for any two potential
treatment exposures
$\mathbf{W}_{\mathcal{N}_{K}},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}$ on the
$K$-neighborhoods

$\bar{y}(1,\mathbf{W}_{\mathcal{N}_{K}})-\bar{y}(1,\mathbf{W}^{\prime}_{\mathcal{N}_{K}})-(\bar{y}(0,\mathbf{W}_{\mathcal{N}_{K}})-\bar{y}(0,\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))=0.$
(28)

While strong, Assumption 2 may be plausible under certain settings. One such
setting could occur if direct and indirect effects of treatment are
multiplicative—i.e. treatment effects may increase response by a certain
percentage. In this case, taking the log of response may lead to model in
which the stronger AME assumption holds. For example, under KNNIM, we may
consider the following model of response:

$Y_{i}=Y(0,\mathbf{0})\beta_{t}^{W_{i}}\prod_{\ell=1}^{K}\beta_{\ell}^{W_{i\ell}},$

where $W_{i\ell}$ indicates whether the $\ell$th nearest neighbor of unit $i$
receives treatment. After taking logs of both sides, we obtain

$\log
Y_{i}=Y(0,\mathbf{0})+W_{i}\log\beta_{t}+\sum_{\ell=1}^{K}W_{i\ell}\log\beta_{\ell},$

thereby satisfying AME—the direct effect is always $\log(\beta_{t})$
regardless of the treatment status of any other unit.

Assumption 2 may also hold when direct treatment effects are weak relative to
the indirect effects. For example, consider an experiment on a social network
designed to assess the efficacy of an advertisement for an event. Treatment
may involve posting a flyer on a user’s “wall” if that user had previously
expressed interest in attending that event, and response would indicate
whether that user attends the event. Since the user is already interested in
the event, the user may not be particularly affected by the flyer on their
wall, suggesting a weak direct treatment effect. However, if the user notices
that their close friends also have flyers on their walls, they all may decide
to attend the event together, suggesting a strong indirect effect.

To demonstrate the power of Assumption 2, consider the case of estimating the
AIE $\delta_{ind}$. When Assumption 2 holds, both
$\bar{Y}_{HT}^{obs}(1,\mathbf{1})-\bar{Y}_{HT}^{obs}(1,\mathbf{0})$ and
$\bar{Y}_{HT}^{obs}(0,\mathbf{1})-\bar{Y}_{HT}^{obs}(0,\mathbf{0})$ unbiasedly
estimate $\delta_{ind}$—the latter quantity unbiasedly estimates
$\bar{y}(0,\mathbf{1})-\bar{y}(0,\mathbf{0})$, and under Assumption 2,
$\bar{y}(0,\mathbf{1})-\bar{y}(0,\mathbf{0})=\bar{y}(1,\mathbf{1})-\bar{y}(1,\mathbf{0})=\delta_{ind}$.
Thus, these two estimators can be combined to yield a more powerful unbiased
estimate of the AIE:

$\hat{\delta}_{ind}=c_{1}\left(\bar{Y}_{HT}^{obs}(1,\mathbf{1})-\bar{Y}_{HT}^{obs}(1,\mathbf{0})\right)+c_{2}\left(\bar{Y}_{HT}^{obs}(0,\mathbf{1})-\bar{Y}_{HT}^{obs}(0,\mathbf{0})\right)$
(29)

where the weights $c_{1},c_{2}$ satisfy $c_{1}+c_{2}=1$. An analogous approach
can be used to derive estimators for the ADE and the A$\ell$NNIEs.

Given strong prior knowledge about the values of the potential outcomes,
$c_{1}$ and $c_{2}$ may be adjusted to reduce the variance of the estimator
(similar to Neyman allocation or optimal allocation in Lohr (2019)). However,
when no prior knowledge is available, setting $c_{1}=c_{2}=1/2$ may be
preferable. Additionally, under this choice of weights, estimators can be
decomposed analogously to (22) and (23). Thus, we proceed assuming
$c_{1}=c_{2}=1/2$.

###  6.1 Estimators Under the no Weak Interaction Assumption

We begin by proposing unbiased estimators for the average direct, indirect,
and $\ell^{th}$ nearest neighbor effects:

######  Theorem 2.

Under Assumption 2, the following are unbiased estimators for the ADE, AIE,
and the A$\ell$NNIE:

$\displaystyle\widehat{\delta^{*}}_{{HT,dir}}$
$\displaystyle=1/2\left(\bar{Y}_{HT}^{obs}(1,\mathbf{1})-\bar{Y}_{HT}^{obs}(0,\mathbf{1})\right)+1/2\left(\bar{Y}_{HT}^{obs}(1,\mathbf{0})-\bar{Y}_{HT}^{obs}(0,\mathbf{0})\right),$
(30) $\displaystyle\widehat{\delta^{*}}_{{HT,ind}}$
$\displaystyle=1/2\left(\bar{Y}_{HT}^{obs}(1,\mathbf{1})-\bar{Y}_{HT}^{obs}(1,\mathbf{0})\right)+1/2\left(\bar{Y}_{HT}^{obs}(0,\mathbf{1})-\bar{Y}_{HT}^{obs}(0,\mathbf{0})\right),$
(31) $\displaystyle\widehat{\delta^{*}}_{HT,\ell}$
$\displaystyle=1/2\left(\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell})-\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell-1})\right)+1/2\left(\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell})-\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell-1})\right).$
(32)

###### Proof.

Earlier in this section, we detailed the proof of unbiasedness for
$\widehat{\delta^{*}}_{{HT,ind}}$. The argument for unbiasedness of
$\widehat{\delta^{*}}_{HT,\ell}$ proceeds identically, as does that for
$\widehat{\delta^{*}}_{HT,dir}$ after rewriting (28) as

$\bar{y}(1,\mathbf{W}_{\mathcal{N}_{K}})-\bar{y}(0,\mathbf{W}_{\mathcal{N}_{K}})=\bar{y}(1,\mathbf{W}^{\prime}_{\mathcal{N}_{K}})-\bar{y}(0,\mathbf{W}^{\prime}_{\mathcal{N}_{K}}).$
(33)

∎

For brevity, to distinguish between these estimators and the estimators
defined in Section 5, we will refer to the estimators
$\widehat{\delta}_{{HT,dir}}$, $\widehat{\delta}_{{HT,ind}}$, and,
$\widehat{\delta}_{HT,\ell}$ as estimators “under Assumption 1” and
$\widehat{\delta}^{*}_{{HT,dir}}$, $\widehat{\delta}^{*}_{{HT,ind}}$, and,
$\widehat{\delta}^{*}_{HT,\ell}$ as “under Assumption 2,” even though both
estimators can be used when Assumption 2 holds.

These estimators satisfy a similar decomposition to those derived under
Assumption 1:

$\displaystyle\widehat{\delta}_{HT,tot}$
$\displaystyle=\widehat{\delta^{*}}_{HT,dir}+\widehat{\delta^{*}}_{HT,ind},$
(34) $\displaystyle\widehat{\delta^{*}}_{HT,ind}$
$\displaystyle=\sum_{\ell=1}^{K}\widehat{\delta^{*}}_{HT,\ell}.$ (35)

Additionally, the variance of $\widehat{\delta^{*}}_{{HT,dir}}$,
$\widehat{\delta^{*}}_{HT,ind}$, and $\widehat{\delta^{*}}_{HT,\ell}$ can be
derived from repeated application of (14) and (15). This result generalizes to
similar estimators using any four treatment exposures.

######  Theorem 3.

For any four treatment exposures
$(W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}),(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}}),(W^{*\prime},\mathbf{W}^{*\prime}_{\mathcal{N}_{K}})$,

$\displaystyle\mathbf{Var}\left[1/2\left(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}})-\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})\right)+1/2\left(\bar{Y}_{HT}^{obs}(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}})-\bar{Y}_{HT}^{obs}(W^{*\prime},\mathbf{W}^{*\prime}_{\mathcal{N}_{K}})\right)\right]$
$\displaystyle=$ $\displaystyle
1/4\mathbf{Var}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))+1/4\mathbf{Var}(\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))$
$\displaystyle+1/4\mathbf{Var}(\bar{Y}_{HT}^{obs}(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}}))+1/4\mathbf{Var}(\bar{Y}_{HT}^{obs}(W^{*^{\prime}},\mathbf{W}^{*^{\prime}}_{\mathcal{N}_{K}}))$
$\displaystyle-1/2\mathbf{Cov}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))+1/2\mathbf{Cov}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}}))$
$\displaystyle-1/2\mathbf{Cov}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{*^{\prime}},\mathbf{W}^{*^{\prime}}_{\mathcal{N}_{K}}))-1/2\mathbf{Cov}(\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}}))$
$\displaystyle+1/2\mathbf{Cov}(\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{*^{\prime}},\mathbf{W}^{*^{\prime}}_{\mathcal{N}_{K}}))-1/2\mathbf{Cov}(\bar{Y}_{HT}^{obs}(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{*^{\prime}},\mathbf{W}^{*^{\prime}}_{\mathcal{N}_{K}})).$
(36)

##  7 Variance Estimators

We extend the work provided in (Aronow and Samii, 2013, 2017) and (Lohr, 2019)
to derive conservative estimators for the variance for all considered
estimators under KNNIM. This requires estimating all variance and covariance
terms in (17) and (3).

We begin by estimating the
$\mathbf{Var}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))$ terms. The
standard Horvitz-Thompson estimator for these variances is

$\widehat{\mathbf{Var}}_{HT}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))=\frac{1}{N^{2}}\sum_{i=1}^{N}I_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})[1-\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})]\left[\frac{Y_{i}}{\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}\right]^{2}\\\
+\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j\neq
i}I_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})I_{j}(W,\mathbf{W}_{\mathcal{N}_{K}})\frac{[\pi_{ij}(W,\mathbf{W}_{\mathcal{N}_{K}})-\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})\pi_{j}(W,\mathbf{W}_{\mathcal{N}_{K}})]}{\pi_{ij}(W,\mathbf{W}_{\mathcal{N}_{K}})}\\\
\times\frac{Y_{i}}{\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}\frac{Y_{j}}{\pi_{j}(W,\mathbf{W}_{\mathcal{N}_{K}})}.$
(37)

If the joint probabilities $\pi_{ij}(W,\mathbf{W}_{\mathcal{N}_{K}})>0$ for
all $i$ and $j$, then this estimated variance is unbiased. However, under
KNNIM, there will likely be incompatibility of this exposure for some
$i^{\prime}$ and $j^{\prime}$ (see Section 5.1 for details), and hence
$\pi_{i^{\prime}j^{\prime}}(W,\mathbf{W}_{\mathcal{N}_{K}})=0$ for these
units. These probabilities lead to bias in the estimate of
$\mathbf{Var}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))$:

$\mathbf{E}(\widehat{\mathbf{Var}}_{HT}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}})))=\mathbf{Var}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))+A_{Var}$
(38)

where

$A_{Var}=\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}j\neq i,\\\
\pi_{ij}(W,\mathbf{W}_{\mathcal{N}_{K}})=0\end{subarray}}y_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})y_{j}(W,\mathbf{W}_{\mathcal{N}_{K}}).$
(39)

Young’s inequality as derived in Aronow and Samii (2013, 2017) can be used to
obtain a quantity $\widehat{A}_{Var}$ that satisfies
$\mathbf{E}(\widehat{A}_{Var})\geq|A_{Var}|$. We now give a brief proof of
Young’s inequality as applied to KNNIM, and demonstrate how it can be used to
find $\widehat{A}_{Var}$.

######  Lemma 1.

For any two (not necessarily unique) exposures
$(W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}})$,

$\displaystyle|y_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})y_{j}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}})|\leq\mathbf{E}\left(\frac{I_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})Y_{i}^{2}}{2\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}+\frac{I_{j}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}})Y_{j}^{2}}{2\pi_{j}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}})}\right).$
(40)

###### Proof.

To see this, first note that

$|y_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})y_{j}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}})|=\sqrt{y^{2}_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})y^{2}_{j}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}})}.$
(41)

The upper-bound is then proven by applying the arithmetic-mean geometric-mean
inequality (Steele, 2004)

$\sqrt{y^{2}_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})y^{2}_{j}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}})}\leq\frac{y^{2}_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}{2}+\frac{y^{2}_{j}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}})}{2}$
(42)

and by noting that the right-hand side can be unbiasedly estimated by

$\frac{I_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})Y_{i}^{2}}{2\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}+\frac{I_{j}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}})Y_{j}^{2}}{2\pi_{j}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}})}.$
(43)

∎

Thus, we define

$\widehat{A}_{Var}(W,\mathbf{W}_{\mathcal{N}_{K}})=\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}j\neq
i,\\\
\pi_{ij}(W,\mathbf{W}_{\mathcal{N}_{K}})=0\end{subarray}}\left[\frac{I_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})Y_{i}^{2}}{2\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}+\frac{I_{j}(W,\mathbf{W}_{\mathcal{N}_{K}})Y_{j}^{2}}{2\pi_{j}(W,\mathbf{W}_{\mathcal{N}_{K}})}\right].$
(44)

From Lemma 1, linearity of expectations, and the triangle inequality, it
follows that

$\displaystyle\mathbf{E}\left(\widehat{A}_{Var}(W,\mathbf{W}_{\mathcal{N}_{K}})\right)=~{}$
$\displaystyle\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}j\neq
i,\\\
\pi_{ij}(W,\mathbf{W}_{\mathcal{N}_{K}})=0\end{subarray}}\left[\frac{y^{2}_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}{2}+\frac{y^{2}_{j}(W,\mathbf{W}_{\mathcal{N}_{K}})}{2}\right]$
$\displaystyle\geq~{}$
$\displaystyle\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}j\neq
i,\\\
\pi_{ij}(W,\mathbf{W}_{\mathcal{N}_{K}})=0\end{subarray}}|y_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})y_{j}(W,\mathbf{W}_{\mathcal{N}_{K}})|$
$\displaystyle\geq~{}$
$\displaystyle\left|\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}j\neq
i,\\\
\pi_{ij}(W,\mathbf{W}_{\mathcal{N}_{K}})=0\end{subarray}}y_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})y_{j}(W,\mathbf{W}_{\mathcal{N}_{K}})\right|=|A_{Var}|.$
(45)

Therefore, by setting

$\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))=\widehat{\mathbf{Var}}_{HT}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))+\widehat{A}_{Var}(W,\mathbf{W}_{\mathcal{N}_{K}}),$
(46)

we obtain a conservative estimate of the variance:

$\mathbf{E}\left(\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))\right)\geq\mathbf{Var}_{HT}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))+|A_{Var}|+A_{Var}\geq\mathbf{Var}_{HT}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))$
(47)

We apply a similar approach for estimating covariance components. Define

$\widehat{\mathbf{Cov}}_{HT}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))=\\\
\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}j\in\\{1,\ldots,N\\}\\\
\pi_{ij}((W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))>0\end{subarray}}\left[\frac{I_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})I_{j}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})}{\pi_{ij}((W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))}\frac{Y_{i}}{\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}\frac{Y_{j}}{\pi_{j}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})}\right]\\\
\times\left[\pi_{ij}((W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))-\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})\pi_{j}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})\right]$
(48)

Similar to (38) and (39), we have:

$\displaystyle\mathbf{E}\left(\widehat{\mathbf{Cov}}_{HT}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})\right)$
$\displaystyle=~{}$
$\displaystyle\mathbf{Cov}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))-A_{Cov}$
(49)

where

$A_{Cov}=\sum_{i=1}^{N}\sum_{\begin{subarray}{c}j\in\\{1,\ldots,N\\}\\\
\pi_{ij}((W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))=0\end{subarray}}y_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})y_{j}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}).$
(50)

From repeated application of Lemma 1 in a way that mimics the argument in
(45), we can obtain conservative estimators for the covariance. Specifically,
define:

$\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))=\widehat{\mathbf{Cov}}_{HT}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))\\\
-\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}j\in\\{1,\ldots,N\\}\\\
\pi_{ij}((W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))=0\end{subarray}}\left[\frac{I_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})Y^{2}_{i}}{2\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}+\frac{I_{j}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})Y^{2}_{j}}{2\pi_{j}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})}\right]$
(51)

and

$\widehat{\mathbf{Cov}}_{B}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))=\widehat{\mathbf{Cov}}_{HT}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))\\\
+\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}j\in\\{1,\ldots,N\\}\\\
\pi_{ij}((W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))=0\end{subarray}}\left[\frac{I_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})Y^{2}_{i}}{2\pi_{i}(W,\mathbf{W}_{\mathcal{N}_{K}})}+\frac{I_{j}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})Y^{2}_{j}}{2\pi_{j}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})}\right].$
(52)

Note, the only difference in these two quantities is that the last term is
subtracted in (51) and added in (52). The ensuing Lemma immediately follows.

######  Lemma 2.

$\mathbf{E}(\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})))\leq\mathbf{Cov}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))\\\
\leq\mathbf{E}(\widehat{\mathbf{Cov}}_{B}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))).$
(53)

Thereby, the conservative variance estimators for the estimators in the
previous section under Assumption 1 for any two exposures
$(W,\mathbf{W}_{\mathcal{N}_{K}})$,
$(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})$ and under Assumption 2
for any four treatment exposures
$(W,\mathbf{W}_{\mathcal{N}_{K}}),(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}),(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}}),(W^{*\prime},\mathbf{W}^{*\prime}_{\mathcal{N}_{K}})$
respectively can be derived as follows,

$\widehat{\mathbf{Var}}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}})-\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W^{\prime}}_{\mathcal{N}_{K}}))=\\\
\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))+\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))-2\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})),$
(54)

$\displaystyle\widehat{\mathbf{Var}}\left[1/2\left(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}})-\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}})\right)+1/2\left(\bar{Y}_{HT}^{obs}(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}})-\bar{Y}_{HT}^{obs}(W^{*\prime},\mathbf{W}^{*\prime}_{\mathcal{N}_{K}})\right)\right]$
$\displaystyle=$ $\displaystyle
1/4\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}))+1/4\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))$
$\displaystyle+1/4\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}}))+1/4\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(W^{*^{\prime}},\mathbf{W}^{*^{\prime}}_{\mathcal{N}_{K}}))$
$\displaystyle-1/2\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}))+1/2\widehat{\mathbf{Cov}}_{B}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}}))$
$\displaystyle-1/2\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(W,\mathbf{W}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{*^{\prime}},\mathbf{W}^{*^{\prime}}_{\mathcal{N}_{K}}))-1/2\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}}))$
$\displaystyle+1/2\widehat{\mathbf{Cov}}_{B}(\bar{Y}_{HT}^{obs}(W^{\prime},\mathbf{W}^{\prime}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{*^{\prime}},\mathbf{W}^{*^{\prime}}_{\mathcal{N}_{K}}))-1/2\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(W^{*},\mathbf{W}^{*}_{\mathcal{N}_{K}}),\bar{Y}_{HT}^{obs}(W^{*^{\prime}},\mathbf{W}^{*^{\prime}}_{\mathcal{N}_{K}})).$
(55)

This general form of the variance estimators can be applied for all estimators
considered under Assumptions 1 and 2. For example, the conservative variance
estimators for the A$\ell$NNIE under Assumptions 1 and 2 respectively are as
follows,

$\widehat{\mathbf{Var}}(\widehat{\delta}_{{HT,\ell^{th}}})=\widehat{\mathbf{Var}}(\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell})-\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell-1}))=\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell}))+\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell-1}))\\\
-2\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell}),\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell-1})).$
(56)

$\widehat{\mathbf{Var}}(\widehat{\delta^{*}}_{{HT,\ell^{th}}})=\widehat{\mathbf{Var}}(1/2\left(\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell})-\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell-1})\right)+1/2\left(\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell})-\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell-1})\right))\\\
=\frac{1}{4}\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell}))+\frac{1}{4}\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell-1}))\\\
+\frac{1}{4}\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell}))+\frac{1}{4}\widehat{\mathbf{Var}}_{A}(\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell-1}))\\\
-\frac{1}{2}\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell}),\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell-1}))+\frac{1}{2}\widehat{\mathbf{Cov}}_{B}(\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell}),\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell}))\\\
-\frac{1}{2}\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell}),\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell-1}))-\frac{1}{2}\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell-1}),\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell}))\\\
+\frac{1}{2}\widehat{\mathbf{Cov}}_{B}(\bar{Y}_{HT}^{obs}(1,\mathbf{W}^{*}_{\ell-1}),\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell-1}))-\frac{1}{2}\widehat{\mathbf{Cov}}_{A}(\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell}),\bar{Y}_{HT}^{obs}(0,\mathbf{W}^{*}_{\ell-1})).$
(57)

##  8 Simulation

Model | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---
$\delta_{1}$ | 0 | 0 | 0 | 2 | 2 | 2 | 3 | 3 | 3
$\delta_{2}$ | 0 | 0 | 0 | 1 | 1 | 1 | 2 | 2 | 2
$\delta_{3}$ | 0 | 0 | 0 | 0.5 | 0.5 | 0.5 | 1 | 1 | 1
$\delta_{t}$ | 0 | 1 | 4 | 0 | 1 | 4 | 0 | 1 | 4
Table 1: Interference models. Table 2: Estimates under completely randomized design and Bernoulli randomization model 1. |  | Comp. Rand. | Bern. Rand.
---|---|---|---
Estimator | Effects | Emp. EV, | Emp. Var, | Var Est. | Emp. EV, | Emp. Var, | Var Est.
$\widehat{\delta}_{{HT,tot}}$ | 0 | 0.0779 | 1.1668 | 1.1759 | 0.0059 | 1.2480 | 1.2037
$\widehat{\delta}_{{HT,dir}}$ | 0 | 0.0337 | 0.6078 | 0.6683 | -0.0150 | 0.6850 | 0.7016
$\widehat{\delta^{*}}_{{HT,dir}}$ | 0 | 0.0303 | 0.2894 | 0.5036 | 0.0103 | 0.3116 | 0.5153
$\widehat{\delta}_{{HT,ind}}$ | 0 | 0.0442 | 0.9404 | 0.9393 | 0.0210 | 0.8232 | 0.9258
$\widehat{\delta^{*}}_{{HT,ind}}$ | 0 | 0.0476 | 0.5615 | 0.6619 | -0.0044 | 0.5622 | 0.6696
$\widehat{\delta}_{{HT,1^{st}}}$ | 0 | 0.0346 | 0.6273 | 0.6821 | 0.0502 | 0.6502 | 0.6804
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 0 | 0.0191 | 0.2647 | 0.3950 | 0.0173 | 0.2460 | 0.3922
$\widehat{\delta}_{{HT,2^{nd}}}$ | 0 | -0.0111 | 0.4642 | 0.6023 | -0.0181 | 0.4430 | 0.5985
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 0 | -0.0021 | 0.2694 | 0.4625 | -0.0085 | 0.2702 | 0.4599
$\widehat{\delta}_{{HT,3^{rd}}}$ | 0 | 0.0207 | 0.4541 | 0.6076 | -0.0110 | 0.4489 | 0.6048
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 0 | 0.0306 | 0.3014 | 0.4514 | -0.0132 | 0.3134 | 0.4629
Table 3: Estimates under completely randomized design and Bernoulli randomization model 5. |  | Comp. Rand. | Bern. Rand.
---|---|---|---
Estimator | Effects | Emp. EV, | Emp. Var, | Var Est. | Emp. EV, | Emp. Var, | Var Est.
$\widehat{\delta}_{{HT,tot}}$ | 4.5 | 4.5453 | 3.3664 | 3.9196 | 4.6263 | 4.7542 | 5.3978
$\widehat{\delta}_{{HT,dir}}$ | 1 | 0.9804 | 3.9013 | 4.1313 | 1.1013 | 4.3757 | 4.6502
$\widehat{\delta^{*}}_{{HT,dir}}$ | 1 | 1.0070 | 1.1503 | 1.9781 | 1.0725 | 1.1720 | 2.0765
$\widehat{\delta}_{{HT,ind}}$ | 3.5 | 3.5649 | 1.5750 | 2.3108 | 3.5249 | 1.8276 | 2.5207
$\widehat{\delta^{*}}_{{HT,ind}}$ | 3.5 | 3.5383 | 1.1485 | 1.9574 | 3.5537 | 2.0192 | 2.7832
$\widehat{\delta}_{{HT,1^{st}}}$ | 2 | 2.0414 | 0.8733 | 1.1102 | 2.0532 | 0.9882 | 1.1739
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 2 | 2.0245 | 0.4873 | 0.9667 | 2.0176 | 0.4648 | 0.9666
$\widehat{\delta}_{{HT,2^{nd}}}$ | 1 | 1.0003 | 1.4039 | 1.9204 | 0.9878 | 1.4253 | 2.0036
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 1 | 0.9930 | 0.9747 | 1.8982 | 0.9582 | 1.1082 | 2.0400
$\widehat{\delta}_{{HT,3^{rd}}}$ | 0.5 | 0.5230 | 1.8611 | 2.5928 | 0.4838 | 2.2543 | 2.8237
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 0.5 | 0.5207 | 1.4523 | 2.3927 | 0.5778 | 2.0343 | 2.7262
Table 4: Estimates under completely randomized design and Bernoulli randomization model 9. |  | Comp. Rand. | Bern. Rand.
---|---|---|---
Estimator | Effects | Emp. EV, | Emp. Var, | Var Est. | Emp. EV, | Emp. Var, | Var Est.
$\widehat{\delta}_{{HT,tot}}$ | 10 | 10.0055 | 11.54004 | 14.4106 | 10.2734 | 18.2343 | 21.4468
$\widehat{\delta}_{{HT,dir}}$ | 4 | 3.9257 | 14.7035 | 15.0482 | 4.2456 | 17.5899 | 18.4553
$\widehat{\delta^{*}}_{{HT,dir}}$ | 4 | 3.9896 | 4.0963 | 7.2216 | 4.1566 | 4.2346 | 7.7429
$\widehat{\delta}_{{HT,ind}}$ | 6 | 6.0797 | 2.7909 | 4.8503 | 6.0277 | 3.5973 | 5.4461
$\widehat{\delta^{*}}_{{HT,ind}}$ | 6 | 6.0158 | 3.3206 | 6.0066 | 6.1168 | 7.4354 | 9.8815
$\widehat{\delta}_{{HT,1^{st}}}$ | 3 | 3.0449 | 1.1582 | 1.6049 | 3.0547 | 1.3709 | 1.7503
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 3 | 3.0232 | 1.4675 | 3.0152 | 3.0100 | 1.3786 | 3.0134
$\widehat{\delta}_{{HT,2^{nd}}}$ | 2 | 2.0092 | 2.8500 | 3.9616 | 1.9923 | 2.9090 | 4.1353
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 2 | 1.9838 | 3.2241 | 6.4517 | 1.9150 | 3.8171 | 6.9785
$\widehat{\delta}_{{HT,3^{rd}}}$ | 1 | 1.0256 | 4.5078 | 6.1172 | 0.9806 | 5.3764 | 6.7556
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 1 | 1.0086 | 5.3193 | 8.5285 | 1.1917 | 7.4434 | 9.8241

In this section, we assess the performance of our proposed estimators through
a simulation study. In our simulations, we consider a range of indirect
effects from no indirect effects to moderate indirect effects and direct
effects. These simulations are designed to verify unbiasedness of our proposed
estimators and to assess the tightness of the conservative variance bounds.
Additionally, these simulations will satisfy the assumption of no weak
interaction between direct and indirect effects, allowing us to evaluate the
gain in precision for estimators exploiting this assumption.

In our simulations, we generate responses under the following KNNIM model with
$K=3$ nearest neighbors:

$Y_{i}=X_{i1}+X_{i2}+X_{i3}+\delta_{1}W_{i1}+\delta_{2}W_{i2}+\delta_{3}W_{i3}+\delta_{t}W_{i}$
(58)

where $\delta_{\ell}$ denotes the A$\ell$NNIE, $\ell=1,2,3$, and $\delta_{t}$
denotes the ADE. The $X_{ip}$ are covariates, $X_{ip}\sim N(0,1)$, $p=1,2,3$.
We use the squared Euclidean distance of the covariates
$d(i,j)=\sum_{p=1}^{3}(X_{ip}-X_{jp})^{2}$ as our measure of interaction. In
all models considered, closer neighbors provide stronger indirect effects:
$|\delta_{1}|\geq|\delta_{2}\geq|\delta_{3}|$. We assess our estimators under
two randomization designs: completely randomized designs (CRD) where half of
the $N$ units are assigned to treatment and Bernoulli randomized designs (BRD)
with probability $p=0.5$.

We consider 9 models of interference (see Table 1). The first three models
consider no interference, the next three models consider weak interference,
and the last three models consider moderate interference. Within these
categories of indirect effects, we also vary the strength of the direct
effect, from no direct effect to a strong direct effect.

For each model, we evaluate the performance of the total, direct, indirect and
$\ell_{th}$ nearest neighbor estimators under Assumption 1 and Assumption 2.
The experiment is replicated 1000 times with sample size N = 256. The marginal
and joint probabilities are computed as in Section 5.1. For each run of 1000
replications, for each estimator, we compute the empirical expected value of
the estimates (Emp. EV, the average value of the estimators), empirical
variance (Emp. Var, the variance of the estimators), and the average of the
estimated variances (Var Est.). For brevity, we only include results for the
CRD and BRD designs for Model 1 (Table 2), Model 5 (Table 3) and Model 9
(Table 4). A complete analysis is included in the supplementary material.

##  9 Discussion

Results indicate, first and foremost, that all estimators under all models
accurately estimate all direct and indirect effect parameters—as expected from
our theoretical results. Additionally we also verify the decomposition of
estimation of the total effect into direct and indirect components and of the
indirect effect estimator into nearest-neighbor components given by (22),
(23), (34), and (35). Finally, we confirm that our conservative variance
estimators are larger than the true variance, as estimated by the empirical
variance.

Under CRD, variance estimates tend to be smaller for closer neighbors under
both Assumption 1 and Assumption 2. The one exception occurs for models that
do not exhibit treatment interference. Additionally, the variance estimates
tend to be smaller for estimators under Assumption 2 than under Assumption 1
(Tables 2, 3). However, this may not be the case when direct effects are
large; the variance estimates for indirect effects estimators under Assumption
1 are about 50% smaller than under Assumption 2 (Table 4).

Similar results hold for BRD except that Assumption 2 did not reduce standard
errors of the indirect effects in the moderate direct effect scenario (Table
3). Moreover, all variance estimates for all estimators in weak and moderate
interference scenarios are slightly larger than those under CRD (Tables 3 and
4), with differences becoming larger as direct and indirect effects increase.
Hence, CRD may be preferable to BRD for this type of data.

##  10 An Analysis of Social Network Experiment

We now apply our estimators to our motivating example which assessed the
efficacy of a program designed to promote anti-conflict behaviors in New
Jersey middle schools (see Section 1.1). Recall that the experiment was
explicitly designed to determine whether benefits of the program were
propagated through social interactions between students, and moreover, that
the strength of the connection between students was explicitly recorded when
surveying students in the experiment. For our analysis, we use whether or not
a student wears an orange wristband as our response. A student may be given an
orange wristband by a seed student if the original student is observed to be
exhibiting anti-conflict behaviors.

To better fit the study into our KNNIM framework, we restrict our analysis to
seed-eligible students who listed two other seed-eligible students among their
top 10 closest connections over the prior few weeks. This creates a dataset of
$N=348$ students. We then analyze the data according to a KNNIM model with
$K=2$, where we assume that the seed-student designations were assigned
completely at random to the 348 eligible students. For more discussion of this
choice of $K$, please see Alzubaidi and Higgins (2022). Table 5 gives the
number of people assigned to each of the $2^{K+1}=8$ possible treatment
exposures.

We use our proposed estimators to estimate the effect of treatment under
Assumption 1 and Assumption 2. Results are presented in Table 6. Under
Assumption 1, results suggest that indirect exposure to treatment leads to
about a 21% increase in the probability of wearing a wristband. The bulk of
this effect (about 18%) is due to the effect from a student’s nearest
neighbor. Our analysis did not find a strong effect of either a direct effect
or indirect effects outside of the first nearest neighbor.

Under Assumption 2, direct exposure treatment leads to approximately a 5%
increase in the probability of wearing a wrist band while the indirect
exposure provides about a 13% increase in the probability of wearing a wrist
band. Again, a majority of the indirect effect can be attributed to the
treatment status of the nearest neighbor (about 10%).

Finally, there seems to be differences in the estimates of indirect effect
between Assumption 1 and Assumption 2. This could suggest a potential
violation of Assumption 2 for this dataset; while all estimates of standard
errors are considerably smaller under Assumption 2, more care needs to be
performed prior to moving forward with such an analysis. Rigorous methods for
testing Assumption 2 is an area of future research.

Table 5: Number of units in each exposure of Anti-Conflict Program Experiment
with $K=2$, $N=348$. Indirect
---
Direct | $(0,0)$ | $(0,1)$ | $(1,0)$ | $(1,1)$
Treated | 38 | 42 | 39 | 34
Control | 40 | 59 | 46 | 50
Table 6: Estimates of Anti-Conflict Program with K = 2 for only Treated School N = 348. Estimator | $\widehat{\delta}_{{HT,tot}}$ | $\widehat{\delta}_{{HT,dir}}$ | $\widehat{\delta^{*}}_{{HT,dir}}$ | $\widehat{\delta}_{{HT,ind}}$ | $\widehat{\delta^{*}}_{{HT,ind}}$ | $\widehat{\delta}_{{HT,1^{st}}}$ | $\widehat{\delta^{*}}_{{HT,1^{st}}}$ | $\widehat{\delta}_{{HT,2^{nd}}}$ | $\widehat{\delta^{*}}_{{HT,2^{nd}}}$
---|---|---|---|---|---|---|---|---|---
Estimates | 0.1899 | -0.0254 | 0.0559 | 0.2154 | 0.1340 | 0.1788 | 0.1019 | 0.0365 | 0.0320
S.E. | 0.0985 | 0.1332 | 0.0863 | 0.0927 | 0.0781 | 0.0822 | 0.0683 | 0.1148 | 0.0934

##  11 Conclusion

Substantial effort has been devoted to developing new techniques and models
for causal inference when interference is presence—that is, when the effect of
treatment is not limited to the unit that receives treatment, but can also
impact other units that interact with the original units. One recently-
proposed model for treatment interference is the $K$-nearest neighbors
interference model (KNNIM) (Alzubaidi and Higgins, 2022), in which the
treatment status of one unit can affect the response of the $K$ “closest”
units to that original unit.

We define causal estimands related to this model—including the average
$\ell$th nearest neighbor treatment effect (A$\ell$NNIE)—and show that the
indirect effect of treatment can be expressed as a sum of A$\ell$NNIE effects
That is, we show that the KNNIM model is able to determine how the treatment
statuses of each of the individual neighbors of a unit contribute to that
unit’s response. We derive unbiased estimators for these estimands and derive
conservative variance estimates for these unbiased estimators. We consider a
new assumption on the potential outcomes under treatment interference—no weak
interaction between direct and indirect effects—and extend these estimators
under this assumption. We perform a simulation study to determine the efficacy
of these estimators under various settings.

We apply our methodology to a recent experiment testing the efficacy of a
program designed to reduce conflict in middle schools in New Jersey (Paluck et
al., 2016). This experiment has two appealing characteristics: it was designed
to propagate through peer-to-peer interactions between students and students
were explicitly surveyed about which students were their “closest
connections.” Using our approach, we give evidence that that one measure in
particular—wearing a wristband given for demonstrating anti-conflict
behaviors—is affected primarily by the students’ closest connection.

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SUPPLEMENTARY MATERIAL

##  Appendix A Additional simulation results

We provide complete simulation results for all models of response given in
Table 1.

###  A.1 Estimates under complete randomization

Table 7: Estimates Under Completely Randomized Design Model 2 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 1 | 1.0707 | 1.306 | 1.1430 | 1.3314
$\widehat{\delta^{*}}_{{HT,tot}}$ | 1 | 1.0707 | 1.306 | 1.1430 | 1.3314
$\widehat{\delta}_{{HT,dir}}$ | 1 | 1.0265 | 0.7273 | 0.8528 | 0.8050
$\widehat{\delta^{*}}_{{HT,dir}}$ | 1 | 1.0300 | 0.3377 | 0.5812 | 0.6214
$\widehat{\delta}_{{HT,ind}}$ | 0 | 0.0442 | 0.9404 | 0.9697 | 0.9393
$\widehat{\delta^{*}}_{{HT,ind}}$ | 0 | 0.0406 | 0.6081 | 0.7798 | 0.7381
$\widehat{\delta}_{{HT,1^{st}}}$ | 0 | 0.0346 | 0.6273 | 0.7920 | 0.6821
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 0 | 0.01758 | 0.3009 | 0.5486 | 0.4749
$\widehat{\delta}_{{HT,2^{nd}}}$ | 0 | -0.0111 | 0.4642 | 0.6813 | 0.6023
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 0 | -0.0052 | 0.3328 | 0.5769 | 0.5601
$\widehat{\delta}_{{HT,3^{rd}}}$ | 0 | 0.0207 | 0.4541 | 0.6738 | 0.6076
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 0 | 0.0282 | 0.3525 | 0.5937 | 0.5359
Table 8: Estimates Under Completely Randomized Design Model 3 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 4 | 4.0489 | 2.9225 | 1.7095 | 3.3552
$\widehat{\delta^{*}}_{{HT,tot}}$ | 4 | 4.0489 | 2.9225 | 1.7095 | 3.3552
$\widehat{\delta}_{{HT,dir}}$ | 4 | 4.0047 | 2.2827 | 1.5108 | 2.5258
$\widehat{\delta^{*}}_{{HT,dir}}$ | 4 | 4.0292 | 0.9267 | 0.9626 | 2.1155
$\widehat{\delta}_{{HT,ind}}$ | 4 | 0.0442 | 0.9404 | 0.9697 | 0.9393
$\widehat{\delta^{*}}_{{HT,ind}}$ | 0 | 0.01978 | 1.1883 | 1.0901 | 1.7146
$\widehat{\delta}_{{HT,1^{st}}}$ | 0 | 0.0346 | 0.6273 | 0.7920 | 0.6821
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 0 | 0.01286 | 0.8246 | 0.9081 | 1.4569
$\widehat{\delta}_{{HT,2^{nd}}}$ | 0 | -0.0111 | 0.4642 | 0.6813 | 0.6023
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 0 | -0.01445 | 1.0133 | 1.0066 | 1.7389
$\widehat{\delta}_{{HT,3^{rd}}}$ | 0 | 0.0207 | 0.4541 | 0.6738 | 0.6076
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 0 | 0.0213 | 1.0711 | 1.0349 | 1.6299
Table 9: Estimates Under Completely Randomized Design Model 4 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 3.5 | 3.5526 | 2.5285 | 1.5901 | 2.8557
$\widehat{\delta^{*}}_{{HT,tot}}$ | 3.5 | 3.5526 | 2.5285 | 1.5901 | 2.8557
$\widehat{\delta}_{{HT,dir}}$ | 0 | -0.0123 | 2.9637 | 1.7215 | 3.2323
$\widehat{\delta^{*}}_{{HT,dir}}$ | 0 | 0.0073 | 0.8913 | 0.9441 | 1.5485
$\widehat{\delta}_{{HT,ind}}$ | 3.5 | 3.5649 | 1.5750 | 1.2550 | 2.3108
$\widehat{\delta^{*}}_{{HT,ind}}$ | 3.5 | 3.5453 | 0.9523 | 0.9758 | 1.6828
$\widehat{\delta}_{{HT,1^{st}}}$ | 2 | 2.0414 | 0.8733 | 0.9345 | 1.1102
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 2 | 2.0261 | 0.3901 | 0.6245 | 0.7582
$\widehat{\delta}_{{HT,2^{nd}}}$ | 1 | 1.0003 | 1.4039 | 1.1848 | 1.9204
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 1 | 0.9961 | 0.7423 | 0.8615 | 1.4553
$\widehat{\delta}_{{HT,3^{rd}}}$ | 0.5 | 0.5230 | 1.8611 | 1.3642 | 2.5928
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 0.5 | 0.5230 | 1.1157 | 1.0563 | 1.8726
Table 10: Estimates Under Completely Randomized Design Model 6 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 7.5 | 7.5236 | 7.0767 | 2.6602 | 8.6687
$\widehat{\delta^{*}}_{{HT,tot}}$ | 7.5 | 7.5236 | 7.0767 | 2.6602 | 8.6687
$\widehat{\delta}_{{HT,dir}}$ | 4 | 3.9586 | 7.9109 | 2.8126 | 8.1389
$\widehat{\delta^{*}}_{{HT,dir}}$ | 4 | 4.0061 | 2.3713 | 1.5399 | 4.4075
$\widehat{\delta}_{{HT,ind}}$ | 3.5 | 3.5649 | 1.5750 | 1.2550 | 2.3108
$\widehat{\delta^{*}}_{{HT,ind}}$ | 3.5 | 3.5174 | 2.1777 | 1.4757 | 3.5289
$\widehat{\delta}_{{HT,1^{st}}}$ | 2 | 2.0414 | 0.8733 | 0.9345 | 1.1102
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 2 | 2.0198 | 1.1942 | 1.0928 | 2.3348
$\widehat{\delta}_{{HT,2^{nd}}}$ | 1 | 1.0003 | 1.4039 | 1.1848 | 1.9204
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 1 | 0.9838 | 2.1622 | 1.4704 | 4.1131
$\widehat{\delta}_{{HT,3^{rd}}}$ | 0.5 | 0.5230 | 1.8611 | 1.3642 | 2.5928
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 0.5 | 0.5138 | 3.0275 | 1.7399 | 4.7936
Table 11: Estimates Under Completely Randomized Design Model 7 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 6 | 6.0344 | 4.9972 | 2.2354 | 6.0021
$\widehat{\delta^{*}}_{{HT,tot}}$ | 6 | 6.0344 | 4.9972 | 2.2354 | 6.0021
$\widehat{\delta}_{{HT,dir}}$ | 0 | -0.0452 | 7.4190 | 2.7237 | 7.9636
$\widehat{\delta^{*}}_{{HT,dir}}$ | 0 | -0.0091 | 2.0143 | 1.4192 | 3.4718
$\widehat{\delta}_{{HT,ind}}$ | 6 | 6.0797 | 2.7909 | 1.6705 | 4.850
$\widehat{\delta^{*}}_{{HT,ind}}$ | 6 | 6.0436 | 1.6675 | 1.2913 | 3.5938
$\widehat{\delta}_{{HT,1^{st}}}$ | 3 | 3.0449 | 1.1582 | 1.0762 | 1.6049
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 3 | 3.0295 | 0.5412 | 0.7357 | 1.1811
$\widehat{\delta}_{{HT,2^{nd}}}$ | 2 | 2.0092 | 2.8500 | 1.6882 | 3.9616
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 2 | 1.9961 | 1.4097 | 1.1873 | 2.9671
$\widehat{\delta}_{{HT,3^{rd}}}$ | 1 | 1.0256 | 4.5078 | 2.1231 | 6.1172
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 1 | 1.0179 | 2.6139 | 1.6167 | 4.4043
Table 12: Estimates Under Completely Randomized Design Model 8 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 7 | 7.0272 | 6.3337 | 2.5166 | 7.7149
$\widehat{\delta^{*}}_{{HT,tot}}$ | 7 | 7.0272 | 6.3337 | 2.5166 | 7.7149
$\widehat{\delta}_{{HT,dir}}$ | 1 | 0.9474 | 8.9409 | 2.9901 | 9.4071
$\widehat{\delta^{*}}_{{HT,dir}}$ | 1 | 0.9905 | 2.4239 | 1.5568 | 4.1241
$\widehat{\delta}_{{HT,ind}}$ | 6 | 6.0797 | 2.7909 | 1.6705 | 4.8503
$\widehat{\delta^{*}}_{{HT,ind}}$ | 6 | 6.0366 | 1.9707 | 1.4038 | 4.0101
$\widehat{\delta}_{{HT,1^{st}}}$ | 3 | 3.0449 | 1.1582 | 1.0762 | 1.6049
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 3 | 3.0280 | 0.6690 | 0.8179 | 1.4541
$\widehat{\delta}_{{HT,2^{nd}}}$ | 2 | 2.0092 | 2.8500 | 1.6882 | 3.9616
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 2 | 1.9930 | 1.7408 | 1.3194 | 3.6167
$\widehat{\delta}_{{HT,3^{rd}}}$ | 1 | 1.0256 | 4.5078 | 2.1231 | 6.1172
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 1 | 1.0156 | 3.1489 | 1.7745 | 5.2252

###  A.2 Estimates under Bernoulli randomization

Table 13: Estimates Under Bernoulli Randomization Model 2 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 1 | 1.0326 | 1.4420 | 1.2008 | 1.4406
$\widehat{\delta^{*}}_{{HT,tot}}$ | 1 | 1.0326 | 1.4420 | 1.2008 | 1.4406
$\widehat{\delta}_{{HT,dir}}$ | 1 | 1.0116 | 0.8786 | 0.9373 | 0.8960
$\widehat{\delta^{*}}_{{HT,dir}}$ | 1 | 1.0277 | 0.3573 | 0.5978 | 0.6357
$\widehat{\delta}_{{HT,ind}}$ | 0 | 0.0210 | 0.8232 | 0.9073 | 0.9258
$\widehat{\delta^{*}}_{{HT,ind}}$ | 0 | 0.0049 | 0.6462 | 0.8038 | 0.7994
$\widehat{\delta}_{{HT,1^{st}}}$ | 0 | 0.0502 | 0.6502 | 0.8063 | 0.6804
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 0 | 0.0142 | 0.2858 | 0.5346 | 0.4783
$\widehat{\delta}_{{HT,2^{nd}}}$ | 0 | -0.0181 | 0.4430 | 0.6656 | 0.5985
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 0 | -0.0177 | 0.3250 | 0.5700 | 0.5625
$\widehat{\delta}_{{HT,3^{rd}}}$ | 0 | -0.0110 | 0.4489 | 0.6700 | 0.6048
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 0 | 0.0084 | 0.3846 | 0.6202 | 0.5497
Table 14: Estimates Under Bernoulli Randomization Model 3 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 4 | 4.1129 | 4.0302 | 2.0075 | 4.5346
$\widehat{\delta^{*}}_{{HT,tot}}$ | 4 | 4.1129 | 4.0302 | 2.0075 | 4.5346
$\widehat{\delta}_{{HT,dir}}$ | 4 | 4.0919 | 3.4654 | 1.8615 | 3.6069
$\widehat{\delta^{*}}_{{HT,dir}}$ | 4 | 4.0797 | 0.9582 | 0.9789 | 2.2095
$\widehat{\delta}_{{HT,ind}}$ | 0 | 0.0210 | 0.8232 | 0.9073 | 0.9258
$\widehat{\delta^{*}}_{{HT,ind}}$ | 0 | 0.0331 | 1.8081 | 1.3446 | 2.3816
$\widehat{\delta}_{{HT,1^{st}}}$ | 0 | 0.0502 | 0.6502 | 0.8063 | 0.6804
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 0 | 0.0051 | 0.8466 | 0.9201 | 1.5260
$\widehat{\delta}_{{HT,2^{nd}}}$ | 0 | -0.0181 | 0.4430 | 0.6656 | 0.5985
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 0 | -0.0456 | 1.0226 | 1.0112 | 1.7932
$\widehat{\delta}_{{HT,3^{rd}}}$ | 0 | -0.0110 | 0.4489 | 0.6700 | 0.6048
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 0 | 0.0737 | 1.2557 | 1.1205 | 1.7099
Table 15: Estimates Under Bernoulli Randomization Model 4 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 3.5 | 3.5995 | 3.3899 | 1.8411 | 3.7707
$\widehat{\delta^{*}}_{{HT,tot}}$ | 3.5 | 3.5995 | 3.3899 | 1.8411 | 3.7707
$\widehat{\delta}_{{HT,dir}}$ | 0 | 0.0745 | 3.1656 | 1.7792 | 3.4406
$\widehat{\delta^{*}}_{{HT,dir}}$ | 0 | 0.0552 | 0.9018 | 0.9496 | 1.6046
$\widehat{\delta}_{{HT,ind}}$ | 3.5 | 3.5249 | 1.8276 | 1.3519 | 2.5207
$\widehat{\delta^{*}}_{{HT,ind}}$ | 3.5 | 3.5443 | 1.5038 | 1.2262 | 2.1947
$\widehat{\delta}_{{HT,1^{st}}}$ | 2 | 2.0532 | 0.9882 | 0.9940 | 1.1739
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 2 | 2.0206 | 0.3871 | 0.6222 | 0.7689
$\widehat{\delta}_{{HT,2^{nd}}}$ | 1 | 0.9878 | 1.4253 | 1.1938 | 2.0036
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 1 | 0.9675 | 0.8350 | 0.9137 | 1.5577
$\widehat{\delta}_{{HT,3^{rd}}}$ | 0.5 | 0.4838 | 2.2543 | 1.5014 | 2.8237
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 0.5 | 0.5561 | 1.5671 | 1.2518 | 2.1355
Table 16: Estimates Under Bernoulli Randomization Model 6 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 7.5 | 7.7065 | 10.8531 | 3.2944 | 12.6623
$\widehat{\delta^{*}}_{{HT,tot}}$ | 7.5 | 7.7065 | 10.8531 | 3.2944 | 12.6623
$\widehat{\delta}_{{HT,dir}}$ | 4 | 4.1815 | 10.0122 | 3.1642 | 10.4070
$\widehat{\delta^{*}}_{{HT,dir}}$ | 4 | 4.1245 | 2.4463 | 1.5640 | 4.7047
$\widehat{\delta}_{{HT,ind}}$ | 3.5 | 3.5249 | 1.8276 | 1.3519 | 2.5207
$\widehat{\delta^{*}}_{{HT,ind}}$ | 3.5 | 3.5819 | 4.4755 | 2.1155 | 5.7415
$\widehat{\delta}_{{HT,1^{st}}}$ | 2 | 2.0532 | 0.9882 | 0.9940 | 1.1739
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 2 | 2.0084 | 1.1391 | 1.0672 | 2.3490
$\widehat{\delta}_{{HT,2^{nd}}}$ | 1 | 0.9878 | 1.4253 | 1.1938 | 2.0036
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 1 | 0.9304 | 2.4611 | 1.5687 | 4.4098
$\widehat{\delta}_{{HT,3^{rd}}}$ | 0.5 | 0.4838 | 2.2543 | 1.5014 | 2.8237
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 0.5 | 0.6431 | 4.0935 | 2.0232 | 5.3983
Table 17: Estimates Under Bernoulli Randomization Model 7 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 6 | 6.1664 | 7.4275 | 2.7253 | 8.5832
$\widehat{\delta^{*}}_{{HT,tot}}$ | 6 | 6.1664 | 7.4275 | 2.7253 | 8.5832
$\widehat{\delta}_{{HT,dir}}$ | 0 | 0.1386 | 7.8388 | 2.7997 | 8.5881
$\widehat{\delta^{*}}_{{HT,dir}}$ | 0 | 0.0872 | 2.0487 | 1.4313 | 3.6387
$\widehat{\delta}_{{HT,ind}}$ | 6 | 6.0277 | 3.5973 | 1.8966 | 5.4461
$\widehat{\delta^{*}}_{{HT,ind}}$ | 6 | 6.0791 | 3.2309 | 1.7974 | 5.0241
$\widehat{\delta}_{{HT,1^{st}}}$ | 3 | 3.0547 | 1.3709 | 1.1708 | 1.7503
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 3 | 3.0223 | 0.5510 | 0.7423 | 1.2101
$\widehat{\delta}_{{HT,2^{nd}}}$ | 2 | 1.9923 | 2.9090 | 1.7056 | 4.1353
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 2 | 1.9521 | 1.6716 | 1.2929 | 3.2181
$\widehat{\delta}_{{HT,3^{rd}}}$ | 1 | 0.9806 | 5.3764 | 2.3187 | 6.7556
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 1 | 1.1047 | 3.8051 | 1.9506 | 5.1576
Table 18: Estimates Under Bernoulli Randomization Model 8 Estimator | Effects | Emp.Estimates | Emp.Var | Emp.S.D. | Var Estimate
---|---|---|---|---|---
$\widehat{\delta}_{{HT,tot}}$ | 7 | 7.1931 | 9.6277 | 3.1028 | 11.2033
$\widehat{\delta^{*}}_{{HT,tot}}$ | 7 | 7.1931 | 9.6277 | 3.1028 | 11.2033
$\widehat{\delta}_{{HT,dir}}$ | 1 | 1.1654 | 9.7750 | 3.1265 | 10.5230
$\widehat{\delta^{*}}_{{HT,dir}}$ | 1 | 1.1045 | 2.4792 | 1.5745 | 4.3616
$\widehat{\delta}_{{HT,ind}}$ | 6 | 6.0277 | 3.5973 | 1.8966 | 5.4461
$\widehat{\delta^{*}}_{{HT,ind}}$ | 6 | 6.0885 | 4.0546 | 2.0136 | 5.9402
$\widehat{\delta}_{{HT,1^{st}}}$ | 3 | 3.0547 | 1.3709 | 1.1708 | 1.7503
$\widehat{\delta^{*}}_{{HT,1^{st}}}$ | 3 | 3.0192 | 0.6476 | 0.8047 | 1.4636
$\widehat{\delta}_{{HT,2^{nd}}}$ | 2 | 1.9923 | 2.9090 | 1.7056 | 4.1353
$\widehat{\delta^{*}}_{{HT,2^{nd}}}$ | 2 | 1.9428 | 2.0746 | 1.4403 | 3.9275
$\widehat{\delta}_{{HT,3^{rd}}}$ | 1 | 0.9806 | 5.3764 | 2.3187 | 6.7556
$\widehat{\delta^{*}}_{{HT,3^{rd}}}$ | 1 | 1.1264 | 4.5503 | 2.1331 | 6.0993