Exploiting neighborhood interference with low-order interactions under unit randomized design
-
Mayleen Cortez-Rodriguez
,
Matthew Eichhorn
Abstract
Network interference, where the outcome of an individual is affected by the treatment assignment of those in their social network, is pervasive in real-world settings. However, it poses a challenge to estimating causal effects. We consider the task of estimating the total treatment effect (TTE), or the difference between the average outcomes of the population when everyone is treated versus when no one is, under network interference. Under a Bernoulli randomized design, we provide an unbiased estimator for the TTE when network interference effects are constrained to low-order interactions among neighbors of an individual. We make no assumptions on the graph other than bounded degree, allowing for well-connected networks that may not be easily clustered. We derive a bound on the variance of our estimator and show in simulated experiments that it performs well compared with standard estimators for the TTE. We also derive a minimax lower bound on the mean squared error of our estimator, which suggests that the difficulty of estimation can be characterized by the degree of interactions in the potential outcomes model. We also prove that our estimator is asymptotically normal under boundedness conditions on the network degree and potential outcomes model. Central to our contribution is a new framework for balancing model flexibility and statistical complexity as captured by this low-order interactions structure.
1 Introduction
Accurately estimating causal effects is relevant in numerous applications, from pharmaceutical companies researching the efficacy of a new medication, to policy makers understanding the impact of social welfare programs, to social media companies evaluating the impact of different recommendation algorithms on user engagement across their platforms. Often, the entity interested in understanding a causal effect will design an experiment where they randomly assign subsets of the population to treatment (e.g., new medication) and to control (e.g., a placebo) and draw conclusions based on the observed outcomes of the participants (e.g., health outcomes). Other times, the entity may need to determine causal effects from observational data accrued in a previous study where they did not have full control over the treatment assignment mechanism.
Our work focuses on estimating the total treatment effect (TTE), or the difference between the average outcomes of the population when everyone is treated versus when no one is treated, given data collected from a randomized experiment. This estimand is sometimes referred to as the global average treatment effect, as in ref. [1]. The TTE is a quantity of interest in scenarios where the decision maker must choose between adopting the proposed treatment or sticking with the status quo. For example, a social media company may develop a new recommendation algorithm for suggesting content to their users, and they want to decide whether to roll out this new algorithm across their platform. As another example, suppose that a pharmaceutical company develops a vaccine for some infectious disease. Then, public health experts and officials must decide whether the vaccine is safe and efficacious enough to warrant its recommendation to the general population. As the side effects of the new treatment are unknown, the goal is to determine the efficacy of the treatment relative to the status quo baseline by running a budgeted randomized trial, where the number of treated individuals in the trial is limited for safety reasons.
The techniques and guarantees for estimating causal effects in classical causal inference heavily rely upon the stable unit treatment value assumption (SUTVA), which posits that the outcome of each individual is independent of the treatment assignment of all other individuals [2]. Unfortunately, SUTVA is violated in all the aforementioned applications because people influence and are impacted by their peers. In the presence of this network interference, an individual’s outcome is affected by the treatment assignment of others in their social network, and SUTVA no longer holds. Distinguishing between the direct effect of treatment on an individual and the network effect of others’ treatment on the individual can be challenging. This has resulted in a growing literature on causal inference in the presence of network (interference) effects, sometimes referred to as spillover or peer influence effects. In this work, we consider the task of estimating the TTE from unit randomized trials under neighborhood interference, when an individual is affected by the treatment of its direct neighbors but is otherwise unaffected by the treatment of the individuals outside their neighborhood. Furthermore, we focus on unit randomized designs (RDs), wherein individuals are independently assigned to either treatment or control in a randomized experiment. This is in contrast to cluster RDs, which have been proposed as an approach to address network interference for randomized experimental design but which may not be feasible in practice due to an incompatibility with existing experimental platforms.
Estimating a causal effect from randomized experiments involves two decisions. First, we must decide what kind of experiment to run, i.e., how we choose the individuals to participate in the study, and how we determine which individuals are assigned to receive the new treatment. In this article, we focus on randomization inference, in which we assume the entire population participates in the study, and the randomness arises from the assignment of treatment or control. Second, after the experiment is conducted, we must decide how to analyze the data and construct an estimate from the measured observations. The literature addressing network interference can largely be categorized as either constructing clever RDs to exploit clustering structure in the network, or designing new estimators that exploit structure in the potential outcomes model. Without making assumptions on either the network or the potential outcomes model, it is impossible to estimate the TTE under arbitrary interference [3–5].
Table 1 summarizes how different assumptions on the potential outcomes model or network structure lead to different types of solutions, with a focus on unbiased estimators and neighborhood interference models where the network is known. The columns correspond to assumptions on the network structure (in order of increasing generality from left to right), while the rows correspond to assumptions on the structure in the potential outcomes model (in order of increasing generality from top to bottom). The body of the graph lists proposed solutions for estimating the TTE under the corresponding assumptions on the network/model structure. For example, under a fully general network structure and a linear potential outcomes model, refs [6–11] propose using an ordinary least squares (OLS) estimator with a Bernoulli RD. Since we focus on solutions proposing unbiased estimators, we also list bounds on the variance of the estimators in the table, as a point of comparison. As Table 1 indicates, the literature has either focused on analyzing the Horvitz–Thompson estimator under new RDs that exploit network structure by increasing the correlation between neighbors’ treatment assignments or alternatively using regression-style estimators with Bernoulli RD, exploiting strong functional assumptions on the potential outcomes model. Without assuming any structure beyond neighborhood interference, the baseline solution of using the Horvitz–Thompson estimator under the Bernoulli RD is an unbiased estimator whose variance scales exponentially in the degree of the network.
Literature Summary. Each row corresponds to an assumption on the structure of the potential outcomes model, and each column corresponds to an assumption on the structure of network. We list a proposed solution under the corresponding model/network assumptions in the following order: an
| Model structure | Network structure | ||
|---|---|---|---|
|
|
|
Fully general | |
| Linear | Directions for future work | OLS, Bernoulli RD; [6–11] | |
| Generalized linear | Regression/machine learning, Bernoulli RD; [11] | ||
|
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SNIPE, Bernoulli RD;
|
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| Arbitrary neighborhood interference |
Horvitz–Thompson, Cluster RD;
|
Horvitz–Thompson, Randomized Cluster RD;
|
Horvitz–Thompson, Bernoulli RD;
|
In our work, we propose a hierarchy of model classes that extrapolates between simple linear models and complex general models, such that a practitioner can choose the strength of the model assumptions they are willing to impose. Naturally, assuming a more limited model simplifies the causal inference task. We characterize the complexity of a potential outcomes model with the order of interactions
Assuming a
We derive a bound on the variance of our estimator, which scales polynomially in the degree
For a
We provide experimental results to illustrate that using regression models that do not allow for heterogeneity among the network effects can lead to considerable bias when the anonymous interference assumption is violated. The experiments validate that our estimator is unbiased for
To interpret the upper bound on the variance of our proposed estimator for the TTE, we note that our variance scales as
2 Related literature
While there has been a flurry of recent activity in addressing the challenges that arise from network interference, every proposed solution concept fundamentally hinges upon making key assumptions on the form of network interference. Without any assumptions, the vector of observed outcomes under a particular treatment vector
Early works propose framing assumptions via exposure functions, or constant treatment response [3,4]. This assumes that there is some known exposure mapping,
The majority of works in the literature (along with our work) assume neighborhood interference with respect to a known graph. Notable exceptions include ref. [24], which considers a highly structured spatial interference setting with network effects decaying with distance, and [25], which provides methods for testing hypotheses about interference effects including higher order spillovers. Without imposing any additional assumptions on the potential outcomes besides neighborhood interference, a natural solution is to use the Horvitz–Thompson estimator to estimate the average outcomes under full neighborhood treatment and full neighborhood control [3]. While the estimator is unbiased, the variance of the estimator scales inversely with the probability that a unit’s full neighborhood is in either treatment or control. Under a Bernoulli(
One approach in response to the aforementioned challenge is to consider cleverly constructing RDs that increase overlap, i.e., the probability that a unit’s entire neighborhood is assigned to treatment or control. The earliest literature in this line of work additionally assumes partial interference, also referred to as the multiple networks setting, in which the population can be partitioned into disjoint groups, and network interference only occurs within groups and not across groups [12,14,15,20,21,26,27]. This assumption makes sense in contexts where interference is only expected between naturally clustered groups of individuals, such as households, cities, or countries. Given knowledge of the groups, we can then randomly assign groups to different treatment saturation levels, e.g., jointly assigning groups to either treatment or control, increasing the correlation of treatments within neighborhoods. Then, a difference in means estimator or a Horvitz–Thompson estimator can be used to estimate the TTE. The asymptotic consistency of these estimators relies on the number of groups going to infinity, with a variance scaling inversely with the fraction of treated clusters, i.e.,
For general connected graphs, one can still implement a cluster-based RD on constructed clusters, where the clusters are constructed to minimize the number of edges between clusters [1,7,16,17]. References [1,17] provide guarantees for graphs exhibiting a restricted growth property, which limits the rate at which local neighborhoods of the graph can grow in size, and ref. [1] proves that using randomized cluster RD along with the Horvitz–Thompson estimator achieves a variance of
The alternate approach that has gained traction empirically is to impose additional functional assumptions on the potential outcomes in addition to anonymous neighborhood interference. The most common assumption is that the potential outcomes are linear with respect to a particular statistic of the treatment vector, where the linear function is shared across units [6–11]. Under this assumption, estimating the entire potential outcomes function reduces to linear regression, which one could solve using OLS, as indicated in the upper right most cell of Table 1. After recovering the linear model, one could estimate any desired causal estimand. The results rely on correctly choosing the statistic that governs the linearity, or more generally reduces the task to heuristic feature engineering for generalized linear models [11]. One could plausibly extend the function class beyond linear and instead apply machine learning techniques to estimate the function that generalizes beyond linear regression. While we do not state a variance bound in Table 1, one would expect that when
While we have primarily focused on summarizing the literature for unbiased estimators, there has also been some work considering how structure in the potential outcomes model can be exploited to reduce bias of standard estimators. References [32–34] use application domain-specific structure in two-sided marketplaces and network routing to compare the bias of the difference in means estimators under different experimental designs. Reference [35] uses structure in ridesharing platforms to propose a new estimator with reduced bias. In addition, there has been some limited empirical work studying bias under model misspecification [16].
Complementary to the literature on randomized experiments, there has been a growing literature considering causal inference over observational studies in the presence of network interference. However, the limitations similarly mirror the aforementioned concerns. A majority of the literature assumes partial interference [15,36–39], i.e., the multiple networks setting, which then enables causal inference of a variety of different estimands. In particular, it is commonly assumed that the different groups in the network are sampled from some underlying distribution, and the statistical guarantees are given with respect to the number of groups going to infinity. Alternately, other works assume that the potential outcomes only depend on a simple and known statistic of the neighborhood treatment, most commonly the number or fraction of treated [11,40,41]. Either the neighborhood statistic only takes finite values, or assumptions are imposed on the functional form of the potential outcomes, which imply anonymous interference and reduce inference to a regression task or maximum likelihood calculation. Reference [42] considers a general exposure mapping model alongside an inverse propensity weighted estimator, but the estimator has high variance when the exposure mapping is complex.
In contrast to the majority of the mentioned literature, we neither rely on cluster RDs nor anonymous interference assumptions. We instead propose a potential outcomes model with low-order interactions structure, where the degree of interactions
3 Model
3.1 Causal network
Let
3.2 Potential outcomes model
To each individual
where
where
So far, the potential outcomes model described in (3.1) is completely general. However, it is parameterized by
Assumption 1
(Neighborhood interference)
Next, we note that the degree of each
Assumption 2
(Low polynomial degree) Each potential outcome function
where we define
We remark that while we use the formal mathematical term of “low polynomial degree,” since this describes a function over a vector of binary variables, a low polynomial degree constraint is equivalent to a constraint on the order of interactions amongst the treatments of neighbors. In the simplest setting when
We use the notation in (3.2) to express the potential outcomes model for the remainder of the article. If
Many of our variance bounds utilize an upper bound on the treatment effects for each individual. We define
It follows that
Remark 1
We emphasize that the model in (3.2) captures fully general neighborhood interference when
Remark 2
We can consider the order of interactions or polynomial degree
Remark 3
Assumption 2 implies that the model exhibits a particular type of sparsity in the coefficients with respect to the monomial basis, in which the coefficients corresponding to sets larger than size
3.2.1 Examples of low-degree interaction models
We provide a few examples to illustrate when the polynomial degree of the potential outcomes model may naturally be smaller than the total neighborhood size (i.e.,
Example 1
Consider a potential outcomes model that exhibits the joint assumptions of additivity of main effects and interference effects as defined in ref. [18]. This imposes that the potential outcomes satisfy
where
Example 2
Consider a hypothesized setting where each individual’s neighborhood can be divided into smaller sub-communities. For example, one’s close contacts may include their immediate family, their close friends, their work colleagues, etc. Within each of these sub-communities, there may be nontrivial (higher order) interactions between the treatments of its members. However, it is reasonable to assume that the cumulative effects of each sub-community have an additive effect on the individual’s outcome. That is, there are no nontrivial interactions resulting from the treatment of individuals across different sub-communities. In this case, a natural choice for
Example 3
Suppose a social networking platform is testing a new “hangout room” feature where groups of up to five people can interact in a novel environment on the platform. One can posit that a natural choice for
Example 4
Consider a setting where an individual’s outcome is a low-degree polynomial in some auxiliary quantity, which is itself linear in the treatment assignments of their neighborhood. For example, we might have
A similar setting is explored in our simulated experiments in Section 6.
Example 5
Consider an example in which network effects only arise from pairwise edge interactions and triangular interactions, i.e., individual
Remark 4
We emphasize that any potential outcomes model that takes a binary treatment vector
where
where
3.3 Causal estimand and RD
Throughout most of the article, we concern ourselves with estimating the TTE. This quantifies the difference between the average of individual’s outcomes when the entire population is treated versus the average of individual’s outcomes when the entire population is untreated:
where
Since exposing individuals to treatment can have a deleterious and irreversible effect on their outcomes, we wish to estimate the TTE after treating a small random subset of the population. We focus on a nonuniform Bernoulli design, wherein each individual
4 Estimator
In this section, we introduce the estimator that will serve as our central focus through the rest of the article: the SNIPE. While we restrict our attention to nonuniform Bernoulli design, the techniques to derive this estimator can be generalized to a wide variety of causal estimands and experimental designs. In fact, our estimator is connected to the Horvitz–Thompson estimator and turns out to be a special case of both the pseudoinverse estimator first introduced by Swaminathan et al. [45] and more recently the Riesz estimator of Harshaw et al. [46]. We discuss these connections in Section 4.3.
To provide intuition, we first derive the estimator in the
Recalling that
where the inner product argument in the second equation is the
4.1 Building intuition in the linear setting (
β
=
1
)
To motivate our estimator, we consider the linear heterogeneous potential outcomes model (
As a thought experiment toward estimating
To minimize the sum of squared deviations from the true coefficients, we use OLS, computing
We consider the limit of this estimate as
Similarly, the vector
Now, we return from our thought experiment to the actual experimental setting wherein a single instantiation
Note that the matrix
By applying linearity once more, we may obtain the unbiased estimator for the TTE,
For the specific setting of nonuniform Bernoulli design with
which we can invert to obtain
We calculate
By plugging into equation (4.3), we obtain the explicit form for our estimator:
where SNIPE
4.2 SNIPE in the general polynomial setting
For larger
Working with this matrix is significantly more tedious, so we relegate the details to Appendix A. There, we show that
where we define the coefficient function
for each
Remark 5
We pause here to again emphasize that our technique gives us an unbiased estimate of the coefficient vector
This estimator can be evaluated in
In the special case of a uniform treatment probability
Thus, while our estimation guarantees allow for heterogeneity in the potential outcomes model, when treatment probabilities are uniform, computing the estimator does not depend on the identity of the treated individuals, but only the number of treated neighbors. As a result, the estimator can be evaluated in
4.3 Connection to other estimator classes
Our estimator takes the form of a linear weighted estimator
under specifically constructed weight functions
4.3.1 Horvitz–Thompson estimator
First, we show that in the special case where
Thus,
so we exactly recover the Horvitz–Thompson estimator. As a result, when
4.3.2 Pseudoinverse estimator
The key technical steps of deriving our estimator can be described as constructing unbiased estimators for each unit
A primary goal of ref. [45] is to estimate
is the minimizer of the mean squared error (MSE) with the minimum norm. As the reward distribution is unknown, it is replaced with an empirical estimate of
4.3.3 Riesz estimator
In their recent work, Harshaw et al. [46] consider a very general causal inference framework wherein a treatment
In our setting,
Under two assumptions – correct model specification (needed in our work as well) and positivity (always satisfied in our setting since Bernoulli design ensures each treatment in
where
for each
for each
5 Properties of estimator under Bernoulli design
The following theorem summarizes the key properties of our estimator.
Theorem 1
Under a potential outcomes model satisfying the neighborhood interference assumption with polynomial degree at most
where each
Notably, a sequence of networks with
To understand the variance bounds for our estimator, we can compare against the variance of Horvitz–Thompson under a Bernoulli design. In the simple setting of a
We also remark that when
Then, using an OLS estimator would give an unbiased estimate with lower variance than our estimator. Thus, our model and estimator can be viewed as simply “adding to the toolbox” that practitioners may use depending on how expressive they need their potential outcomes models to be.
In the special setting of uniform Bernoulli design and
5.1 Minimax lower bound on mean squared error
To understand the optimality of our estimator, we construct a lower bound on the minimax optimal mean squared error rate. In particular, we show that for a setting with a
Theorem 2
(Minimax lower bound) For any
For a
The proof of Theorem 2 (given in Appendix C) uses a generalized variation of LeCam’s method for fuzzy hypothesis testing [47], [48, Sec. 2.7.4]. We reduce the problem of
While our lower bound clearly indicates that the exponential dependence on
5.2 Central limit theorem
In this section, we show
[53, Theorem 3.6]. Let
where
Our estimator can be written in the form of
Assumption 3
For some constant
This is a typical assumption made in the literature [3,51,54] that rules out degenerate cases, such as all the potential outcomes being 0, that would result in the estimator having an unnaturally low variance scaling smaller than
Theorem 3
(Central limit theorem) Under Assumptions
1–3, and assuming that
Theorem 3 states that our estimator is asymptotically normal, assuming the boundedness of the magnitude of the potential outcomes, the polynomial degree, and the network degree. For the complete proof, refer to Appendix D. The proof follows from a straightforward application of Theorem 3.6 from ref. [53] with appropriate bounds for the moments
5.3 Variance estimator
The central limit theorem result in Theorem 3 implies that we can construct asymptotically valid confidence intervals by
Note that both
If
Otherwise, if
The expectation of this estimate will be higher than the true term in the variance via Cauchy–Schwarz inequality. Therefore, a conservative variance estimator
where
In the Section 6, we compare the empirical variance of the SNIPE estimator
6 Experimental results
Using computational experiments on simulated data, we compare the performance of our estimator with existing estimators. By using an Erdös–Rényi model, we generate random directed graphs of
where
Plots visualizing the performance of various
6.1 Other estimators
We compare the performance of SNIPE with the performance of least-squares regression and difference-in-means estimators, also as in ref. [43]. Although the Horvitz–Thompson estimator is unbiased in this setting, under unit Bernoulli RD its variance is very high in practice. Thus, we omit this estimator from our experimental results and comparison. Another related estimator is the Hájek estimator, which is only approximately unbiased but with lower variance than the Horvitz–Thompson estimator. However, under our RD, its variance is still very high in practice. In addition, as we implemented a Bernoulli RD on a graph with expected network degree of 10, both the Hájek and Horvitz–Thompson estimators consistently took a value of 0, giving us no meaningful results to consider. For these reasons, we chose to omit these two estimators from our experimental results and comparison. The simplest difference-in-means estimator is the difference between the average outcome of individuals assigned to treatment and the average outcome of individuals assigned to control, given by
This estimator does not take into account any information about each individual’s neighborhood and is biased under network interference. We also consider a modified version of this estimator that uses information about the number of treated neighbors of each individual. Let
for some user-defined tolerance
We also compare with least-squares regression models of degree
for some covariate
When we set
6.2 Results and discussion
For each population size
Figures 1 and 2 visualize the effects of various network or estimator parameters on the performance of each of the four TTE estimators described in Section 6.1 and
Plots visualizing the MSE of various
The top row of plots in Figure 1 features results for a linear (
6.3 Variance estimator experiments
We compare the empirical variance of SNIPE with the variance bound from Theorem 1 and the variance estimator constructed from Aronow–Samii’s method described in Section 5.3. As mentioned earlier, for each population size
Table presenting the empirical variance of the SNIPE
| Experimental variance | Variance estimate | Variance bound | |
|---|---|---|---|
| Results for SNIPE(1) | |||
|
|
|||
|
|
17.63 | 16327.43 | 488300.11 |
|
|
8.19 | 3724.33 | 245279.16 |
|
|
3.34 | 2270.81 | 140688.10 |
|
|
2.33 | 1408.50 | 106981.33 |
|
|
1.95 | 1437.32 | 105354.62 |
|
|
|||
|
|
2.93 | 5439.83 | 272939.86 |
|
|
3.71 | 2375.13 | 174599.20 |
|
|
4.33 | 1259.70 | 106516.48 |
|
|
6.10 | 1660.36 | 122405.92 |
|
|
8.06 | 2197.06 | 93585.34 |
| r (n = 5,000, p = 0.2) | |||
|
|
1.06 | 1480.23 | 19410.59 |
|
|
1.69 | 1793.46 | 44527.96 |
|
|
2.50 | 3802.44 | 112805.72 |
|
|
3.91 | 2080.84 | 142140.47 |
| Results for SNIPE(2) | |||
|
|
|||
|
|
255.59 | 11046.03 | 470718.02 |
|
|
92.21 | 3873.10 | 231156.58 |
|
|
46.36 | 2962.08 | 147815.75 |
|
|
29.80 | 1495.31 | 95530.25 |
|
|
21.28 | 1463.03 | 81113.89 |
|
|
|||
|
|
114.95 | 8635.44 | 147108.70 |
|
|
43.94 | 2142.06 | 133774.85 |
|
|
24.36 | 1373.10 | 129746.07 |
|
|
13.24 | 1115.97 | 128101.61 |
|
|
9.19 | 2435.29 | 133271.42 |
|
|
|||
|
|
12.99 | 1627.89 | 126414.68 |
|
|
20.32 | 1731.70 | 129161.00 |
|
|
30.94 | 2054.55 | 134921.48 |
|
|
44.50 | 2494.29 | 144292.22 |
The variance bound is computed using the bound in Theorem 1 and the variance estimate is computed using the Aronow–Samii estimator described in section 5.3. The parameters
7 Conclusions and future work
We propose an estimator for the TTE under neighborhood interference and Bernoulli design when the graph is known. Our approach considers a potential outcomes model that is polynomial in the treatment vector
Our work presents many interesting and likely fruitful directions for future work. We summarize a few of these in the following section.
7.1 Optimized experimental designs
In this work, we studied the relationship between the complexity of the potential outcomes model (parameterized by
7.2 Implications to observational studies
While our stated theoretical results only hold for nonuniform Bernoulli designs, there are natural implications to the analysis of observational studies under appropriate unconfoundedness assumptions. In particular, if treatments across individuals are independent from each other conditioned on observed covariates, and if the conditional treatment probabilities could be estimated, then one could plausibly consider a plugin approach to modify our estimator for such observational data. Formalizing how to extend our results to observational settings would be a fruitful and interesting direction for the future work.
7.3 Dealing with model misspecification
Another interesting direction for future work centers around how to use our proposed class of estimators when the model parameter
Acknowledgements
We gratefully acknowledge the financial support from the National Science Foundation grants CCF-1948256 and CNS-1955997 and the National Science Foundation Graduate Research Fellowship grant DGE-1650441. We also acknowledge support from the Air Force Research Laboratory grant AFOSR FA9550-23-1-0301. We also thank Professor Nathan Kallus for his insightful feedback.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Conflict of interest: The authors state no conflict of interest.
-
Data availability statement: The Python scripts for the experiments and the data used in our results are available at: https://github.com/mayscortez/low-order-unitRD.
Appendix A The explicit TTE estimator for general
β
Here, we derive an explicit formula for the
where each
Here, the second equality uses the fact that each
Lemma 1
The matrix
Proof
We will argue that
Next, we consider the off-diagonal entries. By the symmetry of
Here, the last line follows because any non-zero term in the outer sum must correspond to some
By using this lemma, we consider the entries in the first argument vector of each inner product. We have,
where we define
Finally, we replace
B Proof of Theorem 1
B.1 Unbiasedness
The key insight that we use in our unbiasedness calculations comes from the following lemma.
Lemma 2
If
Proof
By the mutual independence of the
Note that the expectations in the first product each simplify to 0, so this expectation is non-zero only when
The critical feature of this lemma is that this indicator function simplifies sums over arbitrary sets to sums over subsets
We leverage this fact in the following calculation. Given any
By applying the linearity of expectation and the previous result, we calculate
B.2 Variance bound
To bound the variance of this estimator, we make use of the following lemma to bound the magnitude of each
Lemma 3
For any
Proof
First, note that
This next lemma is used to bound the covariance terms that appear in our final calculation.
Lemma 4
Suppose that
where
Proof
We reason separately about the two terms in the covariance expansion. By Lemma 2,
Next, we reason about the expectation of the product term. Since the
We partition the elements of
We are ready to bound the variance. If
By plugging in our bounds from Lemmas 3 and 4, we can simplify this bound:
Via the change of variables
Here, the final inequality makes use of the bound
C Proof of Theorem 2
We use a variation of LeCam’s method, which allows us to recast the hardness of TTE estimation through the lens of hypothesis testing. We consider the following setting.
The causal network is a
The coefficients
where
All units have a uniform treatment probability
We wish to compute a lower bound on the mean squared error of any estimator for
Here, the first inequality lower bounds the conditional expectation by
Now, consider designing a hypothesis test
By rearranging this inequality and plugging into (A1), we may continue the simplification
Here, the second line follows because we have expanded the support of the infimum to all hypothesis tests, not just those that make use of an estimator
Next, we derive an upper bound for this KL-divergence. By applying the definition, we have
Here, the second equality uses the fact that the treatment assignments are independent from the random model coefficients. Now, conditioned on the treatment assignment
where the positive expectation comes from
By plugging into our earlier results, we find that
By taking
This is
D Proof of Theorem 3
Proof of Theorem 3
We apply Theorem 3.6 from ref. [53] to the following defined random variables,
where
such that
First, we note that
The second inequality in (A1) follows from Lemma 3, which upper bounds
By recognizing that
Since
In the same way, we can bound
Since
Let
where recall that
By boundedness of
E Other estimands
E.1 Average treatment effect
The average treatment effect (ATE) measures the average effect that one’s own treatment has on their outcome (assuming no one else is treated).
The estimator for ATE takes the form
where
E.2 Conditional average treatment effect
Given any subdemographic of the population
By the same calculation as mentioned earlier, we estimate the conditional average treatment effect:
E.3 Size-dependent treatment effects
Although not standard in the literature, as its significance is largely brought about by the low-degree polynomial structure of our potential outcomes model, another family of causal estimands can be used to understand the magnitude of the treatment effects that individuals experience as a result of different sized subsets of their neighborhood being treated. We define the
for some parameter
References
[1] Ugander J, Yin H. Randomized graph cluster randomization. 2020. https://arxiv.org/abs/2009.02297. Search in Google Scholar
[2] Rubin DB. Randomization analysis of experimental data: the fisher randomization test comment. J Amer Stat Assoc. 1980;75(371):591–3. http://www.jstor.org/stable/2287653. 10.2307/2287653Search in Google Scholar
[3] Aronow PM, Samii C. Estimating average causal effects under general interference, with application to a social network experiment. Ann Appl Stat. 2017;11(4):1912–47. 10.1214/16-AOAS1005Search in Google Scholar
[4] Manski CF. Identification of treatment response with social interactions. Econom J. 2013;16(1):S1–23. 10.1111/j.1368-423X.2012.00368.xSearch in Google Scholar
[5] Basse GW, Airoldi EM. Limitations of design-based causal inference and A/B testing under arbitrary and network interference. Sociol Methodol. 2018;48(1):136–51. 10.1177/0081175018782569Search in Google Scholar
[6] Toulis P, Kao E. Estimation of causal peer influence effects. In: International Conference on Machine Learning; 2013. p. 1489–97. Search in Google Scholar
[7] Gui H, Xu Y, Bhasin A, Han J. Network a/b testing: From sampling to estimation. In: Proceedings of the 24th International Conference on International Conferences Steering Committee; 2015. p. 399–409. 10.1145/2736277.2741081Search in Google Scholar
[8] Basse GW, Airoldi EM. Model-assisted design of experiments in the presence of network-correlated outcomes. Biometrika. 2018;105(4):849–58. 10.1093/biomet/asy036Search in Google Scholar
[9] Cai J, De Janvry A, Sadoulet E. Social networks and the decision to insure. Amer Econ J Appl Econ. 2015;7(2):81–108. 10.1257/app.20130442Search in Google Scholar
[10] Parker BM, Gilmour SG, Schormans J. Optimal design of experiments on connected units with application to social networks. J R Stat Soc Ser C (Appl Stat.) 2017;66(3):455–80. https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssc.12170. 10.1111/rssc.12170Search in Google Scholar
[11] Chin A. Regression adjustments for estimating the global treatment effect in experiments with interference. J Causal Inference. 2019;7(2):20180026. 10.1515/jci-2018-0026Search in Google Scholar
[12] Sobel ME. What do randomized studies of housing mobility demonstrate? J Amer Stat Assoc. 2006;101(476):1398–407. 10.1198/016214506000000636. Search in Google Scholar
[13] Rosenbaum PR. Interference between units in randomized experiments. J Amer Stat Assoc. 2007;102(477):191–200. 10.1198/016214506000001112. Search in Google Scholar
[14] Hudgens MG, Halloran ME. Toward causal inference with interference. J Amer Stat Assoc. 2008;103:832–42. https://EconPapers.repec.org/RePEc:bes:jnlasa:v:103:y:2008:m:june:p:832-842. 10.1198/016214508000000292Search in Google Scholar PubMed PubMed Central
[15] Tchetgen EJT, VanderWeele TJ. On causal inference in the presence of interference. Stat Meth Med Res. 2012;21(1):55–75. PMID: 21068053. 10.1177/0962280210386779. Search in Google Scholar PubMed PubMed Central
[16] Eckles D, Karrer B, Ugander J. Design and analysis of experiments in networks: reducing bias from interference. J Causal Inference. 2017;5(1):20150021. 10.1515/jci-2015-0021Search in Google Scholar
[17] Ugander J, Karrer B, Backstrom L, Kleinberg J. Graph cluster randomization: Network exposure to multiple universes. In: Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM; 2013. p. 329–37. 10.1145/2487575.2487695Search in Google Scholar
[18] Sussman DL, Airoldi EM. Elements of estimation theory for causal effects in the presence of network interference. 2017. http://arXiv.org/abs/arXiv:170203578. Search in Google Scholar
[19] Bargagli-Stoffi FJ, Tortù C, Forastiere L. Heterogeneous treatment and spillover effects under clustered network interference. 2020. http://arXiv.org/abs/arXiv:200800707. 10.2139/ssrn.3666101Search in Google Scholar
[20] Bhattacharya R, Malinsky D, Shpitser I. Causal inference under interference and network uncertainty. In: Adams RP, Gogate V, editors. Proceedings of The 35th Uncertainty in Artificial Intelligence Conference. vol. 115 of Proceedings of Machine Learning Research. PMLR; 2020. p. 1028–38. https://proceedings.mlr.press/v115/bhattacharya20a.html. Search in Google Scholar
[21] Liu L, Hudgens MG. Large sample randomization inference of causal effects in the presence of interference. J Amer Stat Assoc. 2014;109(505):288–301. PMID: 24659836. 10.1080/01621459.2013.844698. Search in Google Scholar PubMed PubMed Central
[22] Li S, Wager S. Random graph asymptotics for treatment effect estimation under network interference. 2020. https://arxiv.org/abs/2007.13302. Search in Google Scholar
[23] Viviano D. Experimental design under network interference. 2020. https://arxiv.org/abs/2003.08421. Search in Google Scholar
[24] Leung MP. Rate-optimal cluster-randomized designs for spatial interference. 2021. https://arxiv.org/abs/2111.04219. Search in Google Scholar
[25] Athey S, Eckles D, Imbens GW. Exact p-values for network interference. J Amer Stat Assoc. 2018;113(521):230–40. 10.1080/01621459.2016.1241178Search in Google Scholar
[26] VanderWeele TJ, TchetgenTchetgen EJ, Halloran ME. Interference and sensitivity analysis. Statist Sci. 2014 Nov;29(4):687–706. 10.1214/14-STS479. Search in Google Scholar PubMed PubMed Central
[27] Auerbach E, Tabord-Meehan M. The local approach to causal inference under network interference. 2021. http://arXiv.org/abs/arXiv:210503810. Search in Google Scholar
[28] Taljaard M, Weijer C, Grimshaw JM, BelleBrown J, Binik A, Boruch R, et al. Ethical and policy issues in cluster randomized trials: rationale and design of a mixed methods research study. Trials. 2009;10(1):1–10. 10.1186/1745-6215-10-61Search in Google Scholar PubMed PubMed Central
[29] Edwards SJ, Braunholtz DA, Lilford RJ, Stevens AJ. Ethical issues in the design and conduct of cluster randomised controlled trials. Bmj. 1999;318(7195):1407–9. 10.1136/bmj.318.7195.1407Search in Google Scholar PubMed PubMed Central
[30] Hutton JL. Are distinctive ethical principles required for cluster randomized controlled trials? Stat Med. 2001;20(3):473–88. 10.1002/1097-0258(20010215)20:3<473::AID-SIM805>3.0.CO;2-DSearch in Google Scholar
[31] Donner A, Klar N. Pitfalls of and controversies in cluster randomization trials. Amer J Public Health. 2004;94(3):416–22. PMID: 14998805. 10.2105/AJPH.94.3.416. Search in Google Scholar
[32] Johari R, Li H, Liskovich I, Weintraub GY. Experimental design in two-sided platforms: an analysis of bias. Manag Sci. 2022;68:7069–89. 10.1287/mnsc.2021.4247Search in Google Scholar
[33] Li H, Zhao G, Johari R, Weintraub GY. Interference, bias, and variance in two-sided marketplace experimentation: guidance for platforms. In: Proceedings of the ACM Web Conference 2022; 2022. p. 182–92. 10.1145/3485447.3512063Search in Google Scholar
[34] Spang B, Hannan V, Kunamalla S, Huang TY, McKeown N, Johari R. Unbiased experiments in congested networks. In: Proceedings of the 21st ACM Internet Measurement Conference; 2021. p. 80–95. 10.1145/3487552.3487851Search in Google Scholar
[35] Bright I, Delarue A, Lobel I. Reducing marketplace interference bias via shadow prices. 2022. http://arXiv.org/abs/arXiv:220502274. 10.1145/3580507.3597738Search in Google Scholar
[36] Perez-Heydrich C, Hudgens MG, Halloran ME, Clemens JD, Ali M, Emch ME. Assessing effects of cholera vaccination in the presence of interference. Biometrics. 2014;70(3):731–41. 10.1111/biom.12184Search in Google Scholar
[37] Liu L, Hudgens MG, Becker-Dreps S. On inverse probability-weighted estimators in the presence of interference. Biometrika. 2016;103(4):829–42. 10.1093/biomet/asw047Search in Google Scholar
[38] DiTraglia FJ, Garcia-Jimeno C, O’Keeffe-O’Donovan R, Sanchez-Becerra A. Identifying causal effects in experiments with spillovers and non-compliance. 2020. https://arxiv.org/abs/2011.07051. Search in Google Scholar
[39] Vazquez-Bare G. Identification and estimation of spillover effects in randomized experiments. J Econom. 2022;105237. 10.1016/j.jeconom.2021.10.014Search in Google Scholar
[40] Verbitsky-Savitz N, Raudenbush SW. Causal inference under interference in spatial settings: a case study evaluating community policing program in Chicago. Epidemiol Meth. 2012;1(1):107–30. 10.1515/2161-962X.1020Search in Google Scholar
[41] Ogburn EL, Sofrygin O, Diaz I, Van der Laan MJ. Causal inference for social network data. 2017. http://arXiv.org/abs/arXiv:170508527. Search in Google Scholar
[42] Forastiere L, Airoldi EM, Mealli F. Identification and estimation of treatment and interference effects in observational studies on networks. J Amer Stat Assoc. 2021;116(534):901–18. 10.1080/01621459.2020.1768100Search in Google Scholar
[43] Cortez M, Eichhorn M, Yu CL. Staggered rollout designs enable causal inference under interference without network knowledge. In: Koyejo S, Mohamed S, Agarwal A, Belgrave D, Cho K, Oh A, editors. Advances in Neural Information Processing Systems. Curran Associates, Inc.; Vol. 35. 2022.Search in Google Scholar
[44] Yu CL, Airoldi E, Borgs C, Chayes J. Estimating the total treatment effect in randomized experiments with unknown network structure. Proceedings of the National Academy of Sciences. 2022;119(44):e2208975119. 10.1073/pnas.2208975119Search in Google Scholar PubMed PubMed Central
[45] Swaminathan A, Krishnamurthy A, Agarwal A, Dudik M, Langford J, Jose D, et al. Off-policy evaluation for slate recommendation. In: Guyon I, Von Luxburg U, Bengio S, Wallach H, Fergus R, Vishwanathan S, Garnett R, editors. Advances in Neural Information Processing Systems. Curran Associates, Inc.; Vol. 30. 2017.Search in Google Scholar
[46] Harshaw C, Sävje F, Wang Y. A design-based Riesz representation framework for randomized experiments. 2022. http://arXiv.org/abs/arXiv:221008698. Search in Google Scholar
[47] LeCam L. Convergence of estimates under dimensionality restrictions. Ann Stat. 1973;1:38–53. 10.1214/aos/1193342380Search in Google Scholar
[48] Tsybakov AB. Introduction to nonparametric estimation. New York: Springer; 2009. 10.1007/b13794Search in Google Scholar
[49] Chin A. Central limit theorems via Stein’s method for randomized experiments under interference. 2018. http://arXiv.org/abs/arXiv:180403105. Search in Google Scholar
[50] Leung MP. Treatment and spillover effects under network interference. Rev Econ Stat. 2020 May;102(2):368–80. 10.1162/rest_a_00818Search in Google Scholar
[51] Harshaw C, Sävje F, Eisenstat D, Mirrokni V, Pouget-Abadie J. Design and analysis of bipartite experiments under a linear exposure-response model. Electr J Stat. 2023;17(1):464–518. 10.1214/23-EJS2111Search in Google Scholar
[52] Ogburn EL, Sofrygin O, Diaz I, Van der Laan MJ. Causal inference for social network data. J Amer Stat Assoc. 2022:1–15. 10.1080/01621459.2022.2131557Search in Google Scholar
[53] Ross N. Fundamentals of Steinas method. Probabil Surveys. 2011;8:210–93. 10.1214/11-PS182Search in Google Scholar
[54] Leung MP. Rate-optimal cluster-randomized designs for spatial interference. Ann Stat. 2022;50(5):3064–87. 10.1214/22-AOS2224Search in Google Scholar
[55] Aronow PM, Samii C. Conservative variance estimation for sampling designs with zero pairwise inclusion probabilities. Survey Methodol. 2013;39(1):231–41. Search in Google Scholar
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Articles in the same Issue
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- Adaptive normalization for IPW estimation
- Matched design for marginal causal effect on restricted mean survival time in observational studies
- Robust inference for matching under rolling enrollment
- Attributable fraction and related measures: Conceptual relations in the counterfactual framework
- Causality and independence in perfectly adapted dynamical systems
- Sensitivity analysis for causal decomposition analysis: Assessing robustness toward omitted variable bias
- Instrumental variable regression via kernel maximum moment loss
- Randomization-based, Bayesian inference of causal effects
- On the pitfalls of Gaussian likelihood scoring for causal discovery
- Double machine learning and automated confounder selection: A cautionary tale
- Randomized graph cluster randomization
- Efficient and flexible mediation analysis with time-varying mediators, treatments, and confounders
- Minimally capturing heterogeneous complier effect of endogenous treatment for any outcome variable
- Quantitative probing: Validating causal models with quantitative domain knowledge
- On the dimensional indeterminacy of one-wave factor analysis under causal effects
- Heterogeneous interventional effects with multiple mediators: Semiparametric and nonparametric approaches
- Exploiting neighborhood interference with low-order interactions under unit randomized design
- Robust variance estimation and inference for causal effect estimation
- Bounding the probabilities of benefit and harm through sensitivity parameters and proxies
- Potential outcome and decision theoretic foundations for statistical causality
- 2D score-based estimation of heterogeneous treatment effects
- Identification of in-sample positivity violations using regression trees: The PoRT algorithm
- Model-based regression adjustment with model-free covariates for network interference
- All models are wrong, but which are useful? Comparing parametric and nonparametric estimation of causal effects in finite samples
- Confidence in causal inference under structure uncertainty in linear causal models with equal variances
- Special Issue on Integration of observational studies with randomized trials - Part II
- Personalized decision making – A conceptual introduction
- Precise unbiased estimation in randomized experiments using auxiliary observational data
- Conditional average treatment effect estimation with marginally constrained models
- Testing for treatment effect twice using internal and external controls in clinical trials
Articles in the same Issue
- Research Articles
- Adaptive normalization for IPW estimation
- Matched design for marginal causal effect on restricted mean survival time in observational studies
- Robust inference for matching under rolling enrollment
- Attributable fraction and related measures: Conceptual relations in the counterfactual framework
- Causality and independence in perfectly adapted dynamical systems
- Sensitivity analysis for causal decomposition analysis: Assessing robustness toward omitted variable bias
- Instrumental variable regression via kernel maximum moment loss
- Randomization-based, Bayesian inference of causal effects
- On the pitfalls of Gaussian likelihood scoring for causal discovery
- Double machine learning and automated confounder selection: A cautionary tale
- Randomized graph cluster randomization
- Efficient and flexible mediation analysis with time-varying mediators, treatments, and confounders
- Minimally capturing heterogeneous complier effect of endogenous treatment for any outcome variable
- Quantitative probing: Validating causal models with quantitative domain knowledge
- On the dimensional indeterminacy of one-wave factor analysis under causal effects
- Heterogeneous interventional effects with multiple mediators: Semiparametric and nonparametric approaches
- Exploiting neighborhood interference with low-order interactions under unit randomized design
- Robust variance estimation and inference for causal effect estimation
- Bounding the probabilities of benefit and harm through sensitivity parameters and proxies
- Potential outcome and decision theoretic foundations for statistical causality
- 2D score-based estimation of heterogeneous treatment effects
- Identification of in-sample positivity violations using regression trees: The PoRT algorithm
- Model-based regression adjustment with model-free covariates for network interference
- All models are wrong, but which are useful? Comparing parametric and nonparametric estimation of causal effects in finite samples
- Confidence in causal inference under structure uncertainty in linear causal models with equal variances
- Special Issue on Integration of observational studies with randomized trials - Part II
- Personalized decision making – A conceptual introduction
- Precise unbiased estimation in randomized experiments using auxiliary observational data
- Conditional average treatment effect estimation with marginally constrained models
- Testing for treatment effect twice using internal and external controls in clinical trials
