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Causal Identification under Interference: The Role of Treatment Assignment Independence.
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A large share of empirical work in economics relies on identification strategies—such as selection-on-observables (conditional independence), instrumental variables, difference-in-differences, regression discontinuity, and related designs—that are typically justified under the no-interference or individualistic treatment response (ITR) assumption. Under ITR, each unit’s potential outcomes depend only on its own treatment status, so that standard identification formulas admit clear causal interpretations, most commonly the average treatment effect (ATE) or the average treatment effect on the treated (ATT). In many empirical environments, however, units interact, general equilibrium effects exist, and policies generate spillovers. In such settings, outcomes may depend on the treatment assignments of other units, violating ITR in ways that are often unobserved by the researcher.
What, then, do standard identification formulas recover when the ITR assumption fails? A prominent response in the literature reformulates the identification problem by modeling interference through exposure mappings, neighborhood restrictions, or cluster-level structures in order to define and identify mean direct and indirect (spillover) effects. This approach yields well-defined causal estimands under assumptions about the form of interference and the joint distribution of potential outcomes and treatment, and it often requires detailed information about the underlying interaction structure. Early contributions focused primarily on experimental settings (see, for example, halloran1995causal,hudgens2008toward,tchetgen2012causal, manski2013identification,aronow2017estimating). More recent work extends this framework to a wide range of quasi-experimental and observational designs, including difference-in-differences clarke2017estimating,butts2021difference,xu2023difference, synthetic control cao2019estimation, instrumental variables sobel2006randomized,vazquez2023causal, regression discontinuity designs aronow2017regression,auerbach2024regression,torrione2024regression, and selection-on-observables approaches forastiere2021identification.
By contrast, much applied work continues to employ standard estimators developed under the ITR framework while remaining agnostic about the presence or form of interference, even in settings where spillovers are plausible, largely due to limited information about the underlying interference structure.\footnote{Table (ref) in Section (ref) of the Supplementary Material provides an illustrative, though non-exhaustive list of applied econometric studies that follow this practice.} Despite its prevalence, there is little systematic and comprehensive analysis of the causal content of ITR-based identification formulas when interference is present but unmodelled.
This paper fills that gap by characterizing what conventional ITR-based identification formulas recover when outcomes may exhibit arbitrary interference. Our theoretical results yield three main conclusions. First, allowing outcomes to depend on the treatment of other units requires extending the identifying assumptions underlying standard designs, such as unconfoundedness, instrument exogeneity, or parallel trends, beyond their ITR formulations, in line with the interference literature discussed above.
Second, these extensions alone are not sufficient to ensure causal interpretability. Even when interference in potential outcomes is explicitly modeled, identification formulas derived under the ITR framework retain a causal interpretation only under additional restrictions that limit dependence in treatment assignment across units in the population. In the absence of such restrictions, standard ITR-based identification formulas generally combine meaningful causal effects with bias arising from systematic differences in how others’ treatment assignments affect a unit's own treatment values, and vice versa.
Third, the restrictions limiting cross-unit dependence in treatment assignment are not testable from the observed data. To address this concern, we develop a novel sensitivity analysis procedure within the selection-on-observables framework. The procedure evaluates the robustness of statistical inference---in terms of the decision from a randomization test---to violations of independent treatment assignment conditional on covariates by allowing for departures from treatment independence across units. In particular, when interference is considered plausible, but information about its underlying structure is unavailable, the procedure quantifies the degree of dependence in treatment assignment required to overturn a rejection of a proposed sharp null. In this way, it provides a measure of how sensitive an empirical conclusion is to assignment dependence, in the spirit of the sensitivity analysis framework of rosenbaum2002observational. To facilitate implementation, we also provide an easy-to-use R package, caisensitivity, which implements the proposed sensitivity analysis.\footnote{The R package is available upon request.}
To formalize ideas, we adopt a finite-population potential-outcomes framework in which each unit’s outcome may depend arbitrarily on the full vector of treatment assignments. We study the identification formulas underlying common non-experimental designs that, under ITR, recover the ATE or ATT. We show that, under appropriate assumptions, including extensions of ITR-based restrictions on the joint distribution of treatments and potential outcomes and, importantly, restrictions on the dependence structure of treatment assignments, these formulas identify well-defined causal parameters. Thus, when interference is present, the causal object targeted by ITR-based identification formulas is not lost but instead transformed into a well-defined estimand that remains meaningful under appropriate conditions.
The remainder of the paper is organized as follows. Section (ref) introduces the finite-population framework with arbitrary interference and elaborates on the motivation of the paper. Section (ref) examines the interpretability of ITR-based identification formulas in standard non-experimental designs---i.e., selection-on-observable, instrumental variables, difference-in-differences, and regression discontinuity designs. Section (ref) introduces a sensitivity analysis model that quantifies the extent to which statistical inference relies on the restriction imposed on the treatment assignment mechanism, namely, the limitation on dependence across units’ treatment assignments, required for the interpretability of ITR-based identification formulas under interference. We complement the theoretical analysis by assessing the finite-sample performance of the proposed sensitivity procedure in a simulation study and an empirical application. Section (ref) discusses a Monte Carlo study designed to investigate the bias of ITR-based identification formulas when interference is present. The results corroborate our theoretical findings. We conclude in Section (ref). All Proofs are collected in Section (ref) of the Supplementary Material.
We consider a finite population of $N$ units indexed by $i\in [N]:=\{1,\dots, N\}$. Let $D_i\in\{0,1\}$ denote treatment assignment of unit $i$, and $\mathbf{D}=(D_1, \dots, D_N)\in\{0,1\}^N$ denote the vector of treatment assignments of the population. For each unit $i$, potential outcomes are indexed by the full treatment assignment vector, allowing for arbitrary interference across units. Potential outcomes are modeled as stochastic, with unit-level randomness captured by a latent random variable $\epsilon_i$ that is common across treatment states. Thus, for any realization of $\mathbf{D}$, denoted as $\mathbf d=(d_1, \ldots,d_N)\in\{0,1\}^N$, the potential outcome is of the general form: $Y_i(\mathbf d)=f(\mathbf d, \epsilon_i), \,\,\, \text{with the same latent random variable } \epsilon_i\,\,\text{for all } \mathbf d\in\{0,1\}^N, $ where $f(\cdot)$ is some known function. Related formulations appear in leung2020treatment and xu2023difference. Under no interference, angrist2009mostly present a comparable latent structure for potential outcomes. Formally, we say there is arbitrary interference if the following assumption holds.
Assumption (ref) places no restrictions on the form or scope of interference. In particular, it does not require partial interference, neighborhood interference, or a known exposure mapping as in aronow2017estimating. The identity of units whose treatments affect the outcome of unit \(i\), as well as the manner in which such effects operate, are left unrestricted. Indeed, the ITR assumption is a special case of arbitrary interference, where the outcome of unit $i$ depends solely on her treatment status, i.e., for each unit $i\in[N]$ and her treatment condition $d_i\in\{0,1\}$, there exists a potential outcome $Y_i(d_i)$. To simplify notation, the subscript $i$ is suppressed, and we write $d_i=d$ whenever no confusion arises.
Moreover, as emphasized in the causal inference literature (e.g., see rubin1986comment), the foregoing potential outcome under arbitrary interference is well defined only if the following assumption holds:
Assumption (ref) asserts that the treatment is defined solely by the realized assignment vector, not by the mechanism that generated it. Consequently, alternative assignment procedures that produce the same treatment vector $\mathbf d$ do not correspond to distinct treatments.
Let $\mathbf{D}_{-i}=(D_1,\ldots, D_{i-1}, D_{i+1},\ldots, D_N)\in\{0,1\}^{N-1}$ denote the $(N-1)$-dimensional random vector constructed by deleting the $i$th element from $\mathbf{D}.$ Throughout the paper, we refer $\mathbf{D}_{-i}$ as the treatment of others. The corresponding realizations of $\mathbf{D}_{-i}$ are denoted as $\mathbf{d}_{-i}$. Consequently, potential outcome $Y_i(\mathbf{d})$ can then be written as $Y_i(d_i,\mathbf{d}_{-i})=Y_i(d,\mathbf{d}_{-i})$, for $d_i=d\in\{0,1\}$.
We define the random individual direct effect (IDE)\footnote{The IDE parameter was first discussed in halloran1995causal. Moreover, using a framework where potential outcomes are non-stochastic, savje2021average refers to the IDE as assignment-conditional unit-level treatment effect. However, we use the term individual direct effect at $\mathbf{d}_{-i}$ to indicate that it measures the direct effect for unit $i$ with the treatment vector of the other units in the population held fixed at $\mathbf{d}_{-i}$.} at $\mathbf{d}_{-i}$ as \[ \Delta_i(\mathbf{d}_{-i}) := Y_i(1,\mathbf{d}_{-i}) - Y_i(0,\mathbf{d}_{-i}). \] These effects compare treatment and control for unit $i$, holding the treatment status of all other units fixed at $\mathbf{d}_{-i}$. Marginalizing over the joint distribution of $\mathbf D_{-i}$ and $\epsilon_i$, we obtain the average direct effect (ADE): \[ \mathbbm{E}[\Delta_i(\mathbf{D}_{-i})] = \sum_{\mathbf d_{-i}} \mathbbm{E}[\Delta_i(\mathbf d_{-i})]\cdot \Pr(\mathbf D_{-i}=\mathbf d_{-i})= \mathbbm{E}[Y_i(1,\mathbf{D}_{-i}) - Y_i(0,\mathbf{D}_{-i})], \] where the expectation in the first equality is taken with respect to the distribution of the latent random variable, $\epsilon_i$, and the expectation in the second equality is with respect to the joint distribution of $\epsilon_i$ and $\mathbf{D}_{-i}$. Notice that, we write $\sum_{\mathbf d_{-i}}=\sum_{\mathbf d_{-i}\in\{0,1\}^{N-1}}$ to simplify notation. It is worth noting that this ADE estimand is analogous to the expected average treatment effect defined in savje2021average, which they formulated in a framework that treats potential outcomes as fixed rather than stochastic.
We let $W_i\in\mathcal W\subseteq \mathbbm{R}^p$ denote a vector of random pretreatment variables or covariates for unit $i$.\footnote{The results in Section (ref) are presented using notation that suggests covariates are discrete. This is a notational simplification, as our framework accommodates both continuous and discrete covariates.} In empirical studies that invoke the ITR assumption, the researcher observes individual-level data $(Y_i, W_i, D_i)$, jointly distributed according to a law $P$. Under ITR, the identification formulas associated with common observational designs can be written generally as
where $\psi(\cdot;P)$ is a design-specific functional of the observed-data distribution $P$. For example, in designs where identification hinges on conditional independence of potential outcomes given covariates, $ \psi(Y_i,W_i,D_i;P) = \mathbbm{E}[Y_i\mid D_i=1,W_i] -\mathbbm{E}[Y_i\mid D_i=0,W_i].$
In this section, we elaborate on this paper's motivation beyond the discussion in the introduction. To clarify our contribution, we also situate our framework in relation to two studies.
As noted in the introduction, many empirical studies invoke the ITR assumption and implement ITR-based estimators even in settings where outcome interference is plausibly present. We argue that this practice largely reflects limited information on interaction structures within the population, since unit-level data on social or economic linkages are often difficult or infeasible to obtain (see, for example, colpitts2002targeting). In the following example, we revisit the influential study of lalonde1986evaluating to illustrate that it is not uncommon for empirical studies in economics to abstract from interference, even when such effects may be plausible.
The foregoing example motivates two related questions: (i) What is the causal content of ITR-based estimators of existing studies when interference is present, but without information on the underlying structure of the way units interact? In other words, under what conditions are ITR-based causal identification formulas meaningful when arbitrary interference is modelled? (ii) Can the conditions be verified using the available data in the existing studies?
To address these questions, it is important to note that under ITR and consistency assumptions, the observed outcome $Y_i = Y_i(D_i,\mathbf D_{-i}) = Y_i(D_i)= Y_i(1)D_i + Y_i(0)(1-D_i),$ where $(Y_i(0), Y_i(1))$ denotes the canonical pair of potential outcomes under no interference (see, e.g., imbens2015causal). In this setting, the joint assignment of treatments across units is irrelevant for identification: conditional comparisons of outcomes by $D_i$ retain their causal interpretation regardless of the statistical relationship between $D_i$ and $\mathbf D_{-i}$.
By contrast, under arbitrary interference and consistency assumptions, the observed outcome becomes $Y_i=\sum_{\mathbf d_{-i}}\left(Y_i(0,\mathbf d_{-i})+\Delta_i(\mathbf d_{-i})D_i\right)\cdot \mathbbm{1}(\mathbf D_{-i}=\mathbf d_{-i})$. This representation shows that conditional comparisons of outcomes by $D_i$, which underlie standard ITR-based identification formulas, may no longer equal a meaningful causal effect. Instead, they average treatment effects across the distribution of others’ treatments. Consequently, the causal interpretation of ITR-based identification formulas depends on the treatment assignment mechanism, and in particular on the relationship between $D_i$ and $\mathbf D_{-i}$.
We demonstrate in the following sections that the causal interpretation of ITR-based identification formulas under arbitrary interference requires restrictions on the treatment assignment mechanism that govern the relationship between $D_i$ and $\mathbf D_{-i}$. We therefore introduce and discuss the following assumption.
Assumption (ref) requires that, conditional on pretreatment variables $W_i$, a unit’s treatment assignment is independent of the treatment assignments of other units. In designs where the identifying formula does not condition on covariates, (ref) reduces to the stronger unconditional independence restriction $\mathbf D_{-i}\perp\!\!\!\perp D_i$.
Restrictions related to (ref) have been recognized by tchetgen2012causal and forastiere2020identification as key for ensuring that ITR-based estimators retain a causal interpretation in the presence of specific forms of interference. These papers show that, when information about the structure of interference is leveraged, the ITR-based identification formulas in the selection-on-observables framework can be interpreted as direct causal effects.
tchetgen2012causal consider a partial interference setting in which units interact within groups but not across groups, so that each individual’s outcome may depend on the treatment assignments of other members of the same group. They show that when interference is present, the standard ITR-based inverse probability weighted (IPW) estimator of ATE is biased and does not target any meaningful estimand. Unbiasness is recovered under a restriction that eliminates within-group dependence in treatment assignment, namely that an individual’s treatment is independent of other group members’ treatments, conditional on observed covariates. Thus, the expectation of the estimator equals a direct effect.
forastiere2020identification study observational network settings in which interference is determined by a known network and summarized by an exposure mapping $g(\cdot)$ of neighbors’ treatment assignments. They show that, under unconfoundedness and the restriction \[ g(\mathbf D_{-i}) \perp\!\!\!\perp D_i \mid W_i, \vspace{-0.3cm} \] the ITR-based functional in (ref), with $ \psi(Y_i, W_i, D_i; P) = \mathbbm{E}[Y_i\mid D_i=1, W_i] - \mathbbm{E}[Y_i\mid D_i=0, W_i], $ identifies the ADE.\footnote{Note that in the notation of forastiere2020identification, $\mathbf D_{-i}$ here denotes the vector of treatment assignments of unit $i$’s neighbors rather than that of the entire population excluding unit $i$'s treatment.}
The restrictions that deliver interpretability in both papers are related to condition (ref). When group membership of a unit in the partial interference setting or network links in the network setting are treated as fixed or stochastic but independent of treatment, the independence conditions imposed in both papers to obtain interpretability are implied by the condition in (ref). By contrast, when a unit's group assignment or network links are stochastic and depend on her treatment assignment, then there is no universal ordering between the restrictions required for interpretability in these papers and the Conditional Assignment Independence (CAI) condition in (ref).
In practice, empirical studies that employ ITR-based estimators, abstracting away from interference even when it is plausible (see Example (ref)), do so largely because information on the underlying interaction structure---such as networks or group linkages---is not observed. In such settings, although one could impose conditions that rely on the interaction structure to recover interpretability, as proposed by tchetgen2012causal and forastiere2020identification, these conditions cannot be empirically verified. For example, the data used in lalonde1986evaluating contain no information on networks or group membership (see Example (ref)). As a result, the conditions proposed in the related papers to interpret ITR-based estimators as direct causal effects in the presence of interference cannot be empirically assessed using the LaLonde data.
This motivates an investigation into the causal content of standard ITR-based functionals in settings where the interaction structure is unobserved. Our approach, therefore, characterizes interpretability under conditions that do not rely on observed networks or group structure, aligning the identifying assumptions with the informational environment in which ITR-based estimators are typically applied. Relative to tchetgen2012causal and forastiere2020identification, our analysis also considers designs beyond selection on observables. Moreover, because the condition we propose for recovering interpretability (CAI condition) does not depend on the unobserved interaction structure, we also introduce a sensitivity analysis procedure to empirically assess it.
We conclude this section with the following remark on the plausibility of the CAI condition in settings with interference.
In this section, we examine whether identification formulas derived under the ITR assumption retain a causal interpretation in the presence of arbitrary interference. For each observational design, we summarize the identifying assumptions imposed under ITR, state the corresponding identification functional, and characterize the object---potentially involving both observable and unobservable components---that is recovered in the presence of interference, together with the additional conditions required for interpretation. Across designs, we highlight the central role of the CAI condition (Assumption (ref)) and related restrictions.
This section focuses on the selection-on-observables design, one of the most widely used identification strategies under ITR that relies on conditioning on observed covariates. Under ITR and consistency assumptions, rosenbaum1983central show that causal identification primarily relies on the classical strong ignorability assumption written as:
where (ref) means that, conditional on observable covariates, the treatment assignment is as good as random. The overlap condition (ref) ensures that each unit has a nonzero probability of receiving either treatment arm, conditional on the pretreatment variables.
The ITR, consistency, unconfoundedness, and overlap conditions (e.g., see imbens2015causal) ensure that
Several estimators developed under ITR---including outcome regression, inverse probability weighting (IPW), propensity-score-reduced, and augmented inverse probability weighting (AIPW) estimators---are designed to consistently estimate the functional in (ref).
Under arbitrary interference, an analog of the unconfoundedness and overlap conditions in (ref) and (ref) is provided in the following assumption.
Condition (ref) states that for all \(i\) and all \(\mathbf d\in\{0,1\}^N\), $$\Pr(\mathbf D=\mathbf d \,\big|\, W_i,\{Y_i(d_i,\mathbf d_{-i}) : \ d_i\in\{0,1\},\ \mathbf d_{-i}\in\{0,1\}^{N-1}\}) = \Pr(\mathbf D=\mathbf d \mid W_i).$$ Thus, conditional on unit \(i\)'s observed covariates \(W_i\), the distribution of the treatment assignment vector is invariant to unit \(i\)'s potential outcomes. Imposed for all \(i\), Condition (ref) excludes assignment rules for which, after conditioning on \(W_i\), the probability of any assignment vector \(\mathbf d\) varies with unit \(i\)'s potential outcomes. In this sense, it extends the usual unconfoundedness idea to settings with interference: once \(W_i\) is held fixed, neither unit \(i\)'s own treatment nor the treatment assignments of other units depend on unit \(i\)'s potential outcomes.
The relevant comparison with the classical no-interference condition $(Y_i(0),Y_i(1))\perp\!\!\!\perp D_i\mid W_i$ lies in the assignment object with respect to which independence is imposed. Under no interference, it is sufficient to require that own treatment \(D_i\) be independent of unit \(i\)'s potential outcomes given $W_i$. Under interference, however, unit \(i\)'s realized outcome depends not only on \(D_i\) but also on \(\mathbf D_{-i}\). Accordingly, Condition (ref) requires independence between unit \(i\)'s potential outcomes and the entire assignment vector \((D_i,\mathbf D_{-i})\), conditional on \(W_i\).
An important feature of Condition (ref) of Assumption (ref) is that conditioning is on \(W_i\), not on the full covariate matrix \(\mathbf W\). The restriction is therefore local in nature. It does not impose a restriction on the general assignment mechanism that
where $\boldsymbol{\mathcal{Y}}:=\{Y_i(d_i,\mathbf d_{-i}) : i\in [N],\ d_i\in\{0,1\},\ \mathbf d_{-i}\in\{0,1\}^{N-1}\}$, nor does it imply that treatment assignments are independent. The joint distribution of \(\mathbf D\) may still depend on $\mathbf W$ and may exhibit arbitrary cross-unit dependence. The condition does not rule out dependence in assignments, but instead it rules out dependence of the assignment distribution on unit \(i\)'s potential outcomes once \(W_i\) is fixed.
It is worth noting that in tchetgen2012causal and leung2022unconfoundedness, variants of (ref) are imposed to identify causal estimands under different forms of interference. However, for our objective of reinterpreting the ITR-based identification formula, the covariates in (ref) need include only the conditioning variables under no interference, which typically excludes covariates from other units in the population forastiere2021identification.
In sum, Condition (ref) of Assumption (ref) excludes selection on potential outcomes at the unit level while remaining agnostic about the broader dependence structure of the treatment assignment mechanism. It is therefore appropriately viewed as an interference analog of unconfoundedness stated in terms of local conditioning on \(W_i\).
For completeness, we restate the overlap condition in (ref).
Next, we examine the interpretability of the identification formula in (ref) when outcomes may depend on the entire treatment assignment vector, allowing for arbitrary interference across units.
Theorem (ref) clarifies the interpretation of the ITR-based identification functional under the selection-on-observable design under arbitrary interference by distinguishing cases in which CAI fails from those in which it holds. Part (i) describes the situation in which CAI is violated. In this case, treated and untreated units with the same covariates face systematically different distributions of others’ treatment assignments, so that $\Phi_{\mathrm{ITR}}(P)$ no longer reduces to a well-defined estimand. Instead, the functional combines direct effects with the effects of simultaneously changing the assignment mechanism of other units, generating bias.
Equations (ref) and (ref) provide two complementary representations $\Phi_{\mathrm{ITR}}(P)$ under interference. In (ref), $\Phi_{\mathrm{ITR}}(P)$ equals the population average direct effect---where the averaging is taken with respect to the distribution of other units, given the untreated state---plus a bias term reflecting the discrepancy between the conditional distribution of other units’ treatments given the treatment state. Analogously, (ref) expresses $\Phi_{\mathrm{ITR}}(P)$ as the population average direct effect---where the averaging is taken with respect to the distribution of other units’ treatments given the treated state---with a corresponding bias term reflecting the mismatch in the assignment distribution of others in the population when a unit is treated and untreated
Equation (ref) decomposes $\Phi_{\mathrm{ITR}}(P)$ into the ADE plus an overall bias term. The magnitude of this bias depends on the extent to which CAI fails---i.e., how strongly one's own treatment status predicts treatment of others---and on the strength of interference, as reflected in the responsiveness of conditional mean outcomes to changes in others’ treatment assignments. When the distribution of $\mathbf{D}_{-i}$ differ slightly across treatment states of unit $i$, $D_i$, or interference is weak, the bias is small; when both deviations from CAI and interference are large, $\Phi_{\mathrm{ITR}}(P)$ can be substantially different from the ADE.
Part (ii) of Theorem (ref) shows the case in which the aforementioned biases vanish. When CAI holds, treatment assignment of unit $i$ conveys no information about the treatment of other units in the population, conditional on covariates. Thus, treated and untreated units with the same covariates face the same distribution of others' assignments. In this case, the bias terms disappear and $\Phi_{\mathrm{ITR}}(P)$ reduces to the ADE. Consequently, the selection-on-observable ITR-based identification formula retains a causal interpretation under arbitrary interference.
Overall, Theorem (ref) shows that CAI is the key condition separating settings in which the selection-on-observable ITR-based identification formula recovers a meaningful ADE from those in which it does not.
This section discusses observational designs that do not primarily rely on the strong ingnorability assumption for identification. In contrast to the selection on observables setting---where we obtain the common identification formula in (ref) which can be estimated by a relatively unified class of estimators---there is no single identification formula and class of estimators for these designs. Instead, these designs differ substantially in their identifying assumptions and causal inferential targets. In section (ref), we study the instrumental variables (IV) design. Then, in section (ref) we study the regression discontinuity design, and in section (ref) we study the difference-in-differences design.
We study identification under interference using instrumental variables. To highlight the key conceptual issues, we focus on the simplest setting with a single binary instrument, as in angrist1996identification. Thus, we let $Z_i$ represent a binary instrument. Under ITR, the identification formula, which is the population analogue of the Wald estimator wald1940fitting, is
We can re-write (ref) in terms of (ref) with $W_i=Z_i$ such that $\psi(Y_i,Z_i, D_i;P)=(\mathbbm{E}[Y_i\mid Z_i=1]-\mathbbm{E}[Y_i\mid Z_i=0])/ (\mathbbm{E}[D_i\mid Z_i=1]-\mathbbm{E}[D_i\mid Z_i=0]).$ angrist1996identification shows that (ref) equals the local average treatment effect (LATE) for compliers under the following conditions:
where $D_i(z)$ denotes the random counterfactual treatment assignment for unit $i$ under instrument value $z$. Specifically, $D_i(z)=h(z, \nu_i)$, where $\nu_i$ is an independent and identically distributed (i.i.d) latent random variable, and $h(\cdot)$ is some known function (See, e.g., heckman1978dummy and angrist2009mostly where $h(\cdot)$ is a single index indicator function). Hence, the randomness of potential treatment is due to the latent variable $\nu,$ similar to the source of randomness of potential outcomes.
Under arbitrary interference, let $\mathbf Z=(Z_1,\ldots,Z_n)$ denote the vector of instrumental variables and write $\mathbf{Z}_{-i}$ for the subvector excluding unit $i$. For any realization $\mathbf z_{-i}$, let $\mathbf D_{-i}(\mathbf z_{-i}):=(D_1(z_1), \dots, D_{i-1}(z_{i-1}), D_{i+1}(z_{i+1}),\dots, D_{N}(z_N))$ denote the vector of potential treatments of all units other than $i$ when $\mathbf{Z}_{-i}=\mathbf z_{-i}$. We impose the following assumption, which extends the standard instrumental variables conditions to settings with arbitrary interference in outcomes.
This assumption generalizes the independence and exclusion restrictions of angrist1996identification to settings with arbitrary interference. Condition (ref) imposes a no-interference restriction on treatment assignment, requiring unit $i$’s treatment to depend only on its own instrument. Thus, an individual's potential treatment is generated according to the same latent model as in the case of no interference. The unconditional independence condition in (ref) requires the instrument to be as good as randomly assigned with respect to both potential outcomes and potential treatments. Since there is no interference in treatment (Condition (ref)), we do not need to modify the monotonicity assumption, and we restate it in Condition (ref) for completeness. The exclusion restriction in (ref) rules out any direct effect of the instrument on outcomes beyond its effect through own treatment. Condition (ref) is a consistency requirement for others’ treatments: when the vector of instruments for units other than $i$ equals $\mathbf z_{-i}$, the realized treatment vector $\mathbf D_{-i}$ coincides with the corresponding potential treatment vector $\mathbf D_{-i}(\mathbf z_{-i})$.
Together, conditions (ref) and (ref) imply that the population can be partitioned into three principal strata: compliers, for whom $D_i(1) > D_i(0)$; always-takers, for whom $D_i(1)=D_i(0)=1$; and never-takers, for whom $D_i(1)=D_i(0)=0$.
In this IV setting, since the treatment status of unit $i$ is a function of her instrument and an i.i.d. latent variable, we can obtain a CAI-type restriction on treatment assignments by restricting the dependence of instrumental variables across units.
This assumption rules out cross-unit dependence in instrument assignment, so that each unit’s instrument is assigned independently of others. This is an analog of the unconditional version of the CAI assumption.
Next, we study the interpretability of the Wald ratio in (ref) when arbitrary interference is present in the outcomes.
Theorem (ref) characterizes the behavior of the canonical instrumental--variables identification formula when outcomes exhibit arbitrary interference. Part (i) shows that, even under instrument exogeneity and exclusion at the individual level, the Wald ratio fails to identify a causal effect when the instrument is dependent across units. In this case, the reduced-form difference decomposes into the local average direct effect (LADE) plus a bias term that reflects systematic differences in the distribution of others' instruments across values of the own instrument. When a unit’s instrument assignment alters the distribution of instruments among other units, then instrument-induced variation in own treatment is accompanied by changes in the treatments received by others, confounding the effect and inducing bias.
Part (ii) establishes that the bias vanishes when the instrument is independent across units. Under Assumption (ref), the distribution of others' instruments is invariant to the value of one's own instrument, and hence the distribution of others' treatment assignment vector faced by compliers does not differ across instrument realizations. In this case, the Wald ratio recovers the LADE, defined as the local average direct effect evaluated at the realized treatment assignment vector of compliers. Together, these results show that the standard IV identification formula has a causal interpretation in settings with interference only when the instrument assignment is independent across individuals.
The target parameter in the regression discontinuity design (RDD) admits two distinct identification routes: the traditional continuity-based route proposed by hahn2001identification and the local-randomization-based route proposed by cattaneo2015randomization. Using the latter identification strategy, aronow2017regression shows that the ITR-based RDD identification formula equals the ADE under arbitrary interference. As such, in this section, we focus on the former identification approach under a sharp design. Here, $W_i$ denotes a continuous running variable with cutoff $w_0$, i.e., $D_i=\mathbbm{1}(W_i\geq w_0)$. Under ITR and the condition:
the RDD indentification formula \[ \Phi_{\mathrm{ITR}}(P):={\lim_{w\downarrow w_0} \mathbbm{E}[Y_i\mid W_i=w] - \lim_{w\uparrow w_0}\mathbbm{E}[Y_i\mid W_i=w]} \] identifies the ATE at the cutoff. Relative to the generalized formula in (ref), $\psi(Y_i,W_i,D_i;P)={\lim_{w\downarrow w_0} \mathbbm{E}[Y_i\mid W_i=w] - \lim_{w\uparrow w_0}\mathbbm{E}[Y_i\mid W_i=w]}$.
Under arbitrary interference, the consistency assumption (Assumption (ref)) implies that the observed outcome admits the representation \[ Y_i = \sum_{\mathbf d_{-i}\in\{0,1\}^{N-1}} \Big( \alpha_i(\mathbf d_{-i})+\Delta_i(\mathbf d_{-i})D_i \Big)\, \mathbbm{I}(\mathbf D_{-i}=\mathbf d_{-i}), \] where $\alpha_i(\mathbf d_{-i}) := Y_i(0,\mathbf d_{-i})$ and $\Delta_i(\mathbf d_{-i}) := Y_i(1,\mathbf d_{-i})-Y_i(0,\mathbf d_{-i})$.
We consider the following assumption, which extends the standard RDD continuity condition into a setting with arbitrary interference and imposes a conditional independence restriction.
This assumption requires the conditional mean of the untreated potential outcome to be continuous in the running variable at the cutoff for each fixed treatment vector of other units. In addition, it requires that the untreated potential outcome is independent of the assignment of other units, conditional on the running variable. Note that, in the ITR-based RDD with heterogeneous treatment effects, hahn2001identification also implicitly assumes joint unconfoundedness of potential outcomes with respect to the own treatment. Thus, the assumption of unconfoundedness is not new in the RDD literature.
Since $D_i$ is a deterministic function of $W_i$, i.e., $D_i=\mathbbm{1}(W_i\geq w_0)$, the dependence structure of $W_i$ determines the dependence between individual treatments. As a result, we introduce the following assumption.
This assumption rules out cross-unit dependence in the running variable, so that each unit’s running variable is independent of others. Assumption (ref) is sufficient to ensure the unconditional version of the CAI condition that $\mathbf{D}_{-i} \perp\!\!\!\perp D_i$ for all $i\in[N]$. Moreover, it implies that $\mathbf{D}_{-i} \perp\!\!\!\perp W_i$ for all $i\in[N]$.
Theorem (ref) highlights the central role of Assumption (ref) (a CAI-type condition) in the causal interpretation of regression discontinuity designs under arbitrary interference.
Part (i) shows that, in the absence of the independence between the running variable of a unit and the treatment assignments of other units, the sharp RDD identification formula generally combines the ADE with bias terms reflecting the differences between the distribution of others' treatment assignments near both sides of the cutoff.
Part (ii) demonstrates that imposing independence of the running variable across units, which implies independence between one's own treatment and others' treatments, eliminates these sources of bias and restores a causal interpretation of the sharp RDD identification formula. Under this CAI-type condition, the running variable generates exogenous variation in own treatment while holding the distribution of others' treatments fixed, so the discontinuity in outcomes at the cutoff identifies the ADE.
Taken together, these results show that Assumption (ref) (a CAI-type condition) is not merely a technical condition but a key requirement for the ITR-based sharp RDD identification formula to be interpretable in the presence of arbitrary interference.
In this section, we study difference-in-differences (DiD) identification that exploits variation across both time and units. We focus on the canonical two-group (treated and untreated) and two-period ($t=0,1$) design, in which no unit is treated at period 0 and units in the treated group become treated at period 1.
Let $Y_{it}(0)$ and $Y_{it}(1)$ denote unit $i$’s potential outcomes without and with treatment at period $t\in\{0,1\}$. Because treatment is not implemented in period 0 for any unit, the observed outcome in period 0 is $Y_{i0} = Y_{i0}(0)$, so that untreated potential outcomes are observed for all units. This setup rules out anticipation effects, in the sense that treatment assignment in period 1 does not affect outcomes in period 0.\footnote{A setup that allows anticipation effects is discussed in Section (ref) of the Supplementary Material.} In period 1, the observed outcome is $Y_{i1} = Y_{i1}(0)(1-D_i) + Y_{i1}(1)D_i,$ where $D_i \in \{0,1\}$ indicates whether unit $i$ belongs to the treated group and is treated in period 1.
Following abadie2005semiparametric, we allow for covariate-driven differences in outcome dynamics between treated and control units.\footnote{The theoretical results in this section extend trivially to the traditional DiD framework as in card1994minimum, where there are no covariate-driven differences in outcome dynamics.} Therefore, under the ITR framework, the DiD identification formula is
Thus, relative to the generalized identification formula in (ref), $\psi(Y_i,W_i,D_i;P)=D_i\cdot \Pr(D_i=1)^{-1}\cdot\big\{ \left( \mathbbm{E}[Y_{i1}\mid W_i,D_i=1] - \mathbbm{E}[Y_{i0}\mid W_i,D_i=1] \right) - \big( \mathbbm{E}[Y_{i1}\mid W_i,D_i=0] - \mathbbm{E}[Y_{i0}\mid W_i,D_i=0] \big)\big\}$, where $Y_i:=(Y_{i0},Y_{i1})$.
baker2025difference shows that the identification formula in (ref) equals the ATT in period 1, i.e., $\Phi_{\mathrm{ITR}}(P)=\mathbbm{E}[Y_{i1}(1)-Y_{i1}(0)\mid D_i=1]$, under the following conditions:
Condition (ref) requires that, conditional on covariates, treated and control units would have experienced the same average evolution of untreated potential outcomes over time. Condition (ref) asserts that the conditional probability of treatment given observed covariates $W_i$ is bounded away from zero and one.
We now extend this DiD framework to allow for arbitrary interference. Since no unit is treated in period 0, the observed outcome under arbitrary interference in this period is $Y_{i0} = Y_{i0}(\mathbf{0})=Y_{i0}(0,\mathbf{0}_{-i})$, where $\mathbf{0}$ is the $N$-dimensional vector of zeros and $\mathbf{0}_{-i}$ is the $(N-1)$-dimensional vector of zeros. In period 1, the observed outcome under arbitrary interference and consistency assumptions is $Y_{i1}=\sum_{\mathbf{d}_{-i}} \left(Y_{i1}(0, \mathbf{d}_{-i})(1-D_i) + Y_{i1}(1, \mathbf{d}_{-i})D_i\right)\cdot \mathbbm{1}(\mathbf{D}_{-i}=\mathbf{d}_{-i})$. Because the outcome of a unit depends on others' treatment statuses, the classical parallel trends assumption must be strengthened accordingly. We impose the following assumption analogous to conditions (ref) and (ref).
Condition (ref) of Assumption (ref) extends the conditional parallel trends to environments with interference by requiring that, conditional on covariates, the treated and control groups would have exhibited the same average evolution of untreated potential outcomes across periods, even when outcomes may depend on others' treatment conditions. Condition (ref) restates the strong overlap condition for completeness.
Next, we study the interpretability of the ITR-based DiD identification formula under arbitrary interference, with particular emphasis on the role of the CAI restriction.
Theorem (ref) clarifies the behavior of the ITR-based DiD identifying functional under arbitrary interference and states the sources of bias when treatment assignment is arbitrarily dependent. Part (i) shows that even when the parallel-trends-type restriction in Assumption (ref) holds; the DiD functional fails to identify the average direct effect on the treated. The resulting bias arises from two distinct but interacting forces.
First, untreated potential outcomes depend on others' treatment assignments. When interference is present, the counterfactual evolution $Y_{i1}(0,\mathbf d_{-i}) - Y_{i0}(\mathbf 0)$ varies across $\mathbf d_{-i}$. Condition (ref) of Assumption (ref) ensures that, for any fixed treatment vector of other units, treated and control units share the same untreated potential outcome trend in conditional expectation. However, it does not restrict how untreated trends vary across $\mathbf d_{-i}$.
Second, the distribution of others' treatment differs by treatment group, even after conditioning on covariates $W_i$. The bias term in Theorem (ref) aggregates $\mathbf d_{-i}$--specific untreated trends using treatment group-specific weights, $\Pr(\mathbf D_{-i}=\mathbf d_{-i}\mid W_i, D_i=d),$ so that systematic differences in these distributions induce differences in average untreated trends between treated and control groups. As a result, DiD compares weighted averages of untreated potential outcomes taken over different conditional distributions.
These two forces jointly induce the bias. Thus, in general, DiD compares counterfactual trends evaluated under different conditional distributions, which creates a wedge between the DiD estimand and the ADTT.
Part (ii) shows that restricting the post-treatment dependency of assignment eliminates the second source of bias by equalizing treatment asignment distribution of other's across treatment groups conditional on $W_i$. When combined with Assumption (ref), CAI ensures that both the $\mathbf d_{-i}$--specific untreated trends and the weights used to aggregate them coincide across groups, eliminating the bias term. Under these conditions, the DiD functional equals the ADTT. See xu2023difference for a parallel discussion in a setting with network information, exposure mappings, and nonstochastic covariates.
The preceding section highlights the importance of the CAI and its related conditions for the interpretability of ITR-based identification formulas under interference. As with several identifying assumptions in observational studies, the CAI and its related conditions are not testable because they impose restrictions on the dependence structure of the treatment assignment mechanism that are not identified from a single cross-sectional realization. We therefore assess the robustness of the conclusions of a test of a sharp null hypothesis\footnote{See chapter 5 of imbens2015causal for the definition of a sharp null hypothesis.} to violations of CAI through a sensitivity analysis. For brevity, we focus on a selection-on-observables design, though the proposed sensitivity framework may be adapted to other designs.
We study the sensitivity of $P$-value to violations of the CAI condition when testing a Fisher-style sharp null. Specifically, we consider the sharp null hypothesis
which asserts that treatment assignment vector has no effect on outcomes (no interference and no direct treatment effects). Thus, under the alternative hypothesis, treatment assignment vector affects outcomes.
Conditional on the observed data, or equivalently, for a fixed realization of the latent variables underlying the potential outcomes, the only remaining source of randomness is the treatment assignment mechanism. This allows the sharp null hypothesis $H_0$ to be tested using Fisher randomization inference.\footnote{In Section (ref) of the Supplementary Material, we numerically assess the size control and power of the Fisher randomization applied to $H_0$.} Following rosenbaum2002observational, we examine the sensitivity of decisions based on Fisher $P$-values to departures from the CAI condition.
In principle, computing Fisher $P$-values under the sharp null hypothesis requires knowledge of the propensity scores $e(W_i)=\Pr(D_i=1\mid W_i)$ for $W_i\in \mathcal{W}$, $i\in[N]$. In observational studies, the propensity scores are unknown. Under unconfoundedness (condition (ref)) and CAI (Assumption (ref)), however, $e(\cdot)$ is identified from the joint distribution of $(D_i,W_i)$ and can therefore be estimated imbens2015causal.
Formally, if Assumptions (ref) and condition (ref) hold, the assignment mechanism satisfies
where the first equality follows from condition (ref) and the second from Assumption (ref).
By contrast, if Assumption (ref) fails while condition (ref) continues to hold, the assignment mechanism satisfies
so that the treatment condition of unit $i$ may depend on the treatment of other units in the population, even after conditioning on observed covariates.
Our objective is therefore to assess how replacing the benchmark assignment mechanism $\Pr(D_i=1\mid W_i)$ with the more general mechanism $\Pr(D_i=1\mid \mathbf D_{-i}, W_i)$ affects the conclusions of the test of $H_0$. We emphasize that our objective is to quantify sensitivity to violations of CAI rather than to violations of unconfoundedness, and therefore condition (ref) is imposed. While $\Pr(D_i=1\mid W_i)$ is identified and estimable from the observed data, $\Pr(D_i=1\mid \mathbf D_{-i}, W_i)$ is not identified in a single cross-section. We therefore propose a sensitivity analysis framework to model $\Pr(D_i=1\mid \mathbf D_{-i}, W_i)$.
As in Rosenbaum’s framework, we operationalize conditioning on $W_i$ by forming strata of units with similar values of $W_i$ or the propensity score $\Pr(D_i=1|W_i)$. Let $S_i=S(W_i)\in\{1,\ldots,K\}=:[K]$ index strata obtained by coarsening $W_i$ or $\Pr(D_i=1|W_i)$. Thus, when CAI holds, the assignment mechanism satisfies $\Pr(D_i=1\mid S_i)$; on the other hand, when it fails, one's own treatment may additionally depend on the treatment of others, yielding $\Pr(D_i=1\mid \mathbf D_{-i}, S_i)$.
To motivate the sensitivity model introduced in the next section, we describe a two-stage stratified treatment assignment mechanism. Let $n_s$ denote the number of units in stratum $s\in[K]$ and let $M_s=\sum_{i:S_i=s} D_i$ denote the random treated count in stratum $s\in[K]$. Within-stratum treatment is generated as follows:
Hence, for any within stratum assignment vector $\mathbf{d}_s=(d_1,\dots,d_{n_s})$ with $m_s=\sum_{i=1}^{n_s} d_i$, the unconditional within stratum treatment assignment mechanism is
which implies that the within-stratum treatment assignment mechanism is exchangeable and depends only on the number of treated units.
Now, if the treated count is drawn from a binomial distribution with parameters $n_s$ and $e(s)$, i.e, $M_s\sim Binomial(n_s,e(s))$, then the assignment mechanism in (ref) simplifies as
which represents the independent unconditional Bernoulli within-stratum assignment mechanism (a stratified Bernoulli assignment), such that, $D_i \;\perp\!\!\!\perp\; D_j \mid S_i=S_j=s, s\in[K].$ This assignment mechanism satisfies CAI. Thus, using the estimated stratum-level propensity scores $\hat e(s)$ for $s\in[K]$, we can approximate the assignment distribution under unconfoundedness and the CAI conditions.
However, in general, for $\pi_s(m_s)\neq \binom{n_s}{m_s}\cdot e(s)^{m_s}(1-e(s))^{n_s-m_s}$, the resulting assignment mechanism is not indepedent Bernoulli.
In the following Proposition, which holds for each stratum, the probabilities and moments are conditional on $S_i=s$.
Proposition (ref) shows that the two-stage design leads to assignments that are generally dependent. In particular, if $\mathbbm{E}[M_s]=n_se(s)$, then from (ref), the covariance of any pair of treatments is nonzero unless $\mathrm{Var}(M_s)=n_se(s)(1-e(s))$, which corresponds to the case where $M_s\sim Binomial(n_s,e(s))$, i.e., under a stratified Bernoulli assignment.
Therefore, using the two-stage treatment assignment mechanism proposed in Section (ref) where we set $\mathbbm{E}[M_s]=n_se(s)$, dependence in treatment assignments within strata is reflected in changes in the dispersion of the treated count within each stratum, since $\mathrm{Var}(M_s)=n_s(n_s-1)\cdot\mathrm{Cov}(D_i, D_j)+n_se(s)(1-e(s))$. We can therefore use the within-stratum variance of the treated count as a measure of deviations from the CAI. This motivates the sensitivity parameter introduced in the following section.
To index departures from the CAI condition, we measure the extent to which the treated count within each stratum is restricted or dispersed relative to the stratified Bernoulli (CAI) benchmark.
In the spirit of rosenbaum2002observational, we define a one-parameter family of deviations from CAI indexed by $\xi\ge1$ that bounds departures from the stratified Bernoulli assignment within each stratum. Specifically, we allow for violations of CAI by restricting the variance of the number of treated units in each stratum $s\in[K]$ to satisfy
If $\mathbbm{E}[M_s]=n_se(s)$, then the parameter $\xi$ quantifies the maximum proportional departure from the benchmark assignment mechanism under CAI, bounding the extent of dispersion of the treated count, which affects the dependence structure of treatment assignment. Therefore, we say treatment assignment satisfies the CAI Sensitivity Model if $\mathbbm{E}[M_s]=n_se(s)$ and (ref) holds.
Under the two-stage treatment assignment mechanism in Section (ref), where we set $\mathbbm{E}[M_s]=n_se(s)$, $\xi=1$ corresponds to the benchmark case in which CAI holds exactly and $M_s\sim\mathrm{Binomial}(n_s,e(s))$. From (ref), the inequality in (ref) can be written as \[ 0 \;\le\; \operatorname{Cov}(D_i, D_j) \;\le\; (\xi-1)(n_s-1)^{-1}\cdot e(s)(1-e(s)),\,\,\forall\,\, i\neq j\in \{1,\dots, n_s\}. \] Thus, when $\xi>1$, the parameter enlarges the set of admissible treatment assignment distributions within each stratum. In particular, larger values of $\xi$ allow assignment mechanisms that exhibit stronger dependence between units' treatment assignments. The resulting admissible set, therefore, contains assignment laws that satisfy CAI as well as those that violate CAI. For example, when $\xi=2$, the cross-sectional covariance between units' treatment assignments may be as large as $(n_s-1)^{-1}e(s)(1-e(s))$ relative to the benchmark case in which treatment assignments satisfy CAI (i.e., conditional independence).
It is worth noting that all assignment mechanisms under the CAI Sensitivity Model satisfy the unconfoundedness condition for any $\xi \geq 1$. In particular, for any stratum $s \in [K]$ and unit $i \in \{1,\dots,n_s\}$, $\Pr(D_i=1)= \mathbbm{E}[M_s]/n_s = e(s)$. Thus, units within the same stratum have the same probability of receiving treatment regardless of whether CAI holds or is violated. This feature ensures that the proposed sensitivity analysis procedure isolates sensitivity to violations of CAI from sensitivity to hidden bias in treatment assignment.
To implement the sensitivity analysis procedure characterized by the model in (ref), we will consider classes of the two-stage assignment mechanism indexed by $\xi$. Within each stratum $s$, a treated count $M_s$ is drawn with mean $\mathbbm{E}[M_s]= n_s \hat{e}(s)$ and variances satisfying (ref) for a given value of $\xi$. Conditional on the realization of $M_s$, say $M_s=m_s$, treatment is assigned uniformly at random among all vectors with exactly $m_s$ treated units in stratum $s$.
In particular, for a given $\xi>1$, we obtain a set of vectors of stratum-level variance of treated counts that satisfy (ref). Consequently, for each vector of stratum-level variance of treated counts, we can simulate the randomization distribution under $H_0$ of the stratified Wilcoxon rank sum test statistic wilcoxon1945individual defined as
where $q_i$ is the fixed within-stratum ranks of the observed outcome of unit $i$.\footnote{Alternative test statistics could be considered; we focus on the stratified Wilcoxon rank-sum statistic because it is robust to outliers imbens2015causal.} Conditional on the observed data, note that the distribution of the test statistic depends solely on the assignment mechanism, which in turn is governed by vectors of stratum-level variance of treated counts, which depends on the value of the sensitivity parameter $\xi$.
Since each $\xi$ value greater than one produces several randomization tests and their corresponding $P$-values, for each $\xi$, we compute lower and upper bounds of the set of $ P$-values by solving $$\inf_{\mathcal{A} \in \mathcal{A}_\xi} \Pr_{\mathcal{A}}\!\left(T \ge T_{\mathrm{obs}}\right)\quad \text{and} \quad \sup_{\mathcal{A} \in \mathcal{A}_\xi} \Pr_{\mathcal{A}}\!\left(T \ge T_{\mathrm{obs}}\right),$$ where $\mathcal{A}_\xi$ denote all the assignment mechanisms consistent with (ref) for $\xi$ and $\Pr_{\mathcal{A}}(\cdot)$ denotes a probability distribution over the assignment mechanism $\mathcal{A}$. As $\xi$ increases, the feasible set of assignment mechanisms expands, yielding weakly larger upper bounds and weakly smaller lower bounds on the $P$-value.
To summarize, the foregoing sensitivity analysis addresses the following question: When the unconfoundedness condition holds, how large must departures from CAI---measured by the extent to which the dispersion of the treated count within strata differ from the stratified Bernoulli benchmark---be in order to overturn inference based on the sharp null of no direct effect and no interference? As in Rosenbaum’s framework, we summarize robustness by the robustness value\footnote{To the best of our knowledge, the term robustness value was coined by cinelli2020making.} $\xi^*$, defined as the smallest value of $\xi$ for which the upper bound on the $P$-value exceeds a pre-specified significance level $\alpha$. Larger values of $\xi^*$ indicate greater robustness of the conclusions to violations of CAI.
For ease of exposition, we outline the proposed sensitivity analysis in Procedure (ref). Also, see a flow chart of the procedure in Figure (ref) of Section (ref) of the Supplementary Material. The following remark summarizes the use and interpretation of the proposed sensitivity analysis. Technical details on implementation are deferred to Section (ref) of the Supplementary Material.
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In this section, we use the male subsample of the LaLonde job-training dataset lalonde1986evaluating---distributed by the MatchIt package in R ho2018package---to illustrate how the proposed sensitivity analysis procedure can be applied. The data combine male participants in the National Supported Work (NSW) program with a comparison group of non-participants, and include post-program earnings in 1978 as the primary outcome. Treatment status indicates participation in the training program. Based on the specification in Robert Lalonde's paper, we condition on a standard set of pre-treatment covariates: age, years of schooling, race, marital status, an indicator for lacking a high-school degree, and lagged earnings in 1974 and 1975.
As we discussed in Example (ref), interference is plausible in the setting of this study. Thus, we conduct a simulation study calibrated to the data. Moreover, we apply the sensitivity analysis procedure to the observed data.
\paragraph{Simulation Study:} To assess the finite-sample performance of the proposed procedure under realistic conditions, the simulation design is calibrated to the LaLonde sample to preserve key features of the empirical data. Covariates, treatment assignment, and outcomes are generated to closely mimic the observed structure.
Covariates are generated by drawing observations with replacement from the empirical distribution of the LaLonde sample. Each simulated unit is assigned a vector of pre-treatment characteristics sampled from the observed covariate matrix, which preserves the marginal distributions and joint dependence structure of the covariates. Outcomes are generated using a model calibrated to the LaLonde sample. Specifically, a linear regression of the outcome on covariates is estimated using control units to obtain estimates of the regression coefficients on $W_i$, ${\gamma}$, and a residual standard deviation $\hat{\sigma}$. For each simulated unit, the baseline untreated outcome is generated as \[ Y_i(0) = W_i^\top \hat{\gamma} + \varepsilon_i, \vspace{-0.5cm} \] where $\varepsilon_i$ has zero mean and variance $\hat{\sigma}^2$. The observed outcome is then given by $ Y_i = Y_i(0) + \tau D_i + \zeta \Pi_i(\mathbf{D}_{-i}),$ where $\Pi_i(\mathbf{D}_{-i}) = (N-1)^{-1} \sum_{j \neq i} D_j$. $\tau$ denotes the constant direct effect and $\zeta=3000$ captures constant spillover effects. Thus, outcomes depend on $\mathbf{D}$.
To examine the performance of the sensitivity analysis under alternative outcome environments, we consider two specifications for the disturbance term:
Overall, these specifications generate environments with interference, significant direct effects, and varying sensitivity to dispersion in treatment assignment within strata.
To implement the proposed sensitivity analysis procedure for each outcome specification, we estimate propensity scores using a logistic regression of treatment status on the observed covariates, and construct strata by discretizing the estimated propensity scores into $K=6$ quantile bins (reducing the number of bins only if necessary to ensure nonempty strata). Within each stratum, outcomes are ranked, and the test statistic is computed as the sum of the treated ranks across strata. The sensitivity parameter $\xi$ is evaluated over the grid $\{1,1.25,1.5,2,3,5,8\}$. Fisher $P$-value bounds are computed using Monte Carlo simulation. Baseline Fisher $P$-values at $\xi=1$ (under CAI) are obtained using 50000 simulation draws. For $\xi>1$ (departures of CAI), Fisher $P$-value bounds are obtained by first computing the assignment distribution within each stratum that maximizes (or minimizes) the Fisher $P$-value subject to the mean and variance constraints implied by $\xi$. This optimization is implemented iteratively, using Monte Carlo simulation with 2500 draws per iteration to approximate the objective function (See Section (ref) of the Supplementary Material for intuition and details of the optimization algorithm). After convergence, the $P$-value bound is estimated using 30000 draws from the resulting worst-case assignment distribution.
Figure (ref) reports the upper bounds of the Fisher $P$-values as a function of the sensitivity parameter $\xi$ for the two outcome specifications. Under the Gaussian specification (less sensitive design), the Fisher $P$-value at $\xi=1$ is less than 0.01 and exceeds 5% prespecified significance level at $\xi=3$. This implies a robustness value of approximately $\xi \approx 2.9$.
In contrast, under the heavy-tailed specification (sensitive design), the Fisher $P$-value at $\xi=1$ is 0.019 and exceeds the 5% threshold already at $\xi=1.25$, yielding a robustness value close to $\xi \approx 1.1$.
These results show that the test's sensitivity to deviations from CAI depends on the outcome distribution within propensity score strata. When outcomes are normally distributed, inference remains relatively robust to moderate CAI violations. In contrast, when outcomes contain extreme values, the rank-based statistic becomes substantially more sensitive to such deviations, leading to a much smaller robustness value.
\paragraph{Empirical Application:} We apply the proposed sensitivity analysis to the actual male subsample of the LaLonde job-training dataset. The implementation follows the same implementation parameters like the number of randomizations as in the simulation study, with the sensitivity parameter $\xi$ evaluated over the grid $\{1,1.25,1.5\}$. Table (ref) reports the resulting bounds on the $P$-value under the CAI Sensitivity Model.
When $\xi=1$, corresponding to the benchmark assignment mechanism under which CAI holds, the $P$-value is approximately 0.537. Thus, the sensitivity analysis using the observed LaLonde data is uninformative (see Remark (ref)). An application to a different data set, where the procedure is informative, is presented in Section (ref) of the Supplementary Material.
We conduct a Monte Carlo simulation to illustrate the behavior of the IPW estimator, which is unbiased for the selection-on-observables identification formula in Section (ref) in the presence of interference, and under CAI (Assumption (ref)). The data-generating process allows outcomes to depend on both own treatment and the treatment assignments of all other units, thereby violating ITR while preserving a well-defined notion of ADE.
In each replication (1000 replications in total), units $i\in[N]=[500]$ are endowed with covariates $W_i\sim\mathcal N(0,1)$. Treatment assignment follows a logistic model. Specifically, treatment is assigned according to \[ D_i = \mathbbm{1}\!\left\{ U_i \le \frac{\exp(a_0+a_1 W_i + \rho\,\eta)}{1+\exp(a_0+a_1 W_i + \rho\,\eta)} \right\}, \] where $U_i\sim\mathrm{Uniform}(0,1)$, $\eta\sim \mathcal N(4000,1)$ is a common shock shared by all units in the population, $a_0=-0.2$, and $a_1=0.8$. The parameter $\rho$ governs the strength of dependence in treatment assignment across units. When $\rho=0$, treatment is assigned independently across units conditional on $W_i$, so that the distribution of others' treatment assignments $\mathbf D_{-i}$ is independent of one's own treatment given covariates, and CAI (Assumption (ref)) holds. When $\rho\neq 0$, the common shock induces correlation between $D_i$ and $\mathbf D_{-i}$ even after conditioning on $W_i$, violating CAI (Assumption (ref)).
Observed outcome is generated according to the model \[ Y_i = \alpha_0+\alpha_1 W_i+\alpha_2 \Pi_i(\mathbf{D}_{-i}) + (\tau_0+\tau_1 \Pi_i(\mathbf{D}_{-i}))D_i + \varepsilon_i, \] where $\Pi_i(\mathbf{D}_{-i})$ is the leave-one-out mean of treatment assignments, $\alpha_0=0.0$, $\alpha_1=1.0$, $\tau_0=0.5$, $\tau_1=1.0$, and $\varepsilon_i$ is an idiosyncratic error term which follows a standard normal distribution.
The estimand of interest is the ADE, which coincides with the causal object identified by the population analog of the IPW estimator when the CAI condition in Assumption (ref) holds.
We compute the IPW estimator using the true propensity scores $e(W_i)=\Pr(D_i=1\mid W_i)$\footnote{We use the true propensity score to focus on the failure of the CAI condition.}. For each design, we report the Monte Carlo mean of the estimator, the true ADE target, the bias, and the root mean squared error (RMSE).
Table (ref) reports Monte Carlo results of the simulation exercise under varying degrees of violation of the CAI condition.
When Assumption (ref) holds ($\rho=0$), the ITR-based IPW estimator is essentially unbiased for the ADE. The Monte Carlo mean of the estimator (0.960) is very close to the ADE target (0.956), yielding negligible bias and a small RMSE. This confirms the theoretical result that, under arbitrary interference, ITR-based estimators of ATE under the selection on observable design are unbiased for the ADE when treatment assignment is conditionally independent across units.
As Assumption (ref) is progressively violated, the performance of the IPW estimator deteriorates. For $\rho=0.5$, the estimator exhibits a noticeable upward bias of 0.067, despite the ADE target remaining essentially unchanged. Increasing $\rho$ further amplifies this bias: when $\rho=1.0$, the bias rises to 0.257, and when $\rho=1.5$, it exceeds 0.4. The RMSE increases sharply with $\rho$, reflecting both growing bias and increased dispersion of the estimator.
These patterns illustrate that the failure of Assumption (ref) induces systematic differences in others' treatment assignment distributions faced by treated and untreated units that cannot be corrected by conditioning on observed covariates alone. As a result, the IPW identification formula no longer equals the ADE but instead conflates it with bias.
This paper examines the interpretation of standard identification formulas when the ITR assumption is relaxed, and the outcomes may exhibit arbitrary interference. A central message of the analysis is that extending identification arguments developed under no interference to settings with interference necessarily requires additional restrictions on the treatment assignment mechanism for the interpretability of ITR-based identification formulas. In particular, while interference in outcomes may be unrestricted, meaningful interpretation of ITR-based identification formulas hinges on limiting the dependence of one's own treatment assignment on others’ assignments.
We show that when treatment assignment dependence is suitably restricted, the identification formulas underlying common quasi-experimental designs---such as selection-on-observables, instrumental variables, difference-in-differences, and regression discontinuity---continue to identify meaningful causal parameters in the presence of arbitrary interference. Specifically, under (un)conditional assignment independence, the identification formulas derived under ITR recover ADEs. Motivated by this result, we propose a novel sensitivity analysis framework that quantifies deviations from conditional assignment independence under the selection-on-observables design.
The identification results have direct implications for estimation and inference. Estimators developed under ITR---such as outcome regression, matching, IPW, two-stage least squares, and difference-in-differences regressions---remain unbiased for the ADEs identified by their corresponding formulas, provided assignment dependence is restricted in the manner required by each design. These functions of the data, therefore, retain their usual status as valid estimators once interpreted as targeting ADEs rather than ATEs or ATTs.
Inference, however, is more complicated. Arbitrary interference induces unrestricted cross-sectional dependence, invalidating variance formulas and asymptotic approximations derived under ITR. Standard variance estimators need not be consistent in the presence of arbitrary interference, and the limiting distribution of standardized estimators may fail to be normal. Restoring reliable inference, therefore, requires alternative approaches, such as permutation-based inference, exact Hoeffding-type confidence intervals tchetgen2012causal. We leave the development of such procedures for future research.
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