The causal effects of modified treatment policies under network interference

Causal inference methodology commonly assumes no interference, that is, that one unit’s exposure or level does not impact any other units’ outcome (cox1958planning; rubin1980). Oftentimes, though, studies involve units that interact with one another, violating this assumption. This occurs, for instance, in settings where each study unit corresponds to a geographic area, such as a county or ZIP code area. Units often move around—as such, application of an exposure policy to a given county affects not only those residing in that region but also those commuting to and from there.

Drawing reliable conclusions about the effects of exposure policies in such settings, common as they are, necessitates the use of methodology designed to answer causal questions such as, “if one intervened upon a population via exposure $A=a$, how would that have changed the outcome $Y$?” (rubin2005). Ignoring interference between units when estimating causal effects risks invalid identification and overtly biased results (halloranDependentHappeningsRecent2016). Despite these challenges, data from studies that involve interference between study units can be exceedingly useful, even critical, to advancing science and policy in some settings. For instance, environmental epidemiologists may leverage observational spatial data to study the effects of pollution on human health, a case where randomized experiments are often impractical or unethical (elliottSpatialEpidemiologyCurrent2004; reichReviewSpatialCausal2021a; morrisonDefiningSpatialEpidemiology).

However, the current state of analysis techniques in such studies fails to reliably answer questions of causality while appropriately addressing issues posed by interference. As interference frequently occurs in data generated by complex studies with a large number of variables that may confound the treatment–outcome relationship, a promising strategy is to use the large set of measured confounders to account for underlying interference by way of regression adjustment. Doing so requires investigators to employ flexible regression approaches to learn the unknown relationships while relying upon minimal assumptions that are likely to render downstream inference invalid. Compounding this issue, many studies also seek to answer questions about the effects of continuous (or quantitative) exposures; these pose their own challenges for identification and estimation, including possible violations of the positivity (or common support) assumption and construction of asymptotically efficient estimators. The present work aims to provide methods that can obtain inference about the causal effect of a continuous exposure while both accounting for network interference between units and using flexible regression procedures in the estimation process.

Modified Treatment Policies (MTPs) (robins2004; munozPopulationInterventionCausal2012; haneuseEstimationEffectInterventions2013; young2014identification) are a class of interventions that define causal estimands well-suited for formulating counterfactual questions about continuous exposures. An MTP can answer the scientific question, “how much would the average value of $Y$ have changed had the natural value of $A$ been shifted by an increment $\delta$?” (where $\delta$ is chosen by the investigator).

Such questions are often as scientifically relevant than the analogous question answered by the causal dose-response curve—namely, “what would the average value of $Y$ be had every unit been set to the same level of the treatment $A=a$ for all $A\in\mathcal{A}$?”—while being easier to pose and answer. Studies on populations exhibiting interference are frequently conducted precisely because considering setting every unit’s hypothetical exposure to the same level would be unrealistic in practice, making the causal dose-response curve’s assumption of common support more restrictive than necessary. The MTP framework has gained traction for its applicability in settings involving longitudinal, time-varying interventions (diazNonparametricCausalEffects2023; hoffman2024studying); causal mediation analysis (diaz2020causal; hejazi2023nonparametric), including with time-varying mediator–confounder feedback (gilbert2024identification); and causal survival analysis under competing risks (diaz2024causal). However, no work has, to our knowledge, extended the MTP framework to dependent data settings characterized by interference between units.

Contributions. This work presents an inferential framework for estimating the causal effect of an MTP under known network interference (that is, where the interference structure is known to the investigator). First, we introduce the concept of an induced MTP, a new type of intervention that identifies the causal effect of an MTP when network interference is present; this is necessary to account for how the application of an MTP to a given unit affects its neighbors in the network. Next, we prove necessary and sufficient conditions for the induced MTP to yield an identifiable estimand. Then, adapting recent theoretical tools for network causal inference (ogburnCausalInferenceSocial2022), we describe consistent and semi-parametric efficient estimation strategies. Finally, we evaluate how the use of an induced MTP to correct for interference can improve on classical techniques, in both numerical experiments and in a data analysis studying the effect of zero-emission vehicle uptake in California on NO₂ air pollution. The tools we describe allow MTPs to be evaluated in previously intractable settings, including with spatial data and social networks.

Outline. Section 2 reviews MTPs and causal inference under interference. Section 3 describes the induced MTP, including identification of causal effects as well as point and variance estimation. Section LABEL:section:sim-results reports numerical results verifying the proposed methodology, while Section LABEL:section:data-analysis demonstrates application in our motivating data analysis. We conclude and discuss future directions in Section LABEL:section:discussion.

Throughout, we let capital bold letters denote random $n$-vectors; for instance, $\mathbf{Y}=(Y_{1},\ldots,Y_{n})$. Consider data $\mathbf{O}=(\mathbf{L},\mathbf{A},\mathbf{Y})\sim\mathsf{P}\in\mathcal{M}$, where $\mathsf{P}$ is the true and unknown data-generating distribution of $\mathbf{O}$ and $\mathcal{M}$ is a non-parametric statistical model (i.e., set of candidate data-generating probability distributions) that places no restrictions on the data-generating distribution; in principle, $\mathcal{M}$ may be restricted to incorporate any available real-world knowledge about the system under study. Let $O_{i}=(L_{i},A_{i},Y_{i})$ represent measurements on the i^th individual unit, where $Y_{i}$ is an outcome of interest, $A_{i}$ is a continuous exposure with support $\mathcal{A}$, and $L_{i}$ is a collection of baseline (i.e., pre-exposure) covariates. To ease notational burden, we omit subscripts $i$ when referring to an arbitrary unit $i$ (i.e., $Y=Y_{i}$ when the specific unit index is not informative).

We will assume that the data-generating process can be expressed via a structural causal model (SCM, pearl2000models) encoding the temporal ordering between variables: $\mathbf{L}$ is generated first, then $\mathbf{A}$, and finally $\mathbf{Y}$. We denote by $Y(a)$ the counterfactual random variable generated by hypothetically intervening upon $A$ to set it to $a\in\mathcal{A}$ and observing its impact on the component of the SCM that generates $Y$. Our goal will be to reason scientifically about the causal relationship between $\mathbf{A}$ and $\mathbf{Y}$ in spite of the presence of confounders $\mathbf{L}$ and network interference between units $i=1,\ldots,n$.

Let us first formally define the interference structure of interest. Suppose there exists a network describing whether two units are causally dependent with adjacency matrix $\mathbf{F}$, where $F_{i}$ denotes the friends of unit $i$. For each unit $i$, a set of confounders $L_{i}$ is drawn, followed by a treatment $A_{i}$ based on a summary $L_{i}^{s}$ of its own and its friends’ confounders, and finally an outcome based on $L_{i}^{s}$ and a summary of its own and its friends’ treatment, $A_{i}^{s}$. This data-generating process can be defined formally as the SCM in Equation (4):

Following ogburnCausalInferenceSocial2022, we assume error vectors $(\varepsilon_{L_{1}},\ldots,\varepsilon_{L_{n}})$, $(\varepsilon_{A_{1}},\ldots,\varepsilon_{A_{n}})$, and $(\varepsilon_{Y_{1}},\ldots,\varepsilon_{Y_{n}})$ are independent of each other, with entries identically distributed and all $\varepsilon_{i}\mbox{$\perp\!\!\!\perp$}\varepsilon_{j}$ provided $\{i,j\}\not\subseteq F_{k},\forall\,\,k\in 1,\ldots,n$; that is, errors between units are independent provided that the units are neither directly connected nor share ties with a common node in the interference network represented by $\mathbf{F}$.

Interference bias arises when the data arise from the SCM (4) but investigators wrongly assume that $f_{Y}$ is a function only of $A_{i}$ and $L_{i}$, and not of $\{A_{j}\colon j\in F_{i}\}$ or $\{L_{j}\colon j\in F_{i}\}$. Since interference violates the consistency rule (Pearl2010), commonly relied upon for identification of causal effects, ignoring its presence, even inadvertently, risks the drawing of unsound inferences. Under the SCM (4), identifiability of the causal effect of applying an exposure to all units $A_{j}\colon j\in F_{i}$ can be restored by controlling for all $L_{j}\colon j\in F_{i}$ directly or via dimension-reducing summaries (vanderlaanCausalInferencePopulation2014) of a unit’s friends’ confounders and exposures.