Observational Identification With Interference

Why Observational Comparisons Need More Than an Exposure Mapping

An exposure mapping summarizes how other units’ treatments may affect an individual. For example, it might classify a person as having “high peer exposure” when at least half of their treated neighbors received an intervention. That mapping makes the exposure of interest precise, but it does not make observational comparisons causal. People with different mapped exposures may also differ in their own treatment, their networks, or characteristics that predict the outcome.

The central question here is when an effect indexed by own treatment and mapped peer exposure can be identified from observational data. The answer requires both exchangeability, so that relevant differences between exposure groups are accounted for, and positivity, so that the exposure conditions being compared actually occur across the covariate patterns in the target population.

Potential Outcomes and Observed Exposures

Let \(A_i\) denote the treatment received by unit \(i\), and let \(G_i=g_i(A_{-i};\mathcal{N})\) be the mapped exposure determined by the treatments of other units and the network \(\mathcal{N}\). For example, \(G_i\) could be the number or proportion of treated neighbors, or a category such as low versus high neighbor treatment. The mapping is specified before comparing outcomes.

This notation requires the exposure mapping to capture the treatment features of others that matter for the outcome: assignments that give unit \(i\) the same \((a,g)\) must have the same potential outcome for the purpose of the estimand. This is an exposure-sufficiency assumption, not a consequence of observing a network. If two assignment patterns with the same mapped exposure can still affect outcomes differently, then \(Y_i(a,g)\) does not fully represent the relevant potential outcomes.

Let \(X_i\) denote measured pretreatment covariates relevant to assignment and outcomes. In network settings, these may include individual characteristics and observed features of the unit’s network, such as degree or the characteristics of its neighbors. For simplicity, write \(X\) for the covariates of a target unit drawn from the population of interest. A contrast such as the average effect of high rather than low peer exposure at own treatment \(a\) is

The same reasoning applies to contrasts in own treatment at a fixed peer exposure. The target population must be specified: averaging over all units gives a different target from averaging only over a defined subgroup.

Conditions for Observational Identification

Because own treatment and peer exposure are jointly observed, a direct sufficient condition is exchangeability for the joint exposure. It rules out residual differences in potential outcomes between units assigned different \((A_i,G_i)\), once the measured covariates are held fixed.

Joint exchangeability does not say that treatment assignment is randomized. It says that, within levels of the measured covariates, the observed joint exposure behaves as though it were unrelated to the potential outcomes. The covariates must therefore address causes of both exposure and outcome, including relevant network features and peer-selection processes. Conditioning only on a person’s own characteristics may be inadequate when network position or peer characteristics affect both exposure and outcome.

Joint positivity concerns the pair, not just its components separately. It is not enough that some units receive each own treatment and that some units have each mapped peer exposure. The required combinations must be possible within the covariate patterns being standardized over. A particular combination may be structurally impossible in a small network, or absent among people with a certain network structure.

Under these conditions, the average potential outcome at \((a,g)\) is identified by standardization:

The inner expectation uses the observed outcome mean among units with the specified joint exposure and covariates. The outer expectation averages those conditional means over the covariate distribution of the target population. This is important: simply comparing the observed means of two exposure groups generally averages over two different covariate distributions.

Worked Example: Standardizing a Peer-Exposure Contrast

How Network Structure Can Confound Exposure Comparisons

Network characteristics can create confounding even if the exposure mapping itself is correctly computed. For instance, a unit’s degree, neighborhood composition, or position in a network may influence the chance of having many treated neighbors. The same features may also predict the outcome, through access to resources, baseline risk, or other pathways. If they are not adequately measured and addressed, units with high and low peer exposure are not comparable.

Peer Selection and the Need to Measure the Assignment Process

Peer exposure can also be confounded by how peers are selected. People may form ties with others who have similar health, preferences, resources, or behaviors. Those shared characteristics can predict both the treatments received by peers and the focal unit’s outcome. In that case, a high-exposure group may differ from a low-exposure group before the peer treatment contrast is considered.

Why the Identification Formula Holds

The result follows by conditioning on covariates, using exchangeability to replace a potential-outcome mean with an observed conditional mean, and then applying consistency. For each covariate value \(x\) with support in the target population, joint exchangeability gives

Consistency says that among units with \(A_i=a\) and \(G_i=g\), the observed outcome equals \(Y_i(a,g)\). Therefore,

Taking expectations over the target distribution of \(X_i\) yields

Positivity is what makes the observed conditional mean on the right available at each relevant \(x\). If no units with covariates \(x\) have \((A_i,G_i)=(a,g)\), the data cannot supply that conditional mean there without an additional modeling or extrapolation assumption. Such extrapolation is not identification from the observed exposure support alone.

Practical Checks and Common Mistakes

Before interpreting a standardized contrast, check that the exposure pair is well defined, the covariates are measured before the relevant treatment and exposure, and the covariate distribution used for standardization matches the target population. Then examine support for each joint exposure condition across the covariate patterns that matter. Sparse cells are a warning that the estimate may depend heavily on modeling assumptions, even when a formal positivity condition is stated.

• Mistake: “The exposure mapping controls for interference.” A mapping defines which peer-treatment summaries index the potential outcomes; it does not remove confounding. Correction: assess exchangeability and positivity for the resulting joint exposure.
• Mistake: “Own treatment and peer exposure each occur, so joint positivity holds.” Their combination may still be absent within important covariate strata. Correction: check support for each \((A_i,G_i)\) pair conditional on \(X_i\).
• Mistake: “Adjusting for the focal unit’s covariates is enough.” Network position, peer composition, or peer-selection factors may also predict exposure and outcome. Correction: identify and measure relevant pretreatment causes at both the individual and network levels.
• Mistake: “A standardized contrast is automatically causal.” Standardization is an identification formula, not a guarantee that its assumptions hold. Correction: state and defend the exposure mapping, consistency, exchangeability, and positivity assumptions.