Exposure Mappings for Interference

Why Exposure Mappings Are Needed

When one unit’s treatment can affect another unit’s outcome, a unit’s potential outcome may depend on more than that unit’s own treatment. In principle, it could depend on the complete vector of assignments across all units. For \(N\) units receiving binary treatments, that vector is \(\mathbf{z}=(z_1,\ldots,z_N)\), where \(z_j=1\) denotes treatment and \(z_j=0\) denotes control. Keeping every possible assignment distinct can make the causal description unwieldy. An exposure mapping summarizes the parts of the assignment vector considered relevant to a particular unit.

The mapping is not merely a convenient way to shorten notation. It says which different assignments are treated as equivalent for that unit’s outcome. For example, a study might posit that a person’s outcome depends on whether they are treated and on the number of their treated neighbors, but not on the identities of those neighbors. That claim is part of the causal model and needs substantive justification.

Definition: For unit \(i\), an exposure mapping is a specified function \(g_i\) that maps the full treatment assignment vector \(\mathbf{z}\) to an exposure value \(g_i(\mathbf{z})\). A mapped potential outcome \(Y_i(z_i,e)\) represents unit \(i\)’s outcome when its own treatment is \(z_i\) and its mapped exposure is \(e\). It is valid when the outcome depends on \(\mathbf{z}\) only through \(z_i\) and \(g_i(\mathbf{z})\).

A general potential outcome can be written \(Y_i(\mathbf{z})\), allowing dependence on the entire assignment vector. A mapping is useful when it is reasonable to replace this high-dimensional description with \(Y_i(z_i,e)\). The own treatment \(z_i\) is kept separate here so that the mapping can focus on other assignment features; a mapping could instead include own treatment in its output, but that convention should be stated clearly.

The Condition Behind the Reduced Description

The essential condition is that assignments with the same own treatment and mapped exposure give the same potential outcome for unit \(i\). In symbols, for every pair of assignments \(\mathbf{z}\) and \(\mathbf{z}'\),

z_i=z_i' \text{ and } g_i(\mathbf{z})=g_i(\mathbf{z}') \quad\Longrightarrow\quad Y_i(\mathbf{z})=Y_i(\mathbf{z}').

This condition is called exposure sufficiency for the proposed mapping. It says that the mapping retains all assignment information relevant to that unit’s outcome, conditional on own treatment. It does not say that the mapping has been established just because it can be calculated from the treatment assignments.

Why the reduced potential outcome is well-defined

Fix an own-treatment value \(a\) and an exposure value \(e\) that can occur for unit \(i\). Consider all full assignments satisfying \(z_i=a\) and \(g_i(\mathbf{z})=e\). If exposure sufficiency holds, every assignment in this set gives the same value of \(Y_i(\mathbf{z})\). We can therefore define \(Y_i(a,e)\) as that common value:

Y_i(a,e)=Y_i(\mathbf{z}) \quad\text{for any }\mathbf{z}\text{ such that }z_i=a \text{ and }g_i(\mathbf{z})=e.

The definition does not depend on which qualifying assignment is chosen, because exposure sufficiency makes their outcomes equal. If two assignments with the same \(a\) and \(e\) could produce different outcomes, then \(Y_i(a,e)\) would not be a single well-defined potential outcome under that mapping. The mapping would need refinement, or the reduced model would need to be rejected.

Worked Example: Counting Treated Neighbors

Suppose unit 1 has three prespecified neighbors, units 2, 3, and 4. For this example, define the relevant exposure as the number of those neighbors assigned treatment:

g_1(\mathbf{z})=\sum_{j\in\mathcal{N}_1}z_j, \qquad \mathcal{N}_1=\{2,3,4\}.

Consider the hypothetical assignment \(\mathbf{z}=(1,0,1,1,0)\) for five units. Unit 1 is treated, and its neighbor assignments are \(z_2=0\), \(z_3=1\), and \(z_4=1\). Thus

g_1(\mathbf{z})=z_2+z_3+z_4=0+1+1=2.

The mapped description for unit 1 is own treatment \(z_1=1\) and treated-neighbor count \(e=2\). If the study instead uses the proportion of neighbors treated, the corresponding mapping is \(2/3\); that is a different, though related, exposure scale.

Now compare \(\mathbf{z}'=(1,1,0,1,0)\). The identities of the treated neighbors have changed, but \(z_1'=1\) and \(g_1(\mathbf{z}')=1+0+1=2\). The count mapping treats these assignments as equivalent for unit 1. That equivalence is justified only if the identity of the treated neighbor does not matter once the count is fixed. If, for instance, neighbor 2 is much closer to unit 1 than neighbor 3, a count alone may discard important information.

Three Common Mapping Structures

A useful mapping starts from the scientific account of how assignments might affect outcomes. The examples below illustrate three structures: exposure through a specified neighbor set, exposure through treatment saturation in a group, and exposure through a shared environment. They are alternative modeling choices, not universally correct summaries.

Treated-neighbor counts

Let \(\mathcal{N}_i\) be the set of units designated as relevant neighbors of unit \(i\), specified independently of the realized treatment assignments. A treated-neighbor count is

g_i(\mathbf{z})=\sum_{j\in\mathcal{N}_i}z_j.

A proportion mapping divides this sum by the number of designated neighbors, when that number is positive. The neighbor set might be defined by physical proximity, a social relation, or a plausible transmission pathway. It should be chosen for substantive reasons: selecting neighbors after seeing which assignments produce convenient results would change the causal model.

Cluster saturation

Suppose units belong to prespecified clusters \(C_1,\ldots,C_K\), and unit \(i\) belongs to cluster \(C(i)\). One mapping uses the proportion of other units in that cluster assigned treatment:

g_i(\mathbf{z})= \frac{\displaystyle\sum_{j\in C(i),\,j\ne i}z_j}{|C(i)|-1}, \qquad |C(i)|>1.

This is a peer-saturation mapping: it excludes unit \(i\)’s own assignment, which is represented separately by \(z_i\). For a cluster of one unit, this particular proportion is undefined, so the analyst must specify a suitable alternative exposure convention. Another study might define saturation as the proportion treated in the whole cluster, including unit \(i\); that is also possible, but it mixes own assignment into the saturation value and must be interpreted accordingly.

Worked Example: Calculating Peer Saturation

Consider a hypothetical cluster with four units, \(C=\{1,2,3,4\}\), and assignment \((z_1,z_2,z_3,z_4)=(1,0,1,1)\). For unit 2, the peer set is \(\{1,3,4\}\). Two of those three peers are treated, so the peer-saturation mapping gives

g_2(\mathbf{z})= \frac{z_1+z_3+z_4}{3} =\frac{1+1+1}{3} =1.

Unit 2’s mapped description is own treatment \(z_2=0\) and peer saturation \(e=1\): the unit is untreated while all three other cluster members are treated. For unit 1, the other three assignments are \(0,1,1\), so \(g_1(\mathbf{z})=(0+1+1)/3=2/3\). Saturation is unit-specific under this own-excluded definition, even though the cluster is the same, because each unit’s peer set excludes itself.

If the cluster assignment were \((0,1,1,1)\), unit 2 would still have \(g_2=2/3\), but its own treatment would now be 1. Keeping the two pieces separate preserves this distinction in the mapped description.

Shared environments

Some units may be affected through an environmental condition generated by treatment assignments across a group, rather than through a list of direct neighbors. Let \(h(\mathbf{z})\) describe that condition. The mapping for unit \(i\) can then be \(g_i(\mathbf{z})=h(\mathbf{z})\), or it can combine that condition with other relevant features. The environmental function should be explicit: a broad label such as “high exposure” is not enough unless the rule assigning that label is stated.

Worked Example: A Shared Capacity Environment

Imagine four hypothetical service units that share a central appointment system. Treatment at a unit is an intervention that increases its expected demand on that system. Suppose the proposed shared-environment mapping labels the system as high congestion when at least three units are treated. Let

h(\mathbf{z})= \begin{cases} 1,&\sum_{j=1}^{4}z_j\geq 3,\\ 0,&\sum_{j=1}^{4}z_j<3. \end{cases}

For the assignment \(\mathbf{z}=(1,0,1,1)\), the total number treated is \(1+0+1+1=3\), so \(h(\mathbf{z})=1\), the high-congestion category. Unit 2’s mapped description is therefore \((z_2,h(\mathbf{z}))=(0,1)\): it is untreated in a high-congestion shared system.

Under this proposal, two assignments with at least three treated units give the same environmental category, even if different units are treated. The model therefore assumes that the identities of treated units do not affect the outcome through the shared system once its congestion category is fixed. If treatment at one particular unit creates substantially more demand than treatment elsewhere, this threshold mapping may be too coarse; a weighted demand measure or a more detailed environment category may be needed.

Choosing and Checking a Mapping

The mapping defines the resolution at which assignments are distinguished. A detailed mapping can preserve more information but may create many exposure categories. A coarse mapping is simpler, but it makes stronger claims of equivalence. The goal is not to retain every possible detail automatically; it is to retain the details that plausibly matter for the outcome being studied.

Before using a mapping, state its inputs, the rule that transforms assignments into exposure, and the assumptions that make the resulting categories meaningful. In particular, clarify whether the mapping is unit-specific, whether it excludes own treatment, how groups or neighbors are defined, and whether the exposure is a count, proportion, category, or vector of several features. If treatment timing, distance, dose, or identity could matter, say how the proposed mapping represents that information.

Specify the assignment information.
Write down the full assignment vector and identify which units or shared conditions could plausibly affect unit \(i\).

Define the mapping rule.
Give a function \(g_i(\mathbf{z})\) that produces an exposure value for every relevant assignment. Make denominators, thresholds, and group membership explicit.

State the equivalence claim.
Explain why assignments with the same own treatment and mapped exposure are assumed to produce the same potential outcome for unit \(i\).

Check what the mapping omits.
Consider whether identities, locations, timing, treatment versions, or other features could still change outcomes within an exposure category.

Common Mistake: Treating the Mapping as a Fact in the Data

A common error is to calculate a convenient summary—such as the number of treated neighbors—and then assume that this summary fully determines exposure. The calculation only shows that the summary can be computed. It does not show that two assignments with the same count have the same causal consequences. Exposure sufficiency is a substantive restriction on potential outcomes, not an arithmetic property of the mapping.

The correction is to distinguish the proposed exposure variable from the assumption that it is sufficient. A count mapping says which assignments are grouped together; the scientific argument must address why those grouped assignments can be treated as equivalent. When this is implausible, refine the mapping, perhaps by including distance bands or a shared-environment measure, or acknowledge that the reduced description is inadequate. An exposure mapping organizes the causal question; by itself, it does not establish that the effects it describes can be identified or estimated from a particular study.

Key takeaway: An exposure mapping \(g_i(\mathbf{z})\) compresses the full assignment vector into a unit-specific exposure. It supports a reduced potential-outcome description only when assignments with the same own treatment and mapped exposure are assumed to yield the same outcome for that unit.

Check Your Understanding

Use the definitions and assumptions in this tutorial to answer the questions.

Unit 1 has neighbors \(\{2,3,4\}\) with assignments \((z_2,z_3,z_4)=(1,0,1)\). What are its treated-neighbor count and treated-neighbor proportion?
In a cluster of five units, four peers of a particular unit are assigned \((1,0,1,0)\). What is that unit’s peer-saturation exposure?
State exposure sufficiency using two full assignment vectors and explain what it allows an analyst to define.
Two assignments give a unit the same treated-neighbor count, but a treated neighbor is much closer in one assignment. What concern does this raise about a count-only mapping?
A shared environment is defined as high whenever at least two of three units are treated. Under assignment \((1,0,1)\), what is the environment category, and which equivalence claim does this mapping make?