Why One Unit’s Treatment May Affect Another Unit’s Outcome
Many causal questions are first described as if each person’s outcome depended only on that person’s treatment. That simplification can fail when treatment changes what happens to other people. A vaccination may reduce transmission to nearby residents; a student’s tutoring may affect classmates through shared discussion; or one clinic’s new practice may change care available to patients elsewhere. These are examples of interference: an outcome for one unit can depend on treatment assignments to other units.
Interference matters because a treatment contrast can change with the treatment environment. The effect of treating one person may differ when none of that person’s peers are treated than when many peers are treated. A comparison that ignores this possibility may combine an effect of a unit’s own treatment with effects transmitted from others.
The other units’ assignments are sometimes denoted \(\mathbf{z}_{-i}\), so the outcome can also be written \(Y_i(z_i,\mathbf{z}_{-i})\). The no-interference condition is precisely that, for every pair of other-unit assignment vectors \(\mathbf{z}_{-i}\) and \(\mathbf{z}'_{-i}\),
for each possible own treatment \(z_i\). This condition concerns potential outcomes, not merely observed associations. Outcomes can be correlated across people because they share weather, resources, or other causes without treatment assignments interfering. Interference specifically means that changing another unit’s treatment would change the focal unit’s potential outcome.
The earlier material on SUTVA separates the no-interference requirement from the requirement that treatment values describe well-defined interventions. Here the focus is the first requirement. A clear description of treatment versions remains important: before asking whether one person’s treatment affects another’s outcome, the intervention itself must be sufficiently specified.
Direct Effects and Spillover Effects
With interference, a unit’s own treatment effect must be defined relative to a specified assignment of treatment to others. Holding \(\mathbf{z}_{-i}\) fixed, the direct effect for unit \(i\) is the contrast between the potential outcome when the unit is treated and the potential outcome when it is untreated. A spillover effect instead compares outcomes under different assignments to others while holding the unit’s own treatment fixed.
These contrasts need not be the same for all assignment environments. For example, treating one student might help most when few classmates receive tutoring, or vaccination might have a different direct effect when community coverage is high. Consequently, “the effect of treatment” may require a description of what is happening to other units as well.
Worked Example: Vaccination and Protection of Nearby Residents
Consider a hypothetical neighborhood. For one resident, let \(z_i=1\) indicate vaccination and let \(g_i\) be the proportion of that resident’s four nearby contacts who are vaccinated. The outcome is symptom-days during a specified period. Suppose, for illustration, the potential outcomes for this resident are: when \(g_i=0\), 8 days if the resident is unvaccinated and 4 days if vaccinated; when \(g_i=0.75\), 5 days if unvaccinated and 3 days if vaccinated.
Assume in this illustration that vaccination is a well-defined intervention, the specified contacts are the relevant group, and their vaccination proportion captures the relevant exposure from others. These are simplifying assumptions for the example, not facts established by the numbers.
At zero coverage among the contacts, the direct effect is \(4-8=-4\) symptom-days. At 75% contact coverage, it is \(3-5=-2\) symptom-days. Thus, under these stipulated potential outcomes, the direct effect depends on the treatment environment.
For an unvaccinated resident, the spillover contrast from changing contact coverage from zero to 75% is \(5-8=-3\) symptom-days. This is a spillover effect: the resident’s own treatment remains unchanged, while the contacts’ treatment changes. The example distinguishes that effect from the direct vaccination contrast.
The values are hypothetical potential outcomes, not four outcomes observed for the same resident. In an actual study, only the outcome under the assignment that occurred would be observed for that resident.
Representing Exposure to Others’ Treatment
In a group or network, writing every other unit’s assignment into each potential outcome can be cumbersome. A common simplification is to define an exposure mapping, a rule that summarizes the relevant features of others’ assignments. For instance, \(g_i\) might be the proportion of unit \(i\)’s contacts who are treated. The resulting model writes an outcome as \(Y_i(z_i,g_i)\) rather than \(Y_i(z_i,\mathbf{z}_{-i})\).
This notation is useful only if the summary is adequate for the question. Treating two close classmates might matter more than treating two distant classmates; a proportion alone would then discard relevant information. An exposure mapping assumes that assignment patterns summarized by the same \(g_i\) have the same effect on the outcome, at least for the purpose of the analysis. That is a substantive assumption, not a consequence of choosing a convenient formula.
Worked Example: Peer Tutoring in a Classroom
Suppose a hypothetical classroom has eight students. For one focal student, define \(g_i\) as the number of the other seven students assigned to tutoring. The outcome is a quiz score in points. Assume that the tutoring intervention is consistent across students and, for this illustration, that the focal student’s outcome depends on peers’ assignments only through their number, not their identities. Suppose the focal student’s stipulated potential scores are 64 points when untreated with \(g_i=0\), 70 points when untreated with \(g_i=2\), 72 points when treated with \(g_i=0\), and 78 points when treated with \(g_i=2\).
The direct effect at \(g_i=2\) is \(78-70=8\) points. The spillover effect for an untreated student when the number of tutored peers rises from zero to two is \(70-64=6\) points. For a treated student, the corresponding spillover contrast is \(78-72=6\) points.
In this example, the chosen exposure summary makes it possible to state these contrasts compactly. But it also imposes a strong assumption: two assignments with the same number of tutored peers are treated as equivalent, even if the particular peers differ. If peer identity or classroom seating matters, this summary may be inadequate. The arithmetic is valid for the stipulated model; whether the model describes a real setting requires separate justification.
Why an Effect Can Depend on the Assignment Policy
When other units’ assignments can affect a person’s outcome, a mean effect for the population may depend on how treatment is distributed among those other units. To make this precise, consider a specified probability distribution \(Q\) over assignments to the other units. For each unit \(i\), average the direct contrast over that distribution, applying the same distribution of others’ assignments to the treated and untreated potential outcomes:
The corresponding average across the \(n\) units is \(\tau(Q)=\frac{1}{n}\sum_{i=1}^{n}\tau_i(Q)\). This is an effect defined relative to \(Q\); it is not automatically identified from observed data. The definition clarifies the target: it averages direct effects over a stated environment for other units.
The simplification under no interference follows directly. If \(Y_i(z_i,\mathbf{z}_{-i})=Y_i(z_i)\) for every \(\mathbf{z}_{-i}\), then the difference inside the sum is the same for every assignment to others. Since the probabilities in \(Q\) sum to one,
Thus, with no interference, this direct effect does not depend on \(Q\). With interference, changing the policy distribution of others’ treatment can change the average contrast even when the definition of each unit’s own treatment stays fixed.
Worked Example: A Contrast That Mixes Direct and Spillover Effects
Consider hypothetical readmission outcomes for a representative patient, measured as percentages, under a well-specified hospital intervention. Let \(g\) be the proportion of the patient’s relevant peer group treated. Suppose the stipulated potential outcomes are: if \(g=0\), 12% when the patient is untreated and 9% when treated; if \(g=0.5\), 10% when untreated and 8% when treated. Assume for this example that \(g\) adequately summarizes peer exposure and that no other intervention versions are being mixed.
At \(g=0.5\), the direct contrast is \(8\%-10\%=-2\) percentage points. Among untreated patients, the spillover contrast from \(g=0\) to \(g=0.5\) is \(10\%-12\%=-2\) percentage points.
Now compare the treated outcome at \(g=0.5\) with the untreated outcome at \(g=0\). The contrast is \(8\%-12\%=-4\) percentage points. It combines two changes: moving from untreated to treated at \(g=0.5\) changes the outcome by \(-2\) percentage points, and moving from \(g=0\) to \(g=0.5\) among untreated patients changes it by \(-2\) percentage points. In symbols,
This decomposition shows why a contrast between groups at different peer exposures is not automatically a direct treatment effect. It may include a spillover component as well. These stipulated outcomes illustrate the distinction; observed group means alone would not establish the relevant potential outcomes or justify this decomposition as an empirical estimate.
Diagnostics, Alternatives, and Limitations
Interference is often plausible when units share a network, location, institution, or resource. Useful design questions include whether treatment changes contact or behavior, whether treated units can influence untreated units, and whether the proposed unit of analysis is large enough to contain the important interactions. These questions can reveal a threat, but a lack of obvious evidence does not prove no interference. Observed similarity or correlation between peers is not, by itself, proof that one person’s treatment caused another’s outcome.
Several approaches make different assumptions and answer different questions:
| Approach | What it assumes or summarizes | Important limitation |
|---|---|---|
| No-interference analysis | Each unit’s outcome depends only on its own assignment. | Spillovers, if present, are excluded from the stated model. |
| Exposure mapping | Relevant features of others’ assignments can be summarized, such as by a treated-peer proportion. | A poor summary can hide important differences between assignment patterns. |
| Partial interference | Interference may occur within specified groups but not between groups. | Cross-group contact or shared resources can violate the group boundary assumption. |
| Group-level treatment | Groups, rather than individuals, receive an intervention that applies to the group. | The resulting effect is about that group-level intervention and may not isolate an individual direct effect. |
Under partial interference, units are partitioned into groups, and a unit’s outcome may depend on assignments within its own group but not on assignments in other groups. This can be plausible for genuinely separated classrooms or communities, but the boundary must be defended. People may cross group boundaries, and resources or information may spread between groups.
A practical warning sign is a control unit that can receive some benefit from treated units. If such spillovers exist, “untreated” does not necessarily mean “unexposed.” Another warning sign is that effects vary with peer coverage or network position. But diagnostics based on observed data cannot, by themselves, prove that an exposure summary is complete or that no unmeasured pathway of interference remains.
The central limitation is that a fuller potential-outcome model requires more assignment conditions than the no-interference model. With \(n\) binary-treated units, a unit’s potential outcome can in principle vary over many assignments to the other \(n-1\) units. Exposure summaries and group assumptions reduce this complexity, but they do so by adding structural assumptions. The appropriate choice depends on the setting and the causal question, not merely on which model is easiest to calculate.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A resident’s outcome changes when a neighbor is treated, even though the resident’s own treatment is held fixed. Is this interference? Why?
- For a hypothetical outcome, \(Y_i(1,g=0.4)=18\) and \(Y_i(0,g=0.4)=23\), measured in hours. Calculate the direct effect at \(g=0.4\), with units.
- For an untreated student, an outcome is 72 points when no peers are tutored and 76 points when two peers are tutored. Calculate the spillover contrast and state the exposure change.
- Why can a treated-versus-untreated contrast that compares different peer-treatment levels fail to represent a direct effect?
- What extra assumption is made when peer exposure is summarized only by the proportion of peers treated?
Solutions. (1) Yes. By definition, changing another unit’s treatment while holding the resident’s assignment fixed changes the resident’s potential outcome. (2) The direct effect is \(18-23=-5\) hours at \(g=0.4\); the negative sign indicates five fewer outcome-hours under treatment in this hypothetical contrast. (3) The spillover contrast is \(76-72=4\) points, comparing zero tutored peers with two tutored peers while the student remains untreated. (4) The contrast can combine the effect of the unit’s own treatment with the effect of changing others’ assignments; the two components are different causal contrasts. (5) It assumes that assignment patterns with the same treated-peer proportion are equivalent for the outcome, even when the identities or arrangement of treated peers differ.