Why Partition Units Into Clusters?
When one person’s treatment can affect another person’s outcome, a unit’s potential outcome may depend on more than that unit’s own treatment. The earlier material on interference and exposure mappings makes this dependence explicit. A useful next step is to ask whether it can be confined to groups: perhaps pupils affect classmates but not pupils at other schools, or patients’ care affects other patients at the same clinic but not those at other clinics. This restriction is called partial interference.
A cluster partition is not merely a convenient way to organize a data table. It is a substantive claim about which treatment assignments can matter for which outcomes. Within a cluster, interference may be unrestricted; across clusters, the assumption rules it out. The credibility of that claim depends on the setting and on the pathways through which treatment could affect outcomes.
Definition and Notation
Suppose there are \(N\) units, indexed by \(i=1,\ldots,N\), and each unit receives a treatment \(Z_i\). For simplicity, take \(Z_i\in\{0,1\}\), though the definition also applies to treatments with more than two levels. Let \(Y_i(\mathbf z)\) denote unit \(i\)’s potential outcome if the full population’s treatment assignment were \(\mathbf z=(z_1,\ldots,z_N)\).
A proposed partition consists of \(K\) nonempty clusters, \(\mathcal C=\{C_1,\ldots,C_K\}\). The clusters are disjoint and exhaustive: every unit belongs to exactly one cluster. Write \(c(i)\) for the cluster containing unit \(i\), and \(\mathbf z_c=(z_j:j\in C_c)\) for the treatment vector within cluster \(c\).
Under this condition, it is possible to write \(Y_i(\mathbf z_{c(i)})\) instead of \(Y_i(\mathbf z)\): the unit’s potential outcome is a function of its cluster’s treatment vector alone. This notation does not mean that all members of a cluster receive the same treatment. It also does not require outcomes to depend only on a summary such as the number or proportion treated. Such a summary would be an additional exposure mapping.
Partial interference is weaker than no interference. No interference would require \(Y_i(\mathbf z)\) to depend only on \(z_i\). Partial interference permits arbitrary spillovers among units in the same cluster, while excluding spillovers between clusters. The partition specifies where interference may occur; it does not assert that interference actually occurs for every pair within a cluster.
What the Assumption Does—and Does Not—Say
The partition concerns the potential-outcome dependence structure, not the treatment assignment mechanism. Units could be assigned independently, assigned together, or assigned by some other process; none of those possibilities is built into partial interference. In particular, a school, clinic, or village can be a plausible interference cluster even when its members receive different treatments.
The condition also does not rule out shared causes of outcomes across clusters. For example, two schools may face the same regional policy or weather. Such shared conditions can create correlated outcomes without one school’s treatment affecting pupils at the other. The defining restriction is about whether changing an outside cluster’s treatment changes the unit’s potential outcome, not whether outcomes in different clusters are statistically independent.
As with other potential-outcome formulations, treatment and outcomes must be well-defined: the label \(Z_i=1\) should refer to a sufficiently specified treatment, and \(Y_i(\mathbf z)\) should have a clear meaning under each assignment considered. If nominally identical treatments have meaningfully different versions, the potential outcomes may need to distinguish those versions before the interference restriction can be assessed.
Why the Potential-Outcome Notation Reduces
The cluster-specific notation follows directly from the definition. Fix a unit \(i\), and let \(c=c(i)\). Choose any reference assignment \(\mathbf r_{-c}\) for units outside cluster \(c\). For a given within-cluster vector \(\mathbf z_c\), define \(f_i(\mathbf z_c)=Y_i(\mathbf z_c,\mathbf r_{-c})\). The definition says that changing the outside-cluster assignment while holding \(\mathbf z_c\) fixed cannot change the outcome. Therefore, for every full assignment \((\mathbf z_c,\mathbf z_{-c})\),
Because this equality holds for any choice of outside assignment, the reference assignment has no substantive role. The potential outcome is determined by the treatment vector in the unit’s own cluster, which justifies writing \(Y_i(\mathbf z_c)\). This reduction is useful only if the partition is defensible: notation cannot make a false no-cross-cluster-spillover claim true.
Assessing a Proposed Partition
Assessment starts with plausible causal pathways, not with whether cluster labels are readily available. Ask how treatment could travel from one unit to another, where those units are located, and whether the proposed boundary cuts any important pathway. Relevant connections can be social, clinical, institutional, environmental, or economic. A cluster boundary may be plausible for one outcome and implausible for another.
A practical assessment can proceed in four steps:
Decide what receives treatment and whose outcome is being considered. Spillover pathways can differ across outcomes, so “no cross-cluster interference” should be evaluated for the outcome at hand.
State exactly which units belong to each cluster and why the proposed boundaries correspond to limited interaction or transmission.
Consider contact, referral, movement, shared resources, information, and other ways treatment could reach units outside a cluster.
Describe the assumption as plausible or questionable given the substantive evidence. Observed data may reveal likely violations, but cannot establish the counterfactual claim for every assignment.
Worked Example: Tutoring Within Schools
Hypothetical setting. A study includes pupils at three schools. Pupils may receive tutoring individually, and tutoring may affect classmates through shared study materials. Suppose the proposed clusters are the three schools. For a pupil \(i\) at School 1, consider this illustrative potential-outcome model, with achievement measured in points:
This model says the pupil’s outcome increases by 4 points under own tutoring and by 2 points for each tutored peer at the same school. It is only a hypothetical model for checking the partition, not an empirical claim. Hold the pupil’s own treatment at \(z_i=0\). If one of the pupil’s three classmates is treated, the outcome is \(50+4(0)+2(1)=52\) points. If two classmates are treated, it is \(50+4(0)+2(2)=54\) points.
These within-school changes are allowed under partial interference. Now hold all School 1 treatments fixed and change every treatment assignment at Schools 2 and 3 from 0 to 1. The formula still gives 52 points in the first case and 54 points in the second: outside-school assignments do not enter. Thus this model satisfies partial interference for the school partition.
The assumption would be less credible if pupils commonly exchanged materials across schools, or if tutors served pupils from multiple schools. Those pathways would make school boundaries potentially consequential. The fact that the proposed model omits them does not show that they are absent in the real setting.
Worked Example: Patient Referrals Across Clinics
Hypothetical setting. Investigators propose one cluster per clinic because patients often interact with staff and services at the clinic they attend. In a review of 120 patients, 18 had a documented consultation with a clinician at another clinic during the outcome period. The proportion with a documented cross-clinic consultation is \(\frac{18}{120}=0.150\), or 15.0%.
That count alone does not prove that clinic-level partial interference fails: a consultation might be unrelated to the treatment or outcome. But suppose the records also indicate that 11 of the 120 patients had a cross-clinic consultation on a care pathway plausibly affected by the treatment being studied. This is \(\frac{11}{120}=0.0917\), or about 9.17%. It identifies a concrete route by which an assignment at one clinic could matter for a patient assigned to another.
For example, imagine a patient at Clinic A whose potential outcome is measured in recovery points and can depend on whether a clinician at Clinic B received a training intervention. If the patient’s own clinic assignment is held fixed, but the relevant clinician’s training at Clinic B changes, the patient’s outcome could change through the consultation. The proposed clinic partition then does not rule out cross-clinic spillovers for this patient.
The appropriate conclusion is that the partition requires further justification or reconsideration, not that the 11 patients establish a particular causal effect. The review measures observed contacts, whereas partial interference is a claim about potential outcomes under alternative assignments. The review is informative because it reveals a plausible boundary-crossing pathway.
Worked Example: Villages Sharing a River
Hypothetical setting. A program reduces agricultural chemical runoff in some villages. Researchers propose that each village is an interference cluster. However, a river flows from some villages into others. Of eight downstream villages in the study area, three receive water from an upstream village that could receive the program. The share with this documented hydrological link is \(\frac{3}{8}=0.375\), or 37.5%.
For a downstream village \(v\), suppose the outcome is a water-contamination score and, for illustration, follows \(Y_v=10+5z_v+4z_u\), where \(z_v\) indicates the program in village \(v\) and \(z_u\) indicates the program in the upstream village. Hold the downstream village’s assignment at \(z_v=0\). If the upstream village is untreated, the score is \(10+5(0)+4(0)=10\). If the upstream village is treated, the score is \(10+5(0)+4(1)=14\).
Because changing an outside village’s treatment changes the downstream outcome by 4 score points, the village partition violates partial interference under this illustrative model. A larger partition based on connected watersheds might better contain the pathway, although other cross-boundary pathways would still need assessment. A revised partition is a substantive proposal to evaluate, not an automatic guarantee of the assumption.
Common Mistakes and Limits of Evidence
| Mistake | Correction |
|---|---|
| Assuming that “one school per cluster” means pupils cannot affect pupils elsewhere. | School membership is only a proposed boundary. Check cross-school contact, shared staff, and other plausible pathways. |
| Treating partial interference as no interference. | Within-cluster spillovers are allowed and may be arbitrary; only cross-cluster spillovers are ruled out. |
| Assuming every member of a cluster receives the same treatment. | A cluster is a boundary for possible interference, not necessarily a treatment-assignment unit. |
| Concluding the assumption holds because no cross-cluster effect appears in observed outcomes. | The assumption concerns potential outcomes under assignments that may not have been observed. A null observed pattern does not establish invariance. |
The key evidential limitation is that one cannot generally observe the same unit under two assignments that agree within its cluster but differ outside it. Partial interference is therefore not directly verified by comparing observed outcomes alone. Network information, referral records, institutional knowledge, geography, and sensitivity analyses can help assess plausibility, but the conclusion remains conditional on substantive assumptions.
Finally, the partition should be stated at the level relevant to the question. A clinic boundary may be reasonable for an outcome unaffected by outside referrals, yet unreasonable for an outcome tied to those referrals. A village boundary may be suitable for a local program but fail for a river-based outcome. Clear cluster definitions make the scope of the assumption visible and make it possible to identify where it is most vulnerable.
Check Your Understanding
Use the definition to answer each question, making clear which claim concerns within-cluster and which concerns cross-cluster interference.
- In a school partition, can a pupil’s potential outcome depend on how many classmates receive treatment? Explain.
- What must remain unchanged for the partial-interference condition to require the same potential outcome for unit \(i\)?
- A clinic study documents referrals between clinics. Why is that evidence relevant, and why does it not by itself quantify a spillover effect?
- In the river example, which change in assignment contradicts the proposed village partition under the illustrative model?
- Can outcomes be correlated across clusters even if partial interference holds? Explain the difference between outcome correlation and a cross-cluster causal effect.