Why Diagnose Cross-Cluster Spillovers?
Many cluster-based causal analyses allow outcomes to depend on other people’s treatment within the same cluster, while assuming that treatment outside the cluster is irrelevant. That assumption is called partial interference. It can be a useful simplification, but it is not guaranteed by the way a researcher draws cluster boundaries. Patients may share information across clinics, students may interact across schools, and households may be connected across neighborhoods.
A cross-cluster tie does not by itself prove that spillovers occurred. It does show a possible pathway through which an outside assignment could affect an outcome, and therefore identifies where the partial-interference assumption deserves scrutiny. Diagnosis combines information about those pathways, treatment and exposure patterns, and outcomes. It cannot turn observational patterns alone into proof that a causal effect exists or does not exist.
This definition concerns potential outcomes, not merely whether people are connected. A tie matters for the assumption if it allows an outside assignment to change an outcome. Conversely, a measured cross-cluster tie may be harmless for a particular outcome, treatment, or time period. Diagnosis should therefore state what counts as a tie, what treatment could travel along it, and what outcome could respond.
Map Potential Paths Across Cluster Boundaries
Start with the cluster definition used by the design or analysis. Then use relevant data—such as contact surveys, referral records, shared staff, or communication networks—to identify ties whose endpoints fall in different clusters. Choose the unit of counting carefully. An edge-based measure counts cross-cluster ties; a person-based measure counts individuals with at least one such tie. They answer different questions and have different denominators.
Worked Example: Summarizing Cross-Cluster Ties
Suppose a hypothetical study assigns 120 people to six clusters. A network map records 80 unique social ties among them. Twelve ties connect people in different clusters, and 18 people have at least one tie to someone in another cluster.
Edge-based diagnostic. The proportion of recorded ties crossing cluster boundaries is
Thus, 15.0% of the mapped ties cross a cluster boundary. This is a fraction of ties, not a fraction of people.
Person-based diagnostic. The proportion of people with at least one cross-cluster tie is
Here, 15.0% of people have at least one mapped cross-cluster tie. The two percentages happen to be equal, but they need not be: one person can have several cross-cluster ties. These summaries flag potential pathways; neither estimates the proportion whose outcomes were actually affected.
A useful map preserves details that an overall rate can conceal: which clusters are connected, how many ties cross each boundary, whether ties are directed, and whether they are likely to carry the treatment or information relevant to the outcome. Record the timing as well. A tie measured after treatment may itself have been changed by treatment, so treating it as a baseline diagnostic can be misleading.
Use Boundary Outcomes as a Diagnostic, Not a Causal Test
If cross-cluster spillovers are plausible, outcomes among people near a boundary or with cross-cluster ties may differ from outcomes among people in cluster interiors. Such a pattern can motivate further investigation. It is not, on its own, evidence that outside treatment caused the difference: boundary residents may differ in baseline characteristics, access to services, or local conditions. The pattern can also arise by chance.
Define the boundary group and comparison group before examining outcomes when possible. Report their sizes, treatment and exposure distributions, and outcome summaries. If a formal comparison is made, state the assumptions behind it. In particular, the fact that treatment was randomized by cluster does not automatically make boundary status randomized, nor does it remove confounding from comparisons of people with different network ties.
Worked Example: Interpreting an Outcome Pattern Near Boundaries
In a hypothetical evaluation, 24 people with mapped cross-cluster ties have a mean outcome of 7.2 points, while 56 people without such ties have a mean of 6.1 points. The descriptive difference is
The tied group’s mean is 1.1 points higher in this sample. That calculation does not establish that cross-cluster exposure raised outcomes by 1.1 points. For example, people with ties might have had higher outcomes even without the intervention, or their clusters might differ in other ways. A useful next check is to compare treatment assignments and measured baseline characteristics across the groups, and to examine whether the pattern is concentrated where a plausible treatment pathway exists. Even after these checks, a causal interpretation requires an appropriate identification strategy; a boundary comparison alone is not one.
Check What Contrast the Data Identify
When outside treatment can matter, a potential outcome may need to index both own-cluster assignment and an external exposure. Let \(A\) denote the treatment assignment of interest and \(E\) an explicitly defined summary of outside-cluster treatment, such as whether an individual has a treated contact across the boundary. An outcome can then be written \(Y_i(a,e)\), provided the exposure mapping is adequate: assignments grouped under the same \(e\) must be equivalent for the outcome contrast being studied.
A comparison that averages over whatever external exposures occur in each treatment arm is generally not an effect at a fixed exposure. If the distribution of \(E\) differs between arms, the comparison can mix own-treatment and external-exposure differences. Even if the distribution is identical across arms, the result is an average of effects over that common distribution—not necessarily the effect at any particular \(e\).
Worked Example: A Shared Exposure Distribution Gives an Average Effect
Consider a hypothetical setting with two external exposure levels, \(E=0\) and \(E=1\). Suppose the treatment effects at these levels are 0 and 2 outcome units, respectively:
Suppose both treatment arms have the same exposure distribution: half the individuals are at \(E=0\), and half are at \(E=1\). The marginal treatment contrast is the average of the two exposure-specific effects:
The contrast is 1 unit, not the fixed-exposure effect at \(E=0\) (which is 0) or at \(E=1\) (which is 2). Identical exposure distributions remove one source of imbalance but do not turn an average into a fixed-exposure contrast.
For a second illustration, suppose a randomized hypothetical study has the following cell counts and mean outcomes:
| Own assignment \(A\) | External exposure \(E\) | Count | Mean outcome |
|---|---|---|---|
| 0 | 0 | 40 | 10 |
| 0 | 1 | 10 | 12 |
| 1 | 0 | 10 | 12 |
| 1 | 1 | 40 | 16 |
The crude mean under \(A=0\) is \((40(10)+10(12))/50=10.4\). Under \(A=1\), it is \((10(12)+40(16))/50=15.2\). The crude difference is \(15.2-10.4=4.8\) outcome units, but the exposure distribution differs substantially between arms.
Standardizing both arms to a 50:50 exposure distribution gives a control mean of \(0.5(10)+0.5(12)=11\) and a treatment mean of \(0.5(12)+0.5(16)=14\). The standardized contrast is \(14-11=3\) units. It averages the exposure-specific contrasts, which are \(12-10=2\) at \(E=0\) and \(16-12=4\) at \(E=1\). It is not either fixed-exposure effect. Interpreting these cell contrasts causally requires that the design or analysis identifies outcomes within each \(A,E\) condition, that the relevant conditions have positive probability, and that the exposure definition captures the relevant outside assignments. Randomization of \(A\) alone does not automatically guarantee those conditions for \(E\).
Assess Consequences for Identification and Uncertainty
A violation can matter in two distinct ways. First, it can change the causal contrast represented by an analysis. If an outcome responds to assignments outside its nominal cluster, a cluster-policy comparison may include consequences of external treatment as well as the intended within-cluster policy. That may still be a meaningful policy effect, but it is not automatically the effect under partial interference. Second, a violation can invalidate the probability model used for uncertainty calculations. Outcomes in different clusters may be dependent when cross-cluster ties transmit influence.
A short variance calculation shows why ignoring such dependence can understate uncertainty. Suppose, for illustration, an estimator is the mean of two cluster-level contrasts, \(D_1\) and \(D_2\). If each has variance 4 and their covariance is 1, then
The corresponding standard error is \(\sqrt{2.5}=1.581\) contrast units. If the covariance were incorrectly set to zero, the variance would be \((4+4)/4=2\), and the standard error would be \(\sqrt{2}=1.414\). The smaller value reflects an independence assumption, not information that the cross-cluster dependence is absent. In an actual analysis, variance estimation must match the assignment mechanism and dependence structure; this calculation illustrates the issue but is not a general variance formula for network interference.
Diagnostics should therefore inform both the estimand and the uncertainty procedure. Depending on the design and data, investigators may refine clusters, define and analyze external exposure conditions, or conduct sensitivity analyses under alternative exposure mappings. Each choice answers a specific question and requires its own identification and variance assumptions. A generic adjustment for “boundary status” does not, by itself, restore partial interference.
State the cluster definition, relevant time period, and the outcomes and assignments for which outside-cluster influence is assumed absent.
Use relevant network, contact, referral, or shared-resource data. Report tie-based and person-based measures with their denominators.
Compare boundary and interior groups descriptively, while distinguishing a warning signal from a causal estimate.
Ask whether the contrast averages over different external exposures and whether the variance calculation permits dependence across clusters.
Common Mistake
Wrong reasoning: “The treated and control arms have the same distribution of external exposure, so their contrast is the effect at \(E=e\).” Equal distributions do not imply that. The contrast averages exposure-specific effects over the common distribution; as the 0-and-2 example shows, that average can be 1 even though no fixed-exposure effect is 1. To target a fixed \(e\), compare or standardize both arms at \(E=e\), or justify conditions under which the average equals that fixed-exposure effect.
A related error is to treat the absence of a visible boundary outcome difference as proof of no spillover. Limited power, inaccurate tie measurement, or an outcome insensitive to the pathway can hide spillovers. Report what was measured and what the diagnostics can—and cannot—rule out.
Check Your Understanding
Use these questions to check whether you can distinguish a diagnostic from an identification claim.
- What is the difference between the proportion of ties crossing cluster boundaries and the proportion of people with at least one cross-cluster tie?
- Why does a higher mean outcome among boundary residents not, by itself, establish a causal spillover?
- If both treatment arms have the same external-exposure distribution, what kind of contrast does their marginal comparison generally represent?
- In the variance illustration, what assumption produces the smaller standard error of 1.414?
- Name one way cross-cluster spillovers can affect the causal estimand and one way they can affect uncertainty.