Cluster-Randomized Trials With Within-Cluster Spillovers

Why Cluster Randomization Changes the Causal Question

A cluster-randomized trial assigns groups—such as clinics, schools, or villages—to treatment or control. When people can affect one another, a participant’s outcome may depend both on their own treatment and on the treatments received by peers in the same cluster. Assigning the whole cluster together therefore changes both at once.

The design can still answer an important causal question: what is the effect of assigning a cluster to the treatment policy rather than the control policy? But that contrast is not, by itself, an estimate of the effect of treating an individual while holding peer treatment fixed. Keeping those questions separate is essential for interpreting cluster trials.

Potential Outcomes Under Whole-Cluster Policies

Suppose there are \(J\) clusters, with \(n_j\) individuals in cluster \(j\). Let \(Z_j=1\) mean that cluster \(j\) is assigned to the treatment policy and \(Z_j=0\) mean assignment to control. In a simple two-arm cluster trial, all members of a cluster are assigned together.

More generally, let \(Y_{ij}(\mathbf z_j)\) be the potential outcome for individual \(i\) in cluster \(j\) under the cluster’s assignment vector \(\mathbf z_j\). The all-control vector is \(\mathbf 0\); the all-treated vector is \(\mathbf 1\). Write \(Y_{ij}(0)=Y_{ij}(\mathbf 0)\) and \(Y_{ij}(1)=Y_{ij}(\mathbf 1)\) for these two policy outcomes. The observed outcome is the potential outcome corresponding to the assignment actually received.

Definition: The whole-cluster policy effect for an individual is \(Y_{ij}(1)-Y_{ij}(0)\): the contrast between that individual’s outcome if their entire cluster were assigned to treatment and their outcome if their entire cluster were assigned to control. A population policy effect averages this contrast over a specified target population, using weights that define that population.

This definition allows interference within clusters: under either policy, a person’s outcome may depend on the assignments of their peers. To interpret the contrast as a causal effect of the cluster policy, the analysis also needs the following conditions.

Random assignment: cluster assignment is generated by the stated randomization scheme, with positive probability for the policies being compared.
Consistency: the observed outcome is the potential outcome under the cluster assignment received, and treatment and control policies are defined clearly enough to be meaningful.
Partial interference: an individual’s outcome may depend on assignments within their cluster, but not on assignments in other clusters.
Outcome observation: outcomes are recorded as intended, or missingness is addressed under appropriate additional assumptions.

The partial-interference condition is particularly important: if a treated cluster affects outcomes in a control cluster, then the control outcomes may not represent the all-control policy. Randomization alone does not remove that problem.

Worked Example: Estimating a Cluster-Policy Contrast

Suppose six clinics are randomized, with three assigned to a treatment program and three to control. The outcome is the proportion of patients in each clinic who meet a health target; the target population for this example weights the six clinics equally. The treatment-clinic proportions are 0.60, 0.70, and 0.50. The control-clinic proportions are 0.30, 0.40, and 0.50. Assume assignment was randomized, both policies were well defined, there was no interference between clinics, and outcomes were observed for all clinics.

The treatment-arm mean across clinics is

\bar Y_{\mathrm{treated}}=\frac{0.60+0.70+0.50}{3} =\frac{1.80}{3}=0.60.

The control-arm mean across clinics is

\bar Y_{\mathrm{control}}=\frac{0.30+0.40+0.50}{3} =\frac{1.20}{3}=0.40.

The estimated policy contrast is \(0.60-0.40=0.20\), or 20 percentage points. It compares assigning a clinic and its members to the treatment policy with assigning them to control. If treatment affects patients partly by changing how their peers behave, that pathway is included in this contrast. The estimate is a randomized comparison, not proof that every clinic or every patient would experience a 20-percentage-point effect.

Which Population Average Does the Trial Target?

A trial’s target can weight clusters equally or weight individuals equally. These are different estimands when cluster sizes vary. Define the potential outcome mean in cluster \(j\) under policy \(a\) as

\bar Y_j(a)=\frac{1}{n_j}\sum_{i=1}^{n_j}Y_{ij}(a), \qquad a\in\{0,1\}.

The equal-cluster policy effect is the average of cluster-specific mean contrasts. The individual-weighted policy effect is the average contrast across all individuals in the target population:

\tau_C=\frac{1}{J}\sum_{j=1}^{J}\left\{\bar Y_j(1)-\bar Y_j(0)\right\}, \qquad \tau_I=\frac{1}{N}\sum_{j=1}^{J}\sum_{i=1}^{n_j}\left\{Y_{ij}(1)-Y_{ij}(0)\right\},

where \(N=\sum_{j=1}^{J}n_j\). The first gives each cluster equal weight; the second gives each individual equal weight. A study should state which target it intends to describe before choosing how to summarize outcomes.

Worked Example: Unequal Cluster Sizes, Different Targets

Consider two clusters with sizes 2 and 6. Suppose their policy effects on the outcome are 0.40 and 0.20, respectively. These values are hypothetical potential-outcome contrasts for illustration; assume the randomized design and outcome measurement are otherwise valid.

For the equal-cluster target, each cluster receives weight \(1/2\):

\tau_C=\frac{0.40+0.20}{2}=0.30.

For the individual-weighted target, the first cluster contributes two individuals and the second contributes six:

\tau_I=\frac{2(0.40)+6(0.20)}{2+6} =\frac{0.80+1.20}{8}=0.25.

Thus the same cluster-level effects imply a 0.30 average effect per equally weighted cluster and a 0.25 average effect per equally weighted individual. The larger cluster has a smaller effect, so it receives more weight in the individual target and pulls that average downward. These are not competing calculations of one uniquely defined quantity; they answer different questions.

Randomization identifies policy means in expectation under the assignment scheme, but the estimator should match the target. For example, let \(T_j(a)=\sum_iY_{ij}(a)\) be the total outcome in cluster \(j\) under policy \(a\), and let \(\pi_j=P(Z_j=1)\). An inverse-probability estimator of the individual-weighted all-treated mean is

\widehat{\mu}_I(1)=\frac{1}{N}\sum_{j=1}^{J}\frac{Z_jT_j^{\mathrm{obs}}}{\pi_j},

where \(T_j^{\mathrm{obs}}\) is the observed cluster total. The corresponding estimator for the all-control mean replaces \(Z_j/\pi_j\) with \((1-Z_j)/(1-\pi_j)\).

Why the Direct and Spillover Effects Are Not Separated

An individual direct effect compares that person’s outcomes under different own-treatment values while holding peer treatment fixed. A spillover effect compares outcomes under different peer-treatment conditions while holding the individual’s own treatment fixed. These contrasts require potential outcomes under assignments that change one component but keep the other fixed.

In a two-arm trial that assigns every cluster member together, the observed assignment vectors are only \(\mathbf 0\) and \(\mathbf 1\). A person’s own treatment and their peers’ treatments therefore change together. The trial can compare the all-treated and all-control policies, but it does not directly observe conditions such as “this person treated, peers untreated” or “this person untreated, peers treated.” Without additional design variation or assumptions, the separate components are not identified.

Worked Example: The Same Policy Contrast, Different Decompositions

Suppose the outcome is a risk, and for simplicity consider one person and one peer. Write \(Y(z,g)\) for the person’s potential outcome when their own treatment is \(z\) and their peer’s treatment is \(g\). A cluster-randomized trial observes the policy endpoints \(Y(0,0)\) and \(Y(1,1)\), not the mixed assignments. Consider two hypothetical outcome schedules:

Schedule	\(Y(0,0)\)	\(Y(1,0)\)	\(Y(0,1)\)	\(Y(1,1)\)
A	0.20	0.30	0.40	0.50
B	0.20	0.45	0.25	0.50

Both schedules have the same all-treated versus all-control contrast:

$$ Y(1,1)-Y(0,0)=0.50-0.20=0.30. $$

But their components differ. Using the direct contrast at untreated peer exposure and the spillover contrast at untreated own treatment, Schedule A gives

Y(1,0)-Y(0,0)=0.30-0.20=0.10, \qquad Y(0,1)-Y(0,0)=0.40-0.20=0.20.

Schedule B instead gives

Y(1,0)-Y(0,0)=0.45-0.20=0.25, \qquad Y(0,1)-Y(0,0)=0.25-0.20=0.05.

The randomized cluster comparison sees the same two policy endpoints in both schedules. It cannot distinguish these different direct and spillover contrasts from those endpoints alone. The example uses a simple two-person cluster to expose the identification issue; the same problem arises in larger clusters.

A Short Justification for Randomized Identification

The reason randomization identifies a policy mean is that assignment makes each cluster’s observed outcome a known-probability sample of its potential outcome under that policy. For the individual-weighted all-treated mean, take the expectation over the random assignment while treating potential outcomes as fixed. Under consistency, \(T_j^{\mathrm{obs}}=T_j(1)\) whenever \(Z_j=1\). Therefore,

\begin{aligned} E\left\{\widehat{\mu}_I(1)\right\} &=\frac{1}{N}\sum_{j=1}^{J} E\left\{\frac{Z_jT_j^{\mathrm{obs}}}{\pi_j}\right\}\\ &=\frac{1}{N}\sum_{j=1}^{J} \frac{P(Z_j=1)T_j(1)}{\pi_j}\\ &=\frac{1}{N}\sum_{j=1}^{J}T_j(1) =\mu_I(1). \end{aligned}

The same argument applies to the all-control mean using assignment probability \(1-\pi_j\). Subtracting the two unbiased mean estimators gives an unbiased estimator of their difference. This reasoning relies on positive assignment probabilities, correct probabilities in the estimator, consistency, and the absence of cross-cluster interference. It establishes identification of the chosen policy means; it does not create observations under mixed own-and-peer assignment conditions.

Common Mistakes and Design Implications

Calling the cluster contrast an individual direct effect. The wrong reasoning is that treatment clusters differ from control clusters only in an individual’s treatment. In fact, peer treatments change too. The contrast is a whole-cluster policy effect.
Claiming that randomization separates mechanisms. Random assignment protects the policy comparison from confounding under the stated conditions, but it does not supply treatment combinations the design never assigns. Direct and spillover effects are not generally separately identified.
Ignoring the target weights. An equal-cluster average and an individual-weighted average can differ, especially when cluster sizes and effects vary. State the target population and use a summary appropriate to it.
Assuming cluster randomization rules out all interference. It permits interference within a cluster. The causal interpretation still requires that outcomes in one cluster are unaffected by assignments in other clusters, or a design and estimand that address cross-cluster effects.

To estimate separate direct and spillover effects, a design must create relevant variation in own treatment and peer treatment—for example, by randomizing treatment assignments within clusters or by randomizing cluster treatment proportions and then individual assignments. Such designs require their own assignment probabilities and estimands. A conventional two-arm cluster-randomized trial remains valuable, but its cleanest causal interpretation is the effect of the cluster-level policy as implemented.

Key takeaway: With within-cluster spillovers, a two-arm cluster-randomized trial can identify a contrast between all-treated and all-control cluster policies under randomization, consistency, and partial interference. Because own and peer treatment move together, that contrast generally cannot be decomposed into individual direct and spillover effects without additional design variation or assumptions.

Check Your Understanding

Use the definitions and reasoning above to answer these questions.

In a cluster-randomized trial, what does the all-treated versus all-control policy contrast compare when peers can affect one another?
Why does randomizing whole clusters not, by itself, identify a direct effect at fixed peer treatment?
Two clusters have sizes 3 and 9, with policy effects 0.10 and 0.30. Calculate the equal-cluster and individual-weighted average effects.
Name one condition needed to interpret a randomized cluster-policy contrast causally, and explain what can go wrong if it fails.
What additional kind of assignment variation could help distinguish direct from spillover effects?