Causal Effects of Cluster Treatment Policies

Why Define Effects of Cluster Policies?

When outcomes can depend on other people’s treatment, “the effect of treatment” does not describe a single comparison. Changing one person’s treatment while holding peer exposure fixed asks a different question from changing the treatment rule for an entire cluster. A cluster policy can change who is treated, how many people are treated, and which treatment combinations occur together.

Policy effects make the comparison explicit: specify the treatment assignment rule under each policy, then compare the outcomes that would result under those rules. This is useful when a real decision concerns a cluster-level rule, such as treating everyone in a clinic or treating a set fraction of households in a community. The target is the effect of adopting one rule rather than another, not necessarily the effect of treating a particular person.

Definition: Let clusters be indexed by $c$, with individuals $i=1,\ldots,n_c$ in cluster $c$. Write $z_c=(z_{c1},\ldots,z_{cn_c})$ for the vector of binary treatment assignments in that cluster, and $Y_{ci}(z_c)$ for individual $i$’s potential outcome under that whole vector. Under partial interference, an individual’s outcome may depend on assignments within their own cluster but not on assignments in other clusters.

The vector notation matters: $Y_{ci}(z_c)$ allows the outcome to depend on both the individual’s own assignment and the assignments of their cluster peers. It is not, in general, enough to write an outcome as a function of the individual’s treatment alone. An exposure mapping can summarize relevant features of the peer assignments, but the policy definitions below can be stated directly using the full vector.

Policies and Their Mean Outcomes

A cluster treatment policy specifies which assignment vectors can occur and, if the rule is randomized, the probability of each vector. Denote the policy for cluster $c$ by $\pi_c$. A deterministic rule is a special case: it assigns probability one to a single vector. For a randomized rule, expected outcomes average over its assignment distribution.

Definition: For a target population of $N=\sum_c n_c$ individuals, define the individual-weighted mean outcome under a collection of cluster policies $\boldsymbol{\pi}$ as $$ \mu_{\mathrm{ind}}(\boldsymbol{\pi}) = \frac{1}{N}\sum_c\sum_{i=1}^{n_c} \mathbb{E}_{Z_c\sim\pi_c}\!\left[Y_{ci}(Z_c)\right]. $$ The expectation is over the assignments generated by the policy, holding the potential outcomes fixed. A policy effect comparing $\boldsymbol{\pi}^{A}$ with $\boldsymbol{\pi}^{B}$ is the difference $$ \Delta_{\mathrm{ind}}(\boldsymbol{\pi}^{A},\boldsymbol{\pi}^{B}) = \mu_{\mathrm{ind}}(\boldsymbol{\pi}^{A})- \mu_{\mathrm{ind}}(\boldsymbol{\pi}^{B}). $$

This definition weights each person equally, so larger clusters contribute more to the target mean. A different target gives every cluster equal weight. For $C$ clusters, its mean is $\mu_{\mathrm{cl}}(\boldsymbol{\pi})=C^{-1}\sum_c n_c^{-1}\sum_i\mathbb{E}_{Z_c\sim\pi_c}[Y_{ci}(Z_c)]$. These targets need not agree when cluster sizes differ or when effects vary across clusters. The weighting rule is part of the estimand and should be stated rather than left implicit.

The policy outcome is an expectation over treatment assignments, not an expectation over uncertainty about whether a particular potential outcome exists. In a finite population, each potential outcome is fixed; randomization by the policy determines which assignment vector is realized. If the target is a superpopulation instead, an additional expectation over the population distribution may be used, but the treatment policy still needs to be specified.

Universal Treatment and Fixed Saturation

Two useful policy comparisons are universal treatment and fixed treatment saturation. Universal treatment assigns everyone in a cluster to treatment; its counterpart, universal control, assigns no one to treatment. Fixed saturation instead sets a specified fraction of cluster members to treatment. With binary treatment and a cluster of size $n_c$, exact saturation $q_c$ means exactly $k_c=q_c n_c$ people are treated, so $k_c$ must be an integer. A policy can, for example, select uniformly among all assignment vectors with exactly $k_c$ treated individuals.

Definition: The universal-treatment policy sets $Z_c=(1,\ldots,1)$ in every cluster; the universal-control policy sets $Z_c=(0,\ldots,0)$. A fixed-saturation policy for cluster $c$ assigns exactly $k_c$ of its $n_c$ members to treatment, according to a specified rule for choosing those members. Its saturation is $q_c=k_c/n_c$.

A saturation value alone does not always fully specify a policy. If different assignment vectors with the same number treated can lead to different outcomes, the rule selecting among those vectors matters. Uniform selection is one possible rule; another could favor particular eligible members. When $q_c n_c$ is not an integer, a policy must say how treatment counts are randomized or rounded. For example, a rule might assign $k$ or $k+1$ treated individuals with probabilities chosen to produce a desired average saturation.

Worked Example: Universal Treatment Versus Universal Control

Hypothetical data. Consider two clusters with five individuals in total. Suppose their potential outcomes under universal control are $(4,6)$ in cluster A and $(5,7,6)$ in cluster B. Under universal treatment, suppose the outcomes are $(7,9)$ in cluster A and $(8,10,9)$ in cluster B. Calculate the individual-weighted policy contrast, universal treatment minus universal control.

Under universal control, the total outcome is $4+6+5+7+6=28$, so the mean is $28/5=5.6$ outcome units per person. Under universal treatment, the total is $7+9+8+10+9=43$, giving a mean of $43/5=8.6$ outcome units per person.

Therefore, $$ \Delta_{\mathrm{ind}}(\text{universal treatment},\text{universal control}) = \frac{43}{5}-\frac{28}{5} = 3.00 $$ outcome units per person.

This is a contrast between two cluster-wide rules. It averages the potential outcomes of all five people under each rule; it does not say that changing any one person’s treatment, with peers held fixed, would increase that person’s outcome by 3.00 units.

Worked Example: Comparing Two Exact Saturations

Hypothetical outcome model. In a cluster of three people, suppose the potential outcome for person $i$ is $Y_i(z)=10+2z_i+\sum_{j\ne i}z_j$, measured in outcome units. Compare policies that treat exactly one person ($q=1/3$) and exactly two people ($q=2/3$), choosing uniformly among assignment vectors with the required number treated.

Under the one-treated policy, each of the three possible vectors has one treated person. For any such vector, the treated person has outcome $10+2=12$; each untreated person has outcome $10+1=11$. The cluster mean for each vector is therefore $(12+11+11)/3=34/3$ outcome units per person. Averaging over the three equally likely vectors leaves the policy mean at $34/3$.

Under the two-treated policy, each vector gives two treated people, each with outcome $10+2+1=13$, and one untreated person with outcome $10+2=12$. The mean for each vector is $(13+13+12)/3=38/3$, so the policy mean is $38/3$ outcome units per person.

The fixed-saturation contrast is $$ \frac{38}{3}-\frac{34}{3} = \frac{4}{3} \approx 1.333 $$ outcome units per person, comparing saturation $2/3$ with saturation $1/3$. The fractions, rather than rounded decimal means, make the subtraction explicit.

Why a Policy Effect Is Not an Individual Direct Effect

An individual direct effect compares two of the same person’s potential outcomes while holding peer exposure fixed. In full-vector notation, choose a fixed assignment $z_{c,-i}$ for everyone else in the cluster. The direct effect for individual $i$ at that peer assignment is $Y_{ci}(1,z_{c,-i})-Y_{ci}(0,z_{c,-i})$. Only the individual’s own treatment changes in this contrast. A cluster policy comparison generally changes more than that: it may change the individual’s treatment, their peers’ treatments, and the distribution of those assignments across the cluster.

The distinction is visible in a simple additive model. Suppose, for a cluster of size $n$, $Y_i(z)=\alpha+\tau z_i+\gamma\sum_{j\ne i}z_j$, where $\tau$ and $\gamma$ are fixed outcome-unit effects. At fixed peer assignments, the direct effect is $\tau$, since subtracting the outcome at $z_i=0$ from the outcome at $z_i=1$ leaves only $\tau$.

Derivation: Under a uniform fixed-saturation policy with exactly $k$ treated people, each individual has probability $k/n$ of being treated. By symmetry, each peer also has probability $k/n$ of being treated. Linearity of expectation gives the mean outcome $$ \mu(k) = \alpha+\tau\frac{k}{n} +\gamma(n-1)\frac{k}{n}. $$ Thus, comparing exact saturations $k_A/n$ and $k_B/n$, $$ \mu(k_A)-\mu(k_B) = \bigl(\tau+\gamma(n-1)\bigr) \left(\frac{k_A-k_B}{n}\right). $$ This policy contrast includes both the change in own treatment and the change in peer treatment; it is not merely the direct effect $\tau$.

For universal treatment versus universal control in this model, the contrast is $\tau+(n-1)\gamma$: each person’s own assignment changes, and all $n-1$ peers’ assignments change. If $\gamma=0$, there is no peer effect in this particular model, but that special case does not make policy effects and direct effects equivalent in general.

Worked Example: Two Randomized Policies With Different Assignment Patterns

Hypothetical potential outcomes. In a two-person cluster, let $Y_1(z)=10+2z_1+4z_2+6z_1z_2$ and $Y_2(z)=12+3z_2+z_1+2z_1z_2$, in outcome units. Policy A selects the all-zero vector or the all-one vector with probability $1/2$ each. Policy B selects exactly one treated person, choosing either person with probability $1/2$. Compare the individual-weighted mean outcomes.

Under the all-zero vector, the outcomes are $(10,12)$, with cluster mean $11$. Under the all-one vector, they are $(22,18)$, with cluster mean $20$. Policy A therefore has mean $\tfrac12(11)+\tfrac12(20)=15.5$ outcome units per person.

Under policy B, vector $(1,0)$ gives outcomes $(12,13)$, with mean $12.5$. Vector $(0,1)$ gives outcomes $(14,15)$, with mean $14.5$. Policy B’s mean is $\tfrac12(12.5)+\tfrac12(14.5)=13.5$ outcome units per person.

The contrast, policy A minus policy B, is $15.5-13.5=2.00$ outcome units per person. Both policies have an average of one treated person per cluster, but their assignment patterns differ: policy A sometimes treats both and sometimes neither, whereas policy B always treats exactly one. The interaction terms in the specified potential outcomes make those patterns consequential.

Interpretation, Assumptions, and a Common Pitfall

A policy contrast is defined by its target population, outcome scale, cluster weighting, and assignment rules. It is not automatically an estimate of an effect: estimating it from observed data additionally requires a design or identification assumptions that connect observed outcomes to the relevant potential outcomes. The definitions alone do not guarantee that both policies are observed or that their outcomes can be recovered from the available data.

The potential-outcome setup also relies on a well-defined treatment and policy: each specified assignment vector must correspond to a meaningful intervention, and the observed outcome under the realized vector must equal the corresponding potential outcome (consistency). Partial interference rules out effects from treatment assignments in other clusters. If cross-cluster effects are plausible, the cluster policies and potential outcomes must be expanded to represent them rather than assuming they vanish.

Common mistake: “The mean outcome is higher under the two-thirds saturation policy, so treating an individual rather than leaving that individual untreated raises their outcome by the policy contrast.” This reasoning changes the intervention being described. A saturation contrast compares averages under two cluster assignment rules; own treatment and peer treatment may both change. The correction is to label the result as a policy effect and use a fixed-peer contrast when the question concerns an individual direct effect.

Specify the target.
State whether people or clusters receive equal weight, and identify the population over which the mean is defined.

Write each policy as a rule.
Give the possible cluster assignment vectors and their probabilities, including the treatment count or saturation where relevant.

Average potential outcomes under each rule.
For a randomized policy, average over its assignment distribution before taking the contrast.

Interpret what changed.
Decide whether the contrast changes only own treatment at fixed peer assignments or changes the cluster-level assignment policy.

Key takeaway: A cluster policy effect compares mean potential outcomes under specified cluster assignment rules. Universal treatment, universal control, and fixed saturation are distinct policies, and their contrasts generally combine changes in own treatment with changes in peer exposure.

Check Your Understanding

Answer each question by stating the policy or contrast precisely.

For a cluster of size six, what treatment count corresponds to exact saturation $1/3$? What must a policy specify if it selects among the eligible assignment vectors?
Why can an individual-weighted policy mean differ from a cluster-weighted policy mean when clusters have unequal sizes?
In the additive model $Y_i(z)=\alpha+\tau z_i+\gamma\sum_{j\ne i}z_j$, what is the direct effect at fixed peer assignments, and why can the saturation contrast differ from it?
Suppose a policy randomly assigns either everyone or no one, while another always assigns half the cluster to treatment. Why might the policies have different mean outcomes even if their average treatment counts match?
What additional assumptions or design features are needed to estimate a defined policy effect from observed outcomes, beyond merely writing down its potential-outcome definition?