Two-Stage Randomized Saturation Designs

Why Randomize Saturation in Two Stages?

When outcomes can depend on other people’s treatment, assigning individuals independently at one common treatment rate may not provide every comparison of interest. A two-stage randomized saturation design deliberately varies the treatment rate across clusters, then assigns individuals within each cluster. The first stage creates clusters with different levels of treatment availability; the second stage creates treated and untreated individuals within at least some of those clusters. Together, the stages can support comparisons of an individual’s own treatment and of their peers’ treatment.

The assigned saturation is a feature of the randomization policy, not necessarily the exact exposure experienced by a particular individual. In a fixed-count design, an individual’s own treatment status affects how many treated peers remain. A careful design therefore specifies both stages and checks the probability of each joint assignment relevant to the intended contrast.

Specifying the Two Stages

Let cluster \(j\) contain \(n_j\) individuals. For example, the first stage might assign clusters to \(S_j \in \{0.25,0.75\}\), and the second stage might treat a fixed number of individuals corresponding to the assigned saturation. Alternatively, conditional on \(S_j=s\), each individual might be assigned treatment independently with probability \(s\). These are different assignment rules: the first fixes the number treated in a cluster, while the second fixes a probability but lets the number treated vary.

For clarity, suppose \(G_{ij}\) is the number of treated peers of individual \(i\), divided by the number of peers. If there are \(n_j-1\) peers, then \(G_{ij}=\sum_{k\ne i} Z_{kj}/(n_j-1)\). Other exposure mappings are possible, but the mapping must be chosen before interpreting a contrast. The cluster-level saturation \(S_j\) and the individual-level peer exposure \(G_{ij}\) need not be equal.

Assume partial interference: an individual’s outcome may depend on assignments within their own cluster, but not on assignments in other clusters. For the exposure-specific comparisons below, also assume the chosen mapping is adequate: outcomes can be represented as \(Y_{ij}(z,g)\), where \(z\) is own treatment and \(g\) is mapped peer exposure. This representation does not claim that all assignments producing the same \(z\) and \(g\) have identical consequences unless that is part of the mapping assumption.

Assignment Probabilities and Supported Comparisons

Write \(\pi_{ij}(z,g)=P(Z_{ij}=z,G_{ij}=g)\), with probability taken over both stages of randomization. A direct contrast at fixed peer exposure \(g\) compares \(Y_{ij}(1,g)\) with \(Y_{ij}(0,g)\). For both conditions to be represented by the design for individual \(i\), the joint assignment probabilities must satisfy \(\pi_{ij}(1,g)>0\) and \(\pi_{ij}(0,g)>0\). A spillover contrast at fixed own treatment \(z\), comparing peer exposures \(g_1\) and \(g_0\), requires \(\pi_{ij}(z,g_1)>0\) and \(\pi_{ij}(z,g_0)>0\).

These are design-support conditions, not guarantees that an effect can be estimated precisely. A very small positive probability may yield few observations in an exposure condition. Also, probabilities must be computed for the joint exposure of interest; showing that a saturation level occurs does not by itself show that every own-treatment and peer-exposure combination occurs.

For a fixed-count design, let a cluster of size \(n\) assigned saturation \(s=m/n\) have exactly \(m\) treated individuals, selected uniformly among the \(\binom{n}{m}\) possible sets. Then \(P(Z_i=1\mid S=s)=m/n\) and \(P(Z_i=0\mid S=s)=(n-m)/n\). If \(Z_i=1\), the number of treated peers is \(m-1\); if \(Z_i=0\), it is \(m\). Thus, for an exact peer count \(k\), both own-treatment conditions do not generally occur at the same fixed saturation.

Under independent Bernoulli assignment within a cluster, conditional on saturation \(s\), each individual is treated independently with probability \(s\). For a cluster of size \(n\), the probability of own treatment \(z\) and exactly \(k\) treated peers is \(\binom{n-1}{k}s^{k+z}(1-s)^{n-1-k+1-z}\), for \(z\in\{0,1\}\) and \(k=0,\ldots,n-1\). Equivalently, this is \(s^z(1-s)^{1-z}\binom{n-1}{k}s^k(1-s)^{n-1-k}\): own assignment and peer assignments are independent conditional on \(s\). If saturation \(s\) is itself assigned with probability \(p(s)\), multiply this conditional probability by \(p(s)\) to obtain the joint probability including the first stage.

Fixed-Count Designs Can Require Multiple Saturations

With exactly \(m\) treated individuals in a cluster of size \(n\), conditional on \(S=m/n\), a particular individual is treated with probability \(m/n\). If untreated, that individual has exactly \(m\) treated peers; if treated, they have exactly \(m-1\). For an exact peer count \(k\), the two own-treatment conditions therefore require different totals: \(m=k\) when the individual is untreated and \(m=k+1\) when treated. If both saturation levels \(k/n\) and \((k+1)/n\) are assigned with positive probability, both conditions can be supported across those levels.

This design feature matters when the target is a direct effect holding realized peer exposure fixed. A contrast that instead holds the assigned saturation fixed is a different policy comparison: under fixed-count assignment, the peer exposure accompanying each own-treatment state differs. State which quantity is held fixed before treating a comparison as “direct.”

Spillover support can also be checked directly. For example, under fixed-count assignment at \(S=2/6\), an untreated individual has two treated peers, while a treated individual has one treated peer. For an untreated individual, this particular saturation does not vary peer count at all; to compare spillovers among untreated individuals, the design needs multiple saturation levels that allow the individual to remain untreated while peer exposure changes.

A Design Check Before the Experiment

Common Mistake: Saturation Is Not Individual Exposure

A frequent error is to reason that a cluster assigned saturation \(s\) gives every individual peer exposure \(s\). That is generally false. In a fixed-count cluster, if exactly \(m\) of \(n\) individuals are treated, a treated person has \(m-1\) treated peers and an untreated person has \(m\). Their peer proportions are \((m-1)/(n-1)\) and \(m/(n-1)\), respectively. Under Bernoulli assignment, the realized peer proportion varies from person to person and can differ substantially from \(s\), especially in small clusters.

The correction is to distinguish the assigned saturation, the individual’s own treatment, and the realized peer exposure in the exposure mapping. Then calculate probabilities for the exact joint conditions in the causal contrast. Positive probability of a saturation alone is not enough to establish support for a direct or spillover comparison.