Randomized Saturation Designs and Their Optimal Allocation
To learn how spillovers change with treatment intensity, an experiment can assign different treatment fractions to different clusters. Moving the fractions farther apart strengthens a slope comparison, but can leave few treated or untreated observations at an endpoint. (Baird et al. 2018) studies this tradeoff and the allocation of clusters across fractions. The causal effects are those defined by a two-stage experiment under partial interference.
1. Separate the effects
Write \(\overline Y(d,p)\) for the mean outcome at own treatment \(d\) and cluster saturation \(p\). A pure control is in a cluster with \(p=0\); a within-cluster control is untreated in a cluster with \(p>0\).
The treated effect relative to pure controls is \(\mathrm{ITT}(p)=\overline Y(1,p)-\overline Y(0,0)\), while the untreated spillover is \(\mathrm{SNT}(p)=\overline Y(0,p)-\overline Y(0,0)\). The overall effect is
A slope comparison divides the difference in effects by the difference in saturation. Consequently its variance is divided by the square of that gap. This explains the precision advantage of separating the saturation levels.
2. Derive the precision cost
Suppose each cluster has \(n\) units, there are \(C\) clusters, \(f(p)\) is the fraction assigned saturation \(p\), and \(\psi=f(0)\). Under the paper’s constant individual variance \(\sigma^2+\tau^2\) and within-cluster covariance \(\tau^2\), the variance of a treated group mean is \(\tau^2+\sigma^2/(np)\).
Averaging across \(Cf(p)\) independent clusters and comparing with \(C\psi\) pure-control clusters gives
For untreated spillovers, replace \(p\) by \(1-p\) in the first term. For a slope across two saturations, divide the sum of their arm variances by the squared saturation gap.
3. The optimal-design results
Proposition 1. Under the paper’s mean and covariance model, the symmetric slope-precision objective is minimized by equal cluster allocations to two saturations symmetric around \(1/2\). In the continuous relaxation, their gap \(\Delta\) satisfies
For the stated pooled treated-and-spillover objective with constant effects across positive saturations, the optimal partial-population design uses saturation \(1/2\), with an optimally chosen pure-control fraction between \(\sqrt2-1\) and \(1/2\).
At \(\rho=0\), the continuous slope-optimal gap is \(1/\sqrt2\); as correlation rises, the optimal gap grows. But finite clusters require integer treatment counts and positive counts in both arms. The continuous answer must be compared over admissible allocations, rather than rounded without checking the objective.
4. Why the objective matters
The advantage has a cost: near an endpoint, one treatment cell is small. Pure-control clusters introduce another allocation choice. They improve comparisons with a common untreated baseline, but use clusters that could otherwise be assigned positive saturations.
The design objective must specify which effects are to be learned. Pooling positive saturations can increase precision when the target and model permit it. If effects vary with saturation, pooling changes their weights and hence the target; an allocation optimal for constant effects need not remain optimal.
5. Start with the variance of one cluster mean
Represent the covariance model as \(Y_{ij}=\mu_{ij}+U_j+\varepsilon_{ij}\), with independent cluster shocks of variance \(\tau^2\) and independent idiosyncratic shocks of variance \(\sigma^2\). For a treatment cell containing \(np\) individuals, its average error is \(U_j+(np)^{-1}\sum\varepsilon_{ij}\). Therefore its variance is
A larger treatment cell averages out more individual noise, while every person still shares the same cluster shock. More independent clusters average out both components. The formula separates these two ways of increasing precision.
An untreated mean within the same cluster shares \(U_j\), so its covariance with the treated mean is \(\tau^2\) in this model. A within-cluster direct contrast cancels that common shock and has variance \(\sigma^2(1/(np)+1/[n(1-p)])\). By contrast, a comparison with pure-control clusters retains shocks from both sets of clusters. The estimand determines which variance calculation is appropriate.
6. Deriving the optimal allocation for a fixed comparison
Suppose two independent sets of clusters provide means with per-cluster variances \(A\) and \(B\), and allocate fraction \(f\) to the first. Apart from the common factor \(1/C\), the difference has variance
Differentiating gives \(V'(f)=-A/f^2+B/(1-f)^2\). The unique interior optimum is
This is the usual square-root allocation rule: assign more clusters to the noisier mean. Equal allocation follows when the two per-cluster variances coincide, as in the symmetric slope problem at symmetric saturations.
7. Deriving the symmetric saturation gap
Take two positive saturations \(p=(1-\Delta)/2\) and \(q=(1+\Delta)/2\), with half the available slope-design clusters at each. The variance of a slope in treated outcomes is
To obtain the second term, use \(1/p+1/q=4/(1-\Delta^2)\). The untreated slope has the same variance by symmetry, so summing the two objectives does not change the optimizer.
Let \(x=\Delta^2\), \(A=4\tau^2\), and \(B=8\sigma^2/n\). Minimizing \(A/x+B/[x(1-x)]\) gives
Since \(A/B=n\rho/[2(1-\rho)]\), this reproduces the stated equation. At zero intracluster correlation, \(x=1/2\); as the cluster component grows, the solution moves toward one. A wider gap improves the signal in a slope comparison, but creates small treatment cells whose idiosyncratic variance diverges as \(\Delta\uparrow1\) when \(\sigma^2>0\).
The continuous solution is a guide to the finite design. Evaluate the objective at feasible integer treatment counts, retaining nonempty arms, rather than rounding a saturation without checking its variance. If all noise is at the cluster level, the endpoint behavior also needs separate consideration.
8. Why the pure-control fraction lies between two familiar values
For a single positive saturation of one half and a pure-control fraction \(\psi\), the sum of treated-effect and untreated-spillover variances is proportional to
This objective adds two marginal variances. Although both estimated effects use the same pure-control mean, their covariance does not enter that particular sum. Applying the square-root allocation rule gives
If \(\tau^2=0\), this is \(1/(1+\sqrt2)=\sqrt2-1\). If idiosyncratic noise becomes negligible relative to cluster noise, it approaches \(1/2\). The positive-saturation cluster supplies two smaller cells, making it noisier than a pure-control cluster when individual noise matters.
The shared-control covariance can be calculated explicitly. At a single positive saturation \(p\), let \(T\) and \(U\) be the treated and untreated sample means averaged over \(C(1-\psi)\) partial-population clusters, and \(B\) the mean in \(C\psi\) pure-control clusters. Under the same covariance model,
Since the two cluster groups are independent,
Subtracting the estimated effects gives the direct contrast \(T-U\), whose variance is
The direct contrast removes the common pure-control mean and the common shock in the partially treated clusters. Under this model, pure-control clusters therefore provide no precision gain for the direct effect alone. They remain needed for the specified spillover and total-effect comparisons.
For the overall effect, combine the two estimates as \(p(T-B)+(1-p)(U-B)\). The partial-population component has variance
Adding the pure-control variance gives an allocation optimum \(\psi=1/2\) for this fixed-\(p\) comparison. The algebra cancels \(p\) from the variance under the maintained constant-covariance model, although the target effect can still depend on \(p\). The paper’s different design recommendations reflect different estimands and objectives; the covariance terms explain why they need not coincide.
For an overall effect, the estimates are combined before taking the variance, so their covariance must be included. This gives a different allocation problem from adding marginal variances. Likewise, if effects vary across saturations, pooling must state the weights that define the response being estimated.