Two-Stage Experiments under Partial Interference: Effects, Variance, and Limits
Treating a person and treating more of that person’s peers are different interventions. A two-stage experiment varies both: it assigns treatment strategies to groups, then assigns treatment to people within each group. Under partial interference, outcomes depend on assignments within their own group only. (Hudgens and Halloran 2008) defines effects that match this design.
1. Define the policy before defining the effect
Let \(Y_{gj}(\mathbf z_g)\) be unit \(j\)’s outcome in group \(g\). Under an allocation strategy \(s\), define \(\overline Y_g(a;s)\) by averaging over other assignments according to the strategy conditional on the focal unit receiving \(a\). Let \(\overline Y(a;s)\) average these group means with the chosen group weights.
A direct effect at strategy \(s\) is \(\overline Y(1;s)-\overline Y(0;s)\). An indirect effect compares \(\overline Y(0;s_1)\) with \(\overline Y(0;s_0)\). The total effect satisfies the accounting identity
The overall effect averages over each person’s own assignment as well. It answers what happens to the group as a whole when the allocation strategy changes.
2. The unbiasedness result
Proposition 1. In the paper’s mixed two-stage design, randomly select a fixed number of groups for each strategy, then randomly select a fixed number of treated individuals within each group. The appropriate treated and untreated group means, followed by averages over randomized groups, are unbiased for the corresponding allocation-averaged potential outcomes. Their contrasts are unbiased for the direct, indirect, total, and overall effects.
The choice of weights is part of the definition of the effect. Equal weights give a randomly selected small group the same influence as a large group. Weighting by group size instead corresponds to selecting an individual from the population.
3. Why the expectation works
Unbiasedness follows by averaging in the same order as the design. First fix the group’s strategy. Conditional on a person’s own treatment, each admissible assignment to the peers has the probability used in the potential-outcome average. Then average over which groups received the strategy. The two expectations recover the two levels of the estimand.
This sample-mean argument uses fixed treatment counts with both arms nonempty. A Bernoulli design has random arm sizes and can produce an empty arm, so the ratio-unbiasedness conclusion requires a separate argument in that design.
4. Variance needs additional information
Variance introduces a difficulty that the expectation calculation avoids. Expanding a squared estimator produces products of outcomes under different assignments. Those outcomes are never observed together, and arbitrary interference gives no relation between them. Unbiased estimation of a mean therefore leaves some variance components unidentified.
Stratified interference supplies such a relation: at a given treatment count, only a person’s own treatment matters, not the identities of treated peers. This gives two fixed potential-outcome vectors at that count and permits the usual sampling-without-replacement calculation.
The finite-sample bounds take another route by exploiting bounded outcomes and the assignment design. Large-cluster limits explain what increasing group size can achieve when the number of groups remains small.
5. Unbiasedness through both randomization stages
Within a group of size \(n_g\), suppose strategy \(s\) treats exactly \(K_g\) individuals. The treated sample mean is
Because \(\Pr(Z_{gj}=1\mid s)=K_g/n_g\),
The expectation inside the sum averages over precisely the peer assignments used to define the estimand. The untreated calculation uses \((n_g-K_g)/n_g\) instead. It remains valid under arbitrary within-group interference because the entire assignment vector stays inside the potential outcome.
If exactly \(G_s\) of \(G\) groups receive strategy \(s\), let \(I_g(s)\) be their indicators. Averaging their estimated means gives
At each stage, the inclusion probability cancels a fixed denominator. The resulting average gives equal weight to groups. For unequal group sizes, an individual-average target requires size weights at the group-aggregation stage.
6. The finite-population variance under stratified interference
Fix a strategy and write \(n=n_g\), \(K=K_g\). Under stratified interference, the two potential-outcome vectors \(y_j(1;s),y_j(0;s)\) are fixed at the specified total count. Complete randomization implies
The negative covariance follows from the fixed sum \(\sum_jZ_j=K\): its variance is zero, so the off-diagonal terms must offset the diagonal terms.
Rewrite the direct-effect estimator as
For finite-population variance \(S_v^2=(n-1)^{-1}\sum_j(v_j-\overline v)^2\), the indicator covariances give \(\operatorname{Var}(\sum_jZ_jv_j)=K(n-K)S_v^2/n\). Expanding \(S_v^2\) gives
because \(S_\tau^2=S_1^2+S_0^2-2S_{10}\). The unobserved covariance \(S_{10}\) has been collected into the treatment-effect variance. Dropping its negative contribution yields a conservative target, estimable from within-arm sample variances under the stated design.
7. What a large group makes approximately normal
Theorem 2. Under the finite-population variance and negligibility conditions in (Liu and Hudgens 2014), with both treatment-arm counts growing, the direct-effect estimator centered at its group estimand and divided by its true conditional standard deviation converges to a standard normal variable. Corresponding results hold for allocation-specific means and overall group means.
The transformed \(v_j\) above turns the proof into a finite-population CLT for a simple-random-sample sum. One must rule out a few \(v_j\) carrying a nonnegligible fraction of its variance and ensure a nondegenerate denominator. Conditions only on one potential-outcome vector need not control a contrast whose two components nearly cancel.
Replacing the true variance by a conservative estimate requires one more step. The estimate may converge to a larger quantity, so the studentized statistic need not have variance one. Conservative normal intervals can still follow, but ratio consistency is a stronger conclusion, particularly with heterogeneous individual effects.
8. A few randomized groups leave a mixture
Suppose \(G\) remains fixed and \(G_s\) groups receive a strategy. Conditional on the selected set \(\mathcal G\), independent within-group randomizations give an average centered at
with conditional variance \(G_s^{-2}\sum_{g\in\mathcal G}\operatorname{Var}(\widehat\tau_g\mid s)\). Unconditionally, its law averages over the possible sets \(\mathcal G\).
To see what increasing group sizes can and cannot remove, separate the variance by conditioning on the selected groups:
If group effects are fixed and have finite-population variance \(S_{\tau,g}^2\) across groups, the second term is \((1-G_s/G)S_{\tau,g}^2/G_s\). It need not vanish as group sizes increase. The first term can shrink while the distribution approaches a discrete mixture over different group-average effects.
The first term can also be evaluated. Suppose the within-group estimators are independent conditional on the selected set and have conditional variances \(\nu_g\). Then
so
In the last display, the first term measures uncertainty about which group effects were selected; the second averages the remaining within-group randomization variances. Increasing the numbers treated and untreated within each group can reduce \(\nu_g\). A census of groups makes the group-selection term zero while leaving within-group randomization uncertainty.
The allocation strategy is part of each group effect \(\tau_g\). Changing saturation can therefore change both the target and its variance. The two-stage definitions keep the same strategy in the expectation and variance calculations, so an ordinary no-interference formula cannot be carried over while changing the strategy it refers to.
For instance, selecting one of two groups with effects zero and two leaves an equal mixture at zero and two after within-group noise disappears. Centering at the overall effect one gives deviations of minus one and plus one, not an increasingly precise normal estimate. More individuals within a group and more randomized groups address different variance components. The few-cluster sign test uses a different model to obtain inference from a finite collection of large-cluster scores.