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Stratified-interference variance estimators are conservative

Statement

Assume the two-stage design of Two-stage randomized design and strategy-specific effects, stratified interference as in Stratified interference and source design-based variance estimators, and $2 \leq C \leq N - 2$ . For the population direct-effect variance estimator under $ψ$ , additionally assume $2 \leq K_{i}^{ψ} \leq n_{i} - 2$ for every group. Then $E [{\hat{V}}_{D} (ψ)] = Var (\hat{D E} (ψ)) + \frac{1}{N^{2}} \sum_{i = 1}^{N} \frac{S_{i D}^{2} (ψ)}{n_{i}} .$ For the source arm-wise variance estimators of the indirect, total, and overall contrasts, $E [{\hat{V}}_{I} (ϕ, ψ)] = Var (\hat{I E} (ϕ, ψ)) + \frac{S_{I}^{2}}{N},$ $E [{\hat{V}}_{T} (ϕ, ψ)] = Var (\hat{T E} (ϕ, ψ)) + \frac{S_{T}^{2}}{N}, E [{\hat{V}}_{O} (ϕ, ψ)] = Var (\hat{O E} (ϕ, ψ)) + \frac{S_{O}^{2}}{N},$ where $I_{i} = {\overset{ˉ}{Y}}_{i} (0; ϕ) - {\overset{ˉ}{Y}}_{i} (0; ψ), T_{i} = {\overset{ˉ}{Y}}_{i} (0; ϕ) - {\overset{ˉ}{Y}}_{i} (1; ψ), O_{i} = {\overset{ˉ}{Y}}_{i} (ϕ) - {\overset{ˉ}{Y}}_{i} (ψ),$ and, with $\overset{ˉ}{I} = N^{- 1} \sum_{i} I_{i}$ , $\overset{ˉ}{T} = N^{- 1} \sum_{i} T_{i}$ , and $\overset{ˉ}{O} = N^{- 1} \sum_{i} O_{i}$ , $S_{I}^{2} = \frac{1}{N - 1} \sum_{i} (I_{i} - \overset{ˉ}{I})^{2}, S_{T}^{2} = \frac{1}{N - 1} \sum_{i} (T_{i} - \overset{ˉ}{T})^{2}, S_{O}^{2} = \frac{1}{N - 1} \sum_{i} (O_{i} - \overset{ˉ}{O})^{2} .$ All correction terms are nonnegative. The direct-effect correction is zero exactly when, within every group, the individual direct effects $Y_{i j} (0; ψ) - Y_{i j} (1; ψ)$ are constant across $j$ . Each contrast correction is zero exactly when its corresponding group-level effects are constant across $i$ . Thus the variance estimators are conservative, and equality requires the stated additivity condition. The variance estimators are undefined outside their stated minimum sample sizes; the point-effect estimators remain defined for the wider design range $1 \leq K_{i}^{s} < n_{i}$ .

Facts & Assumptions

Given: Fixed finite-population potential outcomes; the two-stage complete randomization; stratified interference; $2 \leq C \leq N - 2$ ; and, for the direct variance estimator under $ψ$ , $2 \leq K_{i}^{ψ} \leq n_{i} - 2$ for every group.

[F1]

Two-stage randomized design and strategy-specific effects: exactly $C$ groups are selected uniformly for $ψ$ , and conditional on strategy labels, the fixed-size within-group treatment subsets are uniform and independent across groups.

[F2]

Stratified interference and source design-based variance estimators: at a fixed within-group treated count, each unit's outcome is invariant to which peers are treated; it defines the sample-variance estimators used below.

[F3]

Finite simple-random-sampling moment identities: finite-population sample means have the stated covariance identity, and observed sample covariance is unbiased when the sample size is at least two.

[F4]

Finite two-stage sampling variance identities: first-stage strategy means and contrasts have the stated design variance formulas, and the usual arm-wise variance estimator adds the finite-population variance of the group-level contrast divided by $N$ .

[F5]

Variance, standard deviation, and covariance on a finite probability space: design variance is the finite expectation of the squared centered estimator, with covariance the corresponding centered product expectation.

[F6]

Two-stage strategy-specific effect estimators are design-unbiased: the observed group means under either strategy are unbiased for their defined group-level targets.

[F7]

Laws of finite sums and finite products: finite sums distribute over addition and scaling.

[F8]

Expectation is linear for every finite family of random variables, without any independence hypothesis: expectation distributes over finite real linear combinations on a finite probability space.

[F9]

Conditional expectation on positive-probability events in finite spaces: event-conditioned means are finite weighted sums and conditional expectations average over a finite partition to the unconditional expectation.

[F10]

Measurable coordinatewise functions preserve independence: measurable coordinatewise functions of independent random elements remain independent.

Proof

1.1F1F2F3F9given

Fix a group $i$ assigned to $ψ$ , put $k = K_{i}^{ψ}$ , and set $x_{0 j} = Y_{i j} (0; ψ)$ and $x_{1 j} = Y_{i j} (1; ψ)$ . By [F9], conditioning on $S_{i} = ψ$ uses the finite conditional weighted-sum law; [F2] fixes the two potential-outcome lists, while [F1] makes the treated units a uniform $k$ -subset and the controls its uniform $(n_{i} - k)$ -element complement. If ${\overset{ˉ}{x}}_{0, T}$ is the treated-subset mean of $x_{0}$ and ${\overset{ˉ}{x}}_{0}$ is its full-group mean, the control mean is ${\overset{ˉ}{x}}_{0} - \frac{k}{n_{i} - k} ({\overset{ˉ}{x}}_{0, T} - {\overset{ˉ}{x}}_{0})$ . Both arm means have expectations ${\overset{ˉ}{x}}_{0}, {\overset{ˉ}{x}}_{1}$ by [F3], so $E [{\hat{D}}_{i} (ψ) ∣ S_{i} = ψ] = D_{i} (ψ)$ .

1.2F1F2F4F6F8F10given

For the indirect contrast apply [F4] with $a_{i} (ϕ) = {\overset{ˉ}{Y}}_{i} (0; ϕ)$ and $a_{i} (ψ) = {\overset{ˉ}{Y}}_{i} (0; ψ)$ and observed group means $X_{i} (s) = {\hat{\overset{ˉ}{Y}}}_{i} (0; s)$ . These are measurable functions of their groups' second-stage assignments, so [F1] and [F10] supply [F4]'s conditional-independence premise; [F6] supplies the means. The arm-wise estimator's correction is $S_{I}^{2} / N$ . For the total contrast use $a_{i} (ϕ) = {\overset{ˉ}{Y}}_{i} (0; ϕ)$ and $a_{i} (ψ) = {\overset{ˉ}{Y}}_{i} (1; ψ)$ with their corresponding observed group means; [F6] and [F4] give correction $S_{T}^{2} / N$ . For the overall contrast use $a_{i} (s) = {\overset{ˉ}{Y}}_{i} (s)$ and $X_{i} (s) = {\hat{\overset{ˉ}{Y}}}_{i} (s)$ ; the same results give correction $S_{O}^{2} / N$ . The formulas are exactly the arm-wise variance sums in [F2], and expectation linearity [F8] gives each displayed identity.

2.1F2F3F5F7F8F9step 1.1algebra

Write $S_{i 0}^{2}, S_{i 1}^{2}$ for the finite-population variances of $x_{0}, x_{1}$ and $S_{i 01}$ for their covariance. By [F3], $Cov ({\overset{ˉ}{x}}_{0, T}, {\overset{ˉ}{x}}_{1, T}) = (1 - k / n_{i}) S_{i 01} / k$ , so the control-mean covariance with the treated mean is $- \frac{k}{n_{i} - k} (1 - k / n_{i}) S_{i 01} / k = - S_{i 01} / n_{i}$ . Also the control-mean variance is $k S_{i 0}^{2} / (n_{i} (n_{i} - k))$ and the treated-mean variance is $(n_{i} - k) S_{i 1}^{2} / (n_{i} k)$ . Expanding the variance of their difference by [F5], expectation linearity [F8], and [F7], and using $S_{i D}^{2} = S_{i 0}^{2} + S_{i 1}^{2} - 2 S_{i 01}$ , gives $v_{i} = Var ({\hat{D}}_{i} (ψ) ∣ S_{i} = ψ) = S_{i 0}^{2} / (n_{i} - k) + S_{i 1}^{2} / k - S_{i D}^{2} (ψ) / n_{i}$ . The sample-covariance identity [F3] makes $E [s_{i 0}^{2} ∣ S_{i} = ψ] = S_{i 0}^{2}$ and $E [s_{i 1}^{2} ∣ S_{i} = ψ] = S_{i 1}^{2}$ ; hence $E [{\hat{v}}_{i} ∣ S_{i} = ψ] = v_{i} + S_{i D}^{2} (ψ) / n_{i}$ for ${\hat{v}}_{i} = s_{i 0}^{2} / (n_{i} - k) + s_{i 1}^{2} / k$ .

3.1F1F2F4F8F9F10step 1.1step 2.1givenalgebra

Each group estimator ${\hat{D}}_{i} (ψ)$ is a measurable function of that group's second-stage assignment. By the conditional independence in [F1] and [F10], these estimators are conditionally independent across groups, so [F4] applies using the means from step 1.1 and variances from step 2.1. With $S_{D}^{2} = (N - 1)^{- 1} \sum_{i} (D_{i} (ψ) - D E (ψ))^{2}$ , [F4] gives $Var (\hat{D E} (ψ)) = (1 - C / N) S_{D}^{2} / C + (C N)^{- 1} \sum_{i} v_{i}$ and $E [s_{D, ψ}^{2}] = S_{D}^{2} + N^{- 1} \sum_{i} v_{i}$ . Therefore the first component of ${\hat{V}}_{D}$ has expectation $(1 - C / N) S_{D}^{2} / C + (1 - C / N) (C N)^{- 1} \sum_{i} v_{i}$ . By [F8], [F9], and $P (S_{i} = ψ) = C / N$ , the second component has expectation $N^{- 2} \sum_{i} {v_{i} + S_{i D}^{2} (ψ) / n_{i}}$ . Adding the components and using $(1 - C / N) / (C N) + 1 / N^{2} = 1 / (C N)$ proves the direct-effect identity.

4.1F2F7step 3.1step 1.2algebra∎

Each correction is a positive constant times a finite sum of squared deviations. By [F7], such a correction is zero exactly when every squared deviation is zero: the direct correction requires each within-group direct effect to be constant across units, and each contrast correction requires its corresponding group-level effects to be constant across groups. Thus every correction is nonnegative and the stated equality conditions are exact.

Depends on

Used by

Nonadditive direct effects create positive variance-estimator bias Example

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Sources

Michael G. Hudgens and M. Elizabeth Halloran, Toward Causal Inference With Interference, Theorems 4–9 and Appendices A.4–A.9 (standard reference, not scraped)