Experimentation.Two­Stage­Interference

Second-order inclusion probability of the completely randomized design. Under the completely randomized design that treats exactly K of the N units in the population uniformly over all size-K treated subsets, if i and j are two distinct units then the probability that both i and j are treated equals K(K−1)/(N(N−1)).

Stage-1 / between-group SRS variance term (Hudgens–Halloran 2008, the between-group term of Theorems 4 and 6). Under simple random sampling of m of the N := card ι groups, with {0,1} selection indicators U satisfying the SRS first- and second-order selection moments (hmean, hpair) and the Bernoulli diagonal variance (hvar), the sampling variance of the sample mean (∑ᵢ Uᵢ·μᵢ)/m of the group-level quantities μ equals (1 − m/N)/m times the population sample variance SmuVar μ.

Hudgens–Halloran (2008), Theorem 5 (within-group / Neyman form). For any within-group design whose treatment indicators have first moment K/n (hmean) and pairwise second moment K(K−1)/(n(n−1)) (hpair) — the moments of the completely randomized (mixed) design of Assumption 1, which treats exactly K of n units — with the two-valued potential outcomes a (treated state) and b (control state), the randomization variance of the difference-in-means estimator is S₁/K + S₀/(n−K) − Sτ/n.

Hudgens–Halloran (2008), Theorem 5, for the completely randomized design. For a group of n units with potential outcomes a (treated state) and b (untreated state), consider the completely randomized within-group design that treats exactly K units uniformly at random, where K is positive and strictly less than the group size n. Then the randomization variance of the control-minus-treatment difference-in-means estimator under this design equals S₁/K + S₀/(n−K) − Sτ/n, where S₁ and S₀ are the population sample variances of the treated-state and untreated-state outcomes and Sτ is the population sample variance of the unit-level treatment effects.

Two-stage variance decomposition, abstract per-group statistic. For an arbitrary within-group allocation design ψ and comparison design φ and an arbitrary per-group statistic g, assume the selected-group count C is nonzero, the population has nonzero size and at least two groups, and that the stage-1 selection satisfies first-order inclusion probability C/N for every group and second-order (pairwise) inclusion probability C(C−1)/(N(N−1)) — the moments of a simple random sample of C of the N groups. Then the randomization variance of the aggregate (∑ᵢ 1(Sᵢ=ψ)·g i(wᵢ))/C under the two-stage design jointDesign D₁ ψ φ splits into a between-group SRS term with finite-population correction (1 − C/N)/C applied to the population sample variance of the conditional means (ψ i).E (g i), plus a within-group term (1/(C·N)) times the sum of the conditional variances (ψ i).Var (g i).

Hudgens–Halloran (2008), Theorem 4 (two-stage variance decomposition of Ŷ(z;ψ)). For arbitrary within-group allocation designs ψ and comparison designs φ and potential outcomes Y, assume the treated/control unit counts m i are nonzero, the group sizes n i are nonzero, the population has nonzero size and at least two groups, the selected-group count C is nonzero, that the within-group z-propensity of every unit equals m i / n i, and that the stage-1 selection satisfies first-order inclusion probability C/N and second-order (pairwise) inclusion probability C(C−1)/(N(N−1)) — the moments of a simple random sample of C of the N groups, each then randomized by its within-group design. Then the randomization variance of the population estimator Ŷ(z;ψ) under the two-stage design jointDesign D₁ ψ φ splits into a between-group SRS term with finite-population correction (1 − C/N)/C applied to the population sample variance of the group-level potential outcomes ȳ_i(z;ψ), plus a within-group term (1/(C·N)) times the sum of the per-group conditional variances of the within-group estimator.

Hudgens–Halloran (2008), Theorem 4, for the mixed two-stage design. For arbitrary potential outcomes Y, assume the selected-group count C is positive and strictly less than the number of groups, and that every group's treated-unit count K i is positive and strictly less than the group's size n i. Then, under the mixed two-stage design — stage 1 a simple random sample of C of the groups, each selected group i completely randomized to treat K i of its n i units — Var_popEst's between/within variance decomposition holds for the population estimator Ŷ(z; ·), with all of Var_popEst's design-moment hypotheses discharged by the completely-randomized design's inclusion and propensity facts.

Hudgens–Halloran (2008), Theorem 6 (two-stage variance of the direct-effect contrast estimator). Consider the two-stage design that first allocates each group to strategy ψ or strategy φ and then randomizes the group's units accordingly, with outcomes recorded by the potential-outcome function Y. Assume the target sample size C of selected groups is nonzero, the number N of groups is nonzero, N minus one is nonzero, every group's control-arm unit count m0 is nonzero, every group's treatment-arm unit count m1 is nonzero, and every group's size n is nonzero. Suppose the stage-1 design draws a simple random sample of C of the N groups, so that each group is selected with probability C/N and each pair of distinct groups is jointly selected with probability C(C−1)/(N(N−1)), and that within a selected ψ-group each unit's control propensity is m0/n and each unit's treatment propensity is m1/n. Then the randomization variance of the treatment-minus-control estimator decomposes into a between-group term — the finite-population-corrected sample variance of the group-level treatment-minus-control contrasts, scaled by (1 − C/N)/C — plus a within-group term averaging, over the N groups and scaled by 1/(C·N), the conditional variance of each group's within-group contrast estimator.

Hudgens–Halloran (2008), Theorem 6, for the mixed two-stage design. Consider the completely randomized two-stage design in which stage 1 draws a simple random sample of C groups out of the population of N groups, and each drawn group is completely randomized by treating K of its n units, with outcomes recorded by the potential-outcome function Y. Assume the sample size C is strictly positive, C is strictly less than the number N of groups, every group's treated-unit count K is strictly positive, and every group's treated count K is strictly less than its size n. Then the randomization variance of the treatment-minus-control estimator on this design equals the same between-group/within-group decomposition as Var_estDirect, with control count n−K and treatment count K in each group.

Population unbiasedness, either selection (generalizing Theorem 1). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, fix an arbitrary selection flag pick together with a within-group design ρ meant to govern every group whose stage-1 flag equals pick, where on the event that a group's stage-1 flag equals pick, its conditional within-group design (ψ if flagged true, φ if flagged false) actually equals ρ. Assume the normalizing group count denom is nonzero, every group's unit count m at treatment level z is nonzero, and every group's size n is nonzero. Suppose that within each group governed by ρ each unit's propensity of being assigned treatment level z is m/n, and that the stage-1 design selects each group flagged pick with probability denom/N. Then the population estimator on the groups selected by pick is unbiased for the population average potential outcome at level z computed under design ρ.

Direct-contrast unbiasedness (Theorem 1 contrast). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, assume the target sample size C of ψ-selected groups is nonzero, every group's control-arm unit count m0 is nonzero, every group's treatment-arm unit count m1 is nonzero, and every group's size n is nonzero. Suppose that within each group randomized by ψ each unit's control propensity is m0/n and each unit's treatment propensity is m1/n, and that the stage-1 design selects each group into the ψ arm with probability C/N. Then the Horvitz–Thompson estimator built from the ψ-selected groups is unbiased for the direct-effect contrast — the population average outcome under treatment minus under control, both evaluated under strategy ψ.

Indirect-effect unbiasedness (Theorem 2 contrast). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, assume the target sample size dφ of φ-selected groups is nonzero, the target sample size dψ of ψ-selected groups is nonzero, every group's φ-arm control unit count m0φ is nonzero, every group's ψ-arm control unit count m0ψ is nonzero, and every group's size n is nonzero. Suppose that within each group randomized by φ each unit's control propensity is m0φ/n, that within each group randomized by ψ each unit's control propensity is m0ψ/n, that the stage-1 design selects each group into the φ arm with probability dφ/N, and that it selects each group into the ψ arm with probability dψ/N. Then the indirect-effect estimator is unbiased for the spillover contrast: the population average control outcome under φ minus the population average control outcome under ψ.

Total-effect unbiasedness (Theorem 3 contrast). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, assume the target sample size dφ of φ-selected groups is nonzero, the target sample size dψ of ψ-selected groups is nonzero, every group's φ-arm control unit count m0φ is nonzero, every group's ψ-arm treatment unit count m1ψ is nonzero, and every group's size n is nonzero. Suppose that within each group randomized by φ each unit's control propensity is m0φ/n, that within each group randomized by ψ each unit's treatment propensity is m1ψ/n, that the stage-1 design selects each group into the φ arm with probability dφ/N, and that it selects each group into the ψ arm with probability dψ/N. Then the total-effect estimator is unbiased for the contrast between the population average control outcome under φ and the population average treatment outcome under ψ.

Factorization through the exposure. Under stratified interference of the potential outcomes Y, there is a family g i j : Bool × ℕ → ℝ of exposure-indexed potential outcomes such that every outcome factors as Y i j w = g i j (stratExpo i j w).

Within-group unbiasedness. Fix a per-group design ρ governing the within-group treatment randomization and an outcome recorded for every group, unit, and realized within-group assignment, together with a group i, a treatment state z, and a nonzero real number m used as the treated-count denominator. Assume group i has a nonzero number of units and that every unit of group i receives treatment state z with the same probability m divided by the group's size. Then the expected value, under ρ, of the empirical mean outcome among the z-treated units of group i equals the group's average potential outcome under z.

Population unbiasedness (Theorem 1). Consider the two per-group designs ψ and φ governing the within-group randomization when a group is respectively assigned the ψ-strategy or the φ-strategy at stage 1 and an outcome recorded for every group, unit, and realized within-group assignment. Fix a treatment state z, a nonzero real number C used as the population-estimator denominator, and a family m with every group's value m i nonzero; suppose every group has a nonzero number of units, within every ψ-assigned group every unit receives treatment state z with the same probability m i divided by the group's size, and the marginal probability of each group being assigned the ψ-strategy at stage 1 equals C divided by the number of groups. Then the expected value, under the compound two-stage design, of the population estimator restricted to the ψ-assigned groups equals the population average potential outcome under z computed from the ψ-design.

Mixed-orientation decomposition identity. For per-group two-stage designs ψ (treatment strategy) and φ (control strategy) governing each group's within-group assignment and an outcome recorded for every group, unit, and realized within-group assignment, the total contrast — the population control-state mean under φ minus the population treated-state mean under ψ — equals the indirect contrast — the population control-state mean under φ minus the population control-state mean under ψ — minus the direct contrast — the population treated-state mean under ψ minus the population control-state mean under ψ.

Pointwise nonnegativity of the conservative variance estimator. For a group of n units with potential outcomes a and b, provided at least one unit is treated and at least one unit remains in control (K + 1 ≤ n) — so both within-group sample-variance denominators K and n − K are positive — then for every realized assignment w, the conservative variance estimator v̂ar is nonnegative, being the sum of two nonnegative sample variances each divided by a positive count.

Expected observed treated sample variance. For any within-group design whose treatment indicators satisfy the completely-randomized moment hypotheses — first moment K/n (hmean), pairwise second moment K(K−1)/(n(n−1)) (hpair), and a deterministic treated count K on the support (hsupp) — the expectation of the realized treated-state sample variance equals the population treated-state sample variance S₁: E[Ŝ₁] = S₁.

Expected observed control sample variance. The control analogue: for any design satisfying the same completely-randomized moment hypotheses, the expectation of the realized untreated-state sample variance among the n−K control units equals the population untreated-state sample variance S₀: E[Ŝ₀] = S₀.

Conservativeness of the within-group variance estimator (Hudgens–Halloran 2008, Eq. 9). For any design satisfying the completely-randomized moment hypotheses (hmean/hpair/hsupp) and treating exactly K of n units (with 2 ≤ K and K + 2 ≤ n, so both sample variances are well defined), the conservative estimator overstates the randomization variance, Var(τ̂) ≤ E[v̂ar].

Conservativeness of the within-group variance estimator, for the completely randomized design (Hudgens–Halloran 2008, Eq. 9). For a group of n units with potential outcomes a (treated state) and b (untreated state), consider the completely randomized within-group design that treats exactly K units uniformly at random, where at least two units are treated and at least two units remain in control (K + 2 ≤ n) — so both within-group sample variances are well defined. Then the randomization variance of the control-minus-treatment difference-in-means estimator under this design is at most the expectation, under the same design, of the conservative variance estimator v̂ar.

Expectation of an observed sample variance. Let U be a {0,1}-valued selection family over a design ρ on length-n binary assignments, so that every Uⱼ is idempotent, taking only the values 0 and 1, and suppose the group size n and n − 1 are both nonzero, as are the real-valued selection count M and M − 1. If each Uⱼ has first moment M/n, every two distinct units j and k have second moment M(M−1)/(n(n−1)) for the product Uⱼ·Uₖ, and exactly M units are selected on every assignment the design gives positive weight, then the expectation of the realized sample variance of x over the selected units — (1/(M−1))∑ⱼ Uⱼ(xⱼ − x̄_U)² with x̄_U = (∑ Uⱼxⱼ)/M — equals the population sample variance (1/(n−1))∑ⱼ(xⱼ − x̄)².