Two-stage randomized design and strategy-specific effects

Let $N>1$ groups have fixed sizes $n_i\ge 2$, and let the fixed potential outcomes satisfy partial interference as in Finite-population potential outcomes under partial interference. Let $\varphi$ and $\psi$ be two group-specific treatment-allocation strategies. Under strategy $s\in \{\varphi ,\psi \}$, group $i$ has a fixed treated count $K_i^s\in \{1,\dots ,n_i-1\}$. Fix $C\in \{1,\dots ,N-1\}$. The two-stage design first chooses uniformly a set of exactly $C$ groups for $\psi$, assigning the other $N-C$ groups to $\varphi$. Conditional on these strategy labels, it independently chooses in each group a uniformly random $K_i^s$-element subset for treatment, where $s$ is that group's strategy. Here $\left(\genfrac{}{}{0px}{}{n}{k}\right)$ denotes the number of $k$-element subsets of an $n$-element set. Thus, for any strategy vector $\mathbf{s}$ with exactly $C$ entries equal to $\psi$ and any assignment vectors ${\mathbf{z}}_i$ with ${\sum }_j{z}_{ij}=K_i^{s_i}$, $\mathbb{P}(\mathbf{S}=\mathbf{s},\mathbf{Z}=\mathbf{z})={\left(\genfrac{}{}{0px}{}{N}{C}\right)}^{-1}{\prod }_{i=1}^N{\left(\genfrac{}{}{0px}{}{n_i}{K_i^{s_i}}\right)}^{-1};$ all other pairs have probability zero. In particular, $\mathbb{P}(Z_{ij}=1\mid S_i=s)=K_i^s\mathrm{/}{n}_i$ and $\mathbb{P}(Z_{ij}=0\mid S_i=s)=(n_i-K_i^s)\mathrm{/}{n}_i$, both positive.

Write ${\mathbb{E}}_{i,s}$ for expectation under the finite uniform assignment law within group $i$ under strategy $s$. Conditional means given $Z_{ij}=z$ are the finite conditional weighted sums, defined because that event has probability strictly between zero and one; Conditional expectation on positive-probability events in finite spaces verifies that their piecewise versions satisfy the integral characterization of Conditional expectation given a sigma algebra. Define the strategy-specific group and population mean potential outcomes by ${\bar{Y}}_i(z;s)=\frac{1}{n_i}{\sum }_{j=1}^{n_i}{\mathbb{E}}_{i,s}[Y_{ij}({\mathbf{Z}}_i)\mid Z_{ij}=z],\bar{Y}(z;s)=\frac{1}{N}{\sum }_{i=1}^N{\bar{Y}}_i(z;s),$ for $z\in \{0,1\}$, and ${\bar{Y}}_i(s)=\frac{1}{n_i}{\sum }_{j=1}^{n_i}{\mathbb{E}}_{i,s}[Y_{ij}({\mathbf{Z}}_i)],\bar{Y}(s)=\frac{1}{N}{\sum }_{i=1}^N{\bar{Y}}_i(s)\mathrm{.}$ These are finite-population estimands: the outcomes are fixed, and the expectations average only over the specified within-group assignment strategy. Define $DE(\psi )=\bar{Y}(0;\psi )-\bar{Y}(1;\psi ),IE(\varphi ,\psi )=\bar{Y}(0;\varphi )-\bar{Y}(0;\psi ),$ $TE(\varphi ,\psi )=\bar{Y}(0;\varphi )-\bar{Y}(1;\psi ),OE(\varphi ,\psi )=\bar{Y}(\varphi )-\bar{Y}(\psi )\mathrm{.}$

For an observed design realization, put $m_{i1}^s=K_i^s$ and $m_{i0}^s=n_i-K_i^s$, and define, whenever $S_i=s$, ${\widehat{\bar{Y}}}_i(z;s)=\frac{{\sum }_{j=1}^{n_i}1\{Z_{ij}=z\}{Y}_{ij}^{\mathrm{o}\mathrm{b}\mathrm{s}}}{m_{iz}^s},{\widehat{\bar{Y}}}_i(s)=\frac{1}{n_i}{\sum }_{j=1}^{n_i}{Y}_{ij}^{\mathrm{o}\mathrm{b}\mathrm{s}}\mathrm{.}$ The denominators are the fixed, positive realized arm sizes. With $C_{\psi }=C$ and $C_{\varphi }=N-C$, define the observed population estimators $\widehat{\bar{Y}}(z;s)=\frac{1}{C_s}{\sum }_{i=1}^{N}1\{S_i=s\}{\widehat{\bar{Y}}}_i(z;s),\widehat{\bar{Y}}(s)=\frac{1}{C_s}{\sum }_{i=1}^{N}1\{S_i=s\}{\widehat{\bar{Y}}}_i(s)\mathrm{.}$ The four strategy-contrast estimators $\widehat{DE}(\psi )$, $\widehat{IE}(\varphi ,\psi )$, $\widehat{TE}(\varphi ,\psi )$, and $\widehat{OE}(\varphi ,\psi )$ are obtained by replacing each mean in the corresponding displayed target by its hatted estimator. No stratified-interference or additive-effects assumption is part of this design definition.