Stratified interference and source design-based variance estimators

In the setting of Two-stage randomized design and strategy-specific effects, stratified interference means that, for each group $i$, unit $j$, and treatment count $k\in \{1,\dots ,n_i-1\}$, the value $Y_{ij}({\mathbf{z}}_i)$ is the same for any two assignments ${\mathbf{z}}_i,{\mathbf{z}}_i^{\mathrm{\prime }}$ with ${\sum }_{\mathrm{\ell }}{z}_{i\mathrm{\ell }}={\sum }_{\mathrm{\ell }}{z}_{i\mathrm{\ell }}^{\mathrm{\prime }}=k$ and $z_{ij}=z_{ij}^{\mathrm{\prime }}$. Thus at a fixed group treatment count an outcome may depend on the unit's own assignment, but not on which particular peers are treated. This assumption is additional to partial interference and is not needed for point-effect unbiasedness.

For strategy $s$ and own treatment $z$, write $Y_{ij}(z;s)$ for the common outcome at count $K_i^s$ and own assignment $z$. Define $S_{iz}^2(s)=\frac{1}{n_i-1}{\sum }_{j=1}^{n_i}\{Y_{ij}(z;s)-{\bar{Y}}_i(z;s){\}}^2,d_{ij}(s)=Y_{ij}(0;s)-Y_{ij}(1;s),$ $S_{i01}(s)=\frac{1}{n_i-1}{\sum }_{j=1}^{n_i}\{Y_{ij}(0;s)-{\bar{Y}}_i(0;s)\}\{Y_{ij}(1;s)-{\bar{Y}}_i(1;s)\},$ and $S_{iD}^2(s)=\frac{1}{n_i-1}{\sum }_{j=1}^{n_i}\{d_{ij}(s)-D_i(s){\}}^2,$ where $D_i(s)={\bar{Y}}_i(0;s)-{\bar{Y}}_i(1;s)$. In an observed group assigned $s$, define the estimated group direct effect by ${\widehat{D}}_i(s):={\widehat{\bar{Y}}}_i(0;s)-{\widehat{\bar{Y}}}_i(1;s)\mathrm{.}$ Let $s_{iz}^2(s)$ be the ordinary sample variance of the $m_{iz}^s$ observed outcomes in arm $z$, with denominator $m_{iz}^s-1$, and let $s_{D,s}^2$ be the sample variance of the observed group direct-effect estimates ${\widehat{D}}_i(s)$ across the $C_s$ groups assigned to $s$.

The source direct-effect variance estimator under strategy $s$ is ${\widehat{V}}_D(s)=(1-\frac{C_s}{N})\frac{s_{D,s}^2}{C_s}+\frac{1}{C_{s}N}{\sum }_{i:S_i=s}\{\frac{s_{i0}^2(s)}{n_i-K_i^s}+\frac{s_{i1}^2(s)}{K_i^s}\}\mathrm{.}$ For the indirect, total, and overall effect contrasts define ${\widehat{V}}_I(\varphi ,\psi )=\frac{s_{0,\varphi }^2}{N-C}+\frac{s_{0,\psi }^2}{C},{\widehat{V}}_T(\varphi ,\psi )=\frac{s_{0,\varphi }^2}{N-C}+\frac{s_{1,\psi }^2}{C},$ ${\widehat{V}}_O(\varphi ,\psi )=\frac{s_{\varphi }^2}{N-C}+\frac{s_{\psi }^2}{C},$ where $s_{z,s}^2$ is the sample variance across groups assigned $s$ of the observed group estimates ${\widehat{\bar{Y}}}_i(z;s)$, and $s_s^2$ is the sample variance across such groups of ${\widehat{\bar{Y}}}_i(s)$.

These are design-randomization variance estimators for the fixed finite-population point estimators. The group-arm sample variances in these formulas are defined only if $C\ge 2$ and $N-C\ge 2$. In addition, ${\widehat{V}}_D(s)$ requires $2\le K_i^s\le n_i-2$ for every group, so both within-group arm sample variances have nonzero denominators. The point estimators themselves remain defined under the broader counts $1\le K_i^s<n_i$.