Estimation of causal effects in observational studies with interference between units

Abstract

Causal effects are usually estimated under the assumption of no interference between individuals. This assumption means that the potential outcomes for one individual are unaffected by the treatments received by other individuals. In many situations, this is not reasonable to assume. Moreover, not taking interference into account could result in misleading conclusions about the effect of a treatment. For two-stage observational studies, where treatment assigment is randomized in the first stage but not in the second stage, we propose IPW estimators of direct and indirect causal effects as defined by Hudgens and Halloran (J Am Stat Assoc 103(482):832–842, 2008) for two-stage randomized studies. We illustrate the use of these estimators in an evaluation study of an implementation of Triple P (a parenting support program) within preschools in Uppsala, Sweden.

Appendix: Unbiasedness of the estimators \(\widehat{\textit{CE}}_{i}^{D*}, \widehat{\textit{CE}}^{D*}, \widehat{\textit{CE}}^{I*}\) and \(\widehat{\textit{CE}}^{T*}\)

In the following, we show that the expected value of $\widehat{Y}_{i}^{*}(1;\psi )$ conditional on $S_{i}=1$ equals $\overline{Y}_{i}(1;\psi )$ and that the expected value of $\widehat{Y}^{*}(z;\psi )$ equals $\overline{Y}(z;\psi )$. From this and the definitions of $\overline{\textit{CE}}_{i}^{D}$, $\overline{\textit{CE}}^{D}$, $\overline{\textit{CE}}^{I}$ and $\overline{\textit{CE}}^{T},$ unbiasedness of $\widehat{\textit{CE}}_{i}^{D*}$, $\widehat{\textit{CE}}^{D*}$, $\widehat{\textit{CE}}^{I*}$ and $\widehat{\textit{CE}}^{T*}$follow. We define $R_{k}^{n}$ as the subset of all possible treatment assignment vectors $\varvec{\omega },$ with exactly $k$ of the $n$ individuals receiving treatment $z=1$.

$$\begin{aligned} E(\widehat{Y}_{i}^{*}(1;\psi )\mid S_{i}=1)&= \frac{1}{n_{i}}\sum _{j=1}^{n_{i}}\sum _{\varvec{\omega }\in R_{K_{i}-1}^{n_{i}-1}}Pr_{\psi }(\mathbf {Z}_{i(j)}=\varvec{\omega },Z_{ij}=1)\\&\times w_{ij}Y_{ij}(\mathbf {Z}_{i(j)}=\varvec{\omega },Z_{ij}=1)\\&= \frac{1}{n_{i}}\sum _{j=1}^{n_{i}}\sum _{\varvec{\omega }\in R_{K_{i}-1}^{n_{i}-1}}Pr_{\psi }(\mathbf {Z}_{i(j)}=\omega \mid Z_{ij}=1)Pr(Z_{ij}=1)\\&\times w_{ij}Y_{ij}(\mathbf {Z}_{i(j)}=\varvec{\omega },Z_{ij}=1)\\&= \frac{1}{n_{i}}\sum _{j=1}^{n_{i}}\sum _{\varvec{\omega }\in R_{K_{i}-1}^{n_{i}-1}}Pr_{\psi }(\mathbf {Z}_{i(j)}=\varvec{\omega }\mid Z_{ij}=1)\\&\times Y_{ij}(\mathbf {Z}_{i(j)}=\varvec{\omega },Z_{ij}=1)\\&= \overline{Y}_{i}(1;\psi ). \end{aligned}$$

In the same manner, it can be shown that $E(\widehat{Y}_{i}^{*}(0;\psi )\mid S_{i}=1)=\overline{Y}_{i}(0;\psi )$,

$$\begin{aligned} E(\widehat{Y}^{*}(z;\psi ))&= E\left( \frac{\sum _{i=1}^{N}\widehat{Y}_{i}^{*}(z;\psi )S_{i}}{\sum _{i=1}^{N}S_{i}}\right) =\frac{E\left( \sum _{i=1}^{N}\widehat{Y}_{i}^{*}(z;\psi )S_{i}\right) }{C}\\&= \frac{E\left( \sum _{i=1}^{N}\widehat{Y}_{i}^{*}(z;\psi )S_{i}|S_{i}=1\right) Pr(S_{i}=1)}{C}\\&+\frac{E\left( \sum _{i=1}^{N}\widehat{Y}_{i}^{*}(z;\psi )S_{i}|S_{i}=0\right) Pr(S_{i}=0)}{C}\\&= \frac{\sum _{i=1}^{N}\overline{Y}_{i}(z;\psi )\cdot \frac{C}{N}}{C}+0\\&= \frac{\sum _{i=1}^{N}\overline{Y}_{i}(z;\psi )}{N}=\overline{Y}(z;\psi ). \end{aligned}$$