Introduction
Causal inference has two frameworks. One of them is the Pearlian framework and the other is the potential outcome framework. In this article, some work related to the potential outcome framework will be done. The average treatment effect (ATE) will be decomposed into the average treatment effect for the treated (ATT) and the average treatment effect for the untreated (ATU). The symbols used in the article are like the ones in [1].
Decomposition
The average treatment effect is given by the following: \begin{equation} \text{ATE} = E\left[\delta_{i}\right] \end{equation} $\delta_{i}$ is the unit specific causal effect. It is defined as follows: \begin{equation} \delta_{i} = Y_{i}^{1} - Y_{i}^{0} \end{equation} $Y_{i}^{1}$ is the response of the unit $i$ to the treatment. $Y_{i}^{0}$ is the response of the unit $i$ to the untreatment. If a unit $i$ is treated, then this is represented as \begin{equation} D_{i}=1 \end{equation} On the other hand, if a unit $i$ is untreated or in the control group, then this is denoted by the following: \begin{equation} D_{i}=0 \end{equation} Let the average treatment effect be written in a more detailed way: \begin{gather} \text{ATE} = E\left[Y_{i}^{1}-Y_{i}^{0}\right]=E\left[Y_{i}^{1}\right]- E\left[Y_{i}^{0}\right]=\sum_{i} y_{i}^{1} P\left(y_{i}^{1}\right)- \sum_{i}y_{i}^{0}P\left(y_{i}^{0}\right)= \newline \sum_{i}y_{i}^{1} \left(P\left(y_{i}^{1}, D_{i}=0\right)+P\left(y_{i}^{1}, D_{i}=1\right)\right) - \sum_{i} y_{i}^{0} \left(P\left(y_{i}^{0}, D_{i}=0\right)+P\left(y_{i}^{0}, D_{i}=1\right)\right)= \newline \sum_{i} y_{i}^{1}P\left(y_{i}^{1}, D_{i}=0\right) + \sum_{i} y_{i}^{1}P\left(y_{i}^{1}, D_{i}=1\right) - \sum_{i} y_{i}^{0}P\left(y_{i}^{0}, D_{i}=0\right) - \sum_{i} y_{i}^{0}P\left(y_{i}^{0}, D_{i}=1\right)= \newline \sum_{i}y_{i}^{1} P\left(y_{i}^{1}|D_{i}=0\right) P\left(D_{i}=0\right) + \sum_{i}y_{i}^{1} P\left(y_{i}^{1}|D_{i}=1\right) P\left(D_{i}=1\right) - \sum_{i}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=0\right) P\left(D_{i}=0\right) - \sum_{i}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=1\right) P\left(D_{i}=1\right) \label{eqn1} \end{gather} Each addend in the sum in the equation \eqref{eqn1} is expanded into terms involving the units in the treated group and in the untreated (control) group. Let each expansion be made as follows: \begin{gather} \sum_{i}y_{i}^{1} P\left(y_{i}^{1}|D_{i}=0\right) P\left(D_{i}=0\right) = \sum_{\text{treated}}y_{i}^{1} P\left(y_{i}^{1}|D_{i}=0\right) \underbrace{P\left(D_{i}=0\right)}_{0}+ \sum_{\text{untreated}}y_{i}^{1} P\left(y_{i}^{1}|D_{i}=0\right) \underbrace{P\left(D_{i}=0\right)}_{1-p}= \left(1-p\right)\sum_{\text{untreated}}y_{i}^{1} P\left(y_{i}^{1}|D_{i}=0\right) \newline \sum_{i}y_{i}^{1} P\left(y_{i}^{1}|D_{i}=1\right) P\left(D_{i}=1\right)= \sum_{\text{treated}} y_{i}^{1} P\left(y_{i}^{1}|D_{i}=1\right) \underbrace{P\left(D_{i}=1\right)}_{p}+ \sum_{\text{untreated}}y_{i}^{1} P\left(y_{i}^{1}|D_{i}=1\right) \underbrace{P\left(D_{i}=1\right)}_{0}=p \sum_{\text{treated}} y_{i}^{1} P\left(y_{i}^{1}|D_{i}=1\right) \newline \sum_{i}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=0\right) P\left(D_{i}=0\right)= \sum_{\text{treated}}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=0\right) \underbrace{P\left(D_{i}=0\right)}_{0}+ \sum_{\text{untreated}}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=0\right) \underbrace{P\left(D_{i}=0\right)}_{1-p}=\left(1-p\right) \sum_{\text{untreated}}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=0\right) \newline \sum_{i}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=1\right) P\left(D_{i}=1\right) = \sum_{\text{treated}}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=1\right) \underbrace{P\left(D_{i}=1\right)}_{p} + \sum_{\text{untreated}}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=1\right) \underbrace{P\left(D_{i}=1\right)}_{0}=p \sum_{\text{treated}}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=1\right) \end{gather} Let each expansion result be put into the equation \eqref{eqn1}: \begin{gather} \text{ATE}= \left(1-p\right) \sum_{\text{untreated}} y_{i}^{1} P\left(y_{i}^{1}|D_{i}=0\right) + p \sum_{\text{treated}} y_{i}^{1} P\left(y_{i}^{1}|D_{i}=1\right) - \left(1-p\right) \sum_{\text{untreated}}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=0\right) - p \sum_{\text{treated}}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=1\right) \implies \newline \text{ATE} = p \left(\sum_{\text{treated}} y_{i}^{1} P\left(y_{i}^{1}|D_{i}=1\right) - \sum_{\text{treated}}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=1\right)\right) + \left(1-p\right)\left(\sum_{\text{untreated}} y_{i}^{1} P\left(y_{i}^{1}|D_{i}=0\right)-\sum_{\text{untreated}}y_{i}^{0} P\left(y_{i}^{0}|D_{i}=0\right)\right) \implies \newline \text{ATE}=p\left(\text{ATT}\right)+\left(1-p\right)\left(\text{ATU}\right) \end{gather} As a result, the decomposition of the average treatment effect into the average treatment effect for the treated and the average treatment effect for the untreated has been completed.
Conclusion
The decomposition of the average treatment effect into the average treatment effect for the treated and the average treatment effect for the untreated has been performed.