> [!tldr] Bias-Causation Decomposition > In causal inference on the response $Y$ and treatment $T$, supposed $Y$ takes deterministic values $Y_{0,1}$ given $T=0,1$, all else held constant. > > The **bias-causation decomposition** of the association of $T,Y$ is $\begin{align*} &\underbrace{\mathbb{E}[Y~|~T=1]-\mathbb{E}[Y ~|~ T=0]}_{\text{association}} \\ &~~~= \underbrace{\mathbb{E}[Y_{1}-Y_{0}~|~T=1]}_{\text{causation, ATT}}\\ &~~~+ \underbrace{\mathbb{E}[Y_{0}~|~T=1]-\mathbb{E}[Y_{0} ~|~T=0]}_{\text{bias}}, \end{align*}$where $\mathrm{ATT}$ is **average treatment on treated**. > > With special setups like random assignment of treatment, the bias is $0$ because we can assume $T$ and $(Y_{0},Y_{1})$ being independent. > > Furthermore, we can assume $\mathrm{ATT}=\mathrm{ATE}$ (average treatment effect), as the treated group is representative of the entire population. - Note that $T,(Y_{0},Y_{1})$ being independent (i.e. the assignment of treatment is independent of the potential outcomes) does not mean $T,Y$ are independent (i.e. the treatment has no effect). A real-life example would be measuring the grades $Y$ of a school's pupils, where schools with $T=1$ are (observed to be) handing out tablets during class. - It may have a positive association, even if the causation is negative. - This can be explained by a positive bias -- schools that are handing out tablets are more likely to be rich, granting them more educational resources and/or better applicant pools for admission.