- [[unbiased estimator]], [[conditional average treatment effect]], [[covariate adjustment]] # Idea The average treatment effect (ATE) is also known as the average causal effect (ACE). The letter $D$ is often used to indicate treatment status: $ D_{i}= \begin{cases}1 & \text { if } i \text { is insured } \\ 0 & \text { otherwise }\end{cases} $ For each subject or unit $i$, the causal effect of the treatment $\tau_i$ is defined as the difference between two [[potential outcomes]]: $ \tau_{i} \equiv Y_{i}(1)-Y_{i}(0) $ # ATE $ \mathrm{ATE} \equiv \frac{1}{N} \sum_{i=1}^{N} \tau_{i} $ $ A T E=\mathbb{E}\left[Y_{1}-Y_{0}\right]=\mathbb{E}_{x \sim p(x)}[\operatorname{CATE}(x)] $ Average treatment effect is the expected effect of treatment $T$ on $Y$: $ATE:=\mathbb{E}\left[Y_{1}-Y_{0}\right]$ If we assume [[ignorability assumption|ignorability]], we can derive/use the [[adjustment formula]]. # Averages conditional on treatment condition Average among treatment subjects: $ A v g_{n}\left[Y_{i} \mid D_{i}=1\right] $ Average among control subjects: $ A v g_{n}\left[Y_{i} \mid D_{i}=0\right] $ These quantities are averages conditional on insurance status. # References - [[Gerber 2012 field experiments]]