在这篇文章中，你将会了解到：什么是bias，为什么我们在研究中存在bias，以及如何消除bias。
We sometimes call the potential outcome that happened, factual, and the one that didn’t happen, counterfactual.
$Y_{0i}$ is the potential outcome for unit i without treatment, it can also be written as $Y_{i}(0)$
$Y_{1i}$ is the potential outcome for unit i with treatment, it can also be written as $Y_{i}(1)$
we define the individual effect as $Y_{1i}-Y_{0i}$ , which can not be accumulated for the counterfactual. so we can only accumulate Average treatment effect which is also denoted as ATE:
$$ATE = E[Y_{1}-Y_{0}]$$
we can also estimate Average treatment effect on the treated which is also called ATT
$$ATT=E[Y_{1}-Y_{0}|T=1]$$

Bias

$$\begin{aligned}
E[Y|T=1]-E[Y|T=0]&=E[Y_{1}|T=1]-E[Y_{0}|T=0]
\\&=E[Y_{1}|T=1]-E[Y_{0}|T=0] + E[Y_{0}|T=1] - E[Y_{0}|T=1]
\\&=E[Y_{1}-Y_{0}|T=1]+E[Y_{0}|T=1]-E[Y_{0}|T=0]
\end{aligned}$$
以上公式中，前者为ATT也就是我们所希望估计的, 后者为Bias
也就是说相关性=ATT+Bias。
if $E[Y_{0}|T=1]=E[Y_{0}|T=0]$
then assosiation = causation
furthermore, if we assume that the untreated and the treated only differ in the treatment itself, we can conclude that:
$$E[Y|T=1]-E[Y|T=0]=E[Y_{1}-Y_{0}|T=0]=E[Y_{1}-Y_{0}|T=1]$$
which means ATT=ATE
To eliminate bias, we should Randomized Trial

Thanks for watching! and this my learning note of the blog of Matheus Facure Alves.
感谢观看，这是我学习Matheus Facure Alves博客的笔记。
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