The Average Treatment effect on the Treated is a formula in causal inference which calculates the effect of some treatment on the treated group.
$ATT = E[Y(A{=}1)-Y(A{=}0)| A{=}1]$
I don't understand how the notation of this expression relates to how it is calculated. For instance, given the sample data:
Y(A=1) | Y(A=0)
--------------------
10.0 |
20.0 |
30.0 |
| 1
| 2
| 3
The $ATT = 54=(10+20+30)-(1+2+3)$, where the values under $Y(A=1)$ correspond to the effect for a few samples with treatment $A=1$, and the values under $Y(A=0)$ correspond to the effect for a few samples with treatment $A=0$.
The expression $|A=1$ in the original expression makes me think $({=}1)−({=}0)$ should only be calculated over values in which $A=1$. To me, the following expression makes more sense:
$ATT = E[Y(A{=}1) | A{=}1 - Y(A{=}0)| A{=}0]$
or maybe even
$ATT = E[Y(A{=}1) | A{=}1] - E[Y(A{=}0)| A{=}0]$
If you interpret $E$ as the calculation of the average effect across a distribution. What am I misunderstanding about the relevant notation?