# How can you express the average treatment effect on the treated (ATT) in Pearl's do notation?

How can you express the average treatment effect on the treated (ATT) in Pearl's do notation?

Would it be $$E(Y|X=1,do(X)=1)-E(Y|X=1,do(X)=0)$$?

You cannot express it in "do" notation! The conditioning event $$X = 1$$ would conflict with $$do(X = 0)$$.

Indeed, the ATT is an example of a counterfactual/"rung 3" causal query that is strictly "deeper" than just an interventional "do"-query. See

Pearl, Judea. "The seven tools of causal inference, with reflections on machine learning." Communications of the ACM 62.3 (2019): 54-60.

As Julian and Noah have pointed out, the $$\operatorname{do}$$ notation is not appropriate for expressing the $$\operatorname{ATT}$$ (or $$\operatorname{ETT}$$ - the Effect of Treatment on the Treated - as Pearl calls it). Pearl expresses the $$\operatorname{ETT}$$ as this: $$\operatorname{ETT}:=E[Y_1-Y_0|X=1].$$ The notation $$Y_1$$ and $$Y_0$$ means, respectively, the (possibly counterfactual) values of $$Y$$ had $$X=1$$ or $$0.$$ So, while the $$\operatorname{do}$$ framework is too crude to express the $$\operatorname{ETT},$$ counterfactuals suffice.