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

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

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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.

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  • $\begingroup$ I had no idea about this. It's interesting because the ATT requires fewer assumptions to estimate than the ATE. $\endgroup$
    – Noah
    Sep 27, 2021 at 13:32
  • $\begingroup$ I thought this "The conditioning event X=1 would conflict with do(X=0)." too, but wanted to confirm. So this then begs the question of how to interpret E(Y|do(X)=1) . I do not buy the explanation it means 'force X to be 1'. You cannot force a mathematical object to do anything, so who are we forcing? A subject? A group of subjects? And then how were these subjects selected? It has to be a sample from entire population, in which case it is really a weighted average of do X=1 among subjects who have X=1 and among subjects who have X=0. $\endgroup$ Sep 27, 2021 at 20:50
  • $\begingroup$ This sounds like a separate question you could ask on SO. The do-operator is defined as an operation on a structural causal model, where you change model so that X = f_x(...) becomes X = x. This specifically would be a population-wide operation (everyone will have X = x). There are then no "subjects who have X = 0". The most simple model does not "contain time"; setting X = x is instantaneous. $\endgroup$ Sep 29, 2021 at 7:47
  • $\begingroup$ See also this question: stats.stackexchange.com/questions/529899/… $\endgroup$ Sep 29, 2021 at 7:48
  • $\begingroup$ @stataphobia The idea of forcing a variable to be equal to something is commonplace: this is precisely what is done in an experiment: you control the factors to be certain values in order to eliminate unwanted variation. To answer all your questions about this would require studying experimental design - a course in itself. $\endgroup$ Oct 20, 2021 at 16:14
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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.

For reference, see Causal Inference in Statistics: A Primer, p. 106.

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