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

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


2 Answers 2


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.

  • $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.