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Reading Guo and Fraser's textbook, Propensity Score Analysis: Statistical Methods and Applications (2nd ed.), I came across the formula for the average treatment effect on the treated (ATT) on page 49:

$ATT=E[(Y_1-Y_0)|X,W=1].$

I interpret the ATT formula as the expected treatment outcome for the treated members ($Y_1$) minus the treatment outcome for the untreated members ($Y_0$) given covariates $X$ and given the member received the treatment ($W=1$).

Isn't this a contradictory formula? Perhaps the ATT is meant to measure the difference in the pre-treatment vs. post-treatment outcome ($Y_1-Y_0$) by member given $X$ in a longitudinal dataset? Doesn't this amount to a simple multiple regression?

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  • $\begingroup$ You should re-read the definitions of Y_1 and Y_0; in doing so the ATT may make more sense. Here Y_1 & Y_0 are potential outcomes for the same individual in the potential, possibly counter-to-fact situation in which the individual receives treatment (Y_1) and the counterfactual situation where the individual does not receive treatment (Y_0). These are two distinct potential outcomes, both of which happen after the event of possibly receiving treatment - neither is pre-treatment. The ATT is the expected difference in potential outcomes, stratif'd by X, among individuals who received treatment. $\endgroup$ – BarkleyBG Oct 4 '16 at 18:06
  • $\begingroup$ @BarkleyBG Hmm, I'm still confused. Is the $ATT$ an attempt at estimating the counterfactual outcome, $Y_0$, from a dataset that only contains individuals who received the treatment (hence $W=1$)? Otherwise we're just measuring the $ATE$, which formula makes sense to me since we're utilizing a control dataset. $\endgroup$ – RobertF Oct 4 '16 at 18:17
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    $\begingroup$ The formula you've listed for the ATT is a causal estimand. There are statistical methods to estimate the ATT (including some matching estimators). A researcher might be more interested in the average difference in potential outcomes for anyone in the population, but that's difficult to identify, whereas the ATT (the avg difference for individuals who obtained treatment) is easier to identify and estimate using matching estimators. If you estimate the ATT appropriately (matching each treated indiv to 1+ control indivs) then you can draw inference about the treated population $\endgroup$ – BarkleyBG Oct 4 '16 at 18:20
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    $\begingroup$ @BarkleyBG "If you estimate the ATT appropriately (matching each treated indiv to 1+ control indivs) then you can draw inference about the treated population" that makes sense - but isn't that the ATE? $\endgroup$ – RobertF Oct 4 '16 at 18:35
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    $\begingroup$ Estiamting ATT vs ATE depends on the matching method used. If there are 50 treated and 75 untreated individuals, then matching exactly one untreated individual to each treated individual will provide a 1-to-1 match on the treated and perhaps a good estimate of ATT. But that may/will be a poor estimate of ATE if only because there are 25 non-matched untreated individuals whose outcomes are not considered. However, if you choose 1-to-many matching in certain fashions then you could have better estimates of other estimands (like, say, ATE). $\endgroup$ – BarkleyBG Oct 4 '16 at 18:46
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Saying the propensity score matching estimated quantity is a counterfactual does not make it counterfactual.

According to Pearl, ATT is a counterfactual estimand. This means you cannot find it without knowing ATE (the causal effect) first. Pearl's way to find a counterfactuals is by finding ATE in a SCM setting, replacing the treatment with the alternative treatment (e.g. control) in the equations, and recalculating the outcome from the equations to find the counterfactual effect; i.e., you find the residuals from the causal estimation and use them in the counterfactual.

I think predicting ATT directly from the data using propensity score methods amounts to arbitrarily choosing controls that are not identical to the treatment units; they are just similar, and we do not know how similar. They are similar in the confounding causes, but not similar in the other and unknown causes of the outcome.

A safer method is to find ATE using propensity matching and then use it to find ATT in an independent step using SCM. To me this is more logical. See Pearl's Primer for more details.

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