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:
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?