Propensity score: which treatment effect is easier to infer? I'm currently working on a study where the goal is to estimate the treatment effect of a binary exposure. I want to calculate the Average Treatment Effect (ATE), Average Treatment Effect in the Treated (ATT) and the Average Treatment Effect in the Controls (ATC). For the dataset, I have plotted histograms of the propensity scores for the treated and untreated groups. Now my question is: from this histogram, what could be said about which effect is 'easier' to infer? Say that for treated and untreated, the histogram shows very high propensity scores, would ATC then be easier to infer, since for every untreated sample we could also 'consider it as treated' because of its high propensity score?
 A: The treatment effect for the group about which there is the most information about the causal effect (i.e., for the which the uncertainty of the causal effect is the smallest) is called the average treatment effect in the overlap (ATO), and is described in Li, Morgan, and Zaslavsky (2018). The traditional estimand (i.e., ATT, ATC, or ATE) for which you will have the most certainty is the one for which the covariate distribution of the relevant group is most similar to that of the ATO sample. 
In general, the most certainty is in the area with the most overlap. If there are many control units with high propensity scores, then the ATT will have higher precision. If there are many treated units with low propensity scores, then the ATC will have higher precision. If both groups have a wide propensity score range and there is a lot of overlap, the ATE will have good precision. It sounds like in your case the ATT will be the most precise. 
If you are using propensity score weighting, you should check the effective sample size resulting from the weights for each estimand. The WeightIt package makes it easy to estimate weights and assess their properties, including the effective sample size. The weights with the largest effective sample size are those that should yield the best precision in the effect estimates.

Li, F., Morgan, K. L., & Zaslavsky, A. M. (2018). Balancing Covariates via Propensity Score Weighting. Journal of the American Statistical Association, 113(521), 390–400. https://doi.org/10.1080/01621459.2016.1260466
