Average treatment effect on the treated, under lack of positivity.

 The use of propensity score methods has become ubiquitous in causal inference. At the heart of these methods is the positivity assumption. Violation of the positivity assumption leads to the presence of extreme propensity scoreweights when estimating average causal effects, which affects statistical inference. To circumvent this issue, trimming or truncating methods have been widely used. Unfortunately, these methods require that we pre-specify a threshold. There are anumber of alternative methods to deal with the lack of positivity when we estimate the average treatment effect (ATE). However, no other methods exist beyond trimming and truncation to deal with the same issue when the goal is to estimate theaverage treatment effect on the treated (ATT). In this article, we propose a propensity score weight-based alternative for the ATT, called overlap weighted average treatment effect on the treated. The appeal of our proposed method lies in its abilityto obtain similar or even better results than trimming and truncation while relaxing the constraint to choose an a priori threshold (or related measures). The performance of the proposed method is illustrated via a series of Monte Carlo simulationsand a data analysis on racial disparities in health care expenditures.