  The use of propensity score (PS) 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 PS
weights when estimating average causal effects of interest, such as the average
treatment effect (ATE) or the average treatment effect on the treated (ATT),
which renders invalid related statistical inference. To circumvent this issue,
trimming or truncating the extreme estimated PSs have been widely used.
However, these methods require that we specify a priori a threshold and
sometimes an additional smoothing parameter. While there are a number of
methods dealing with the lack of positivity when estimating ATE, surprisingly
there is no much effort in the same issue for ATT. In this paper, we first
review widely used methods, such as trimming and truncation in ATT. We
emphasize the underlying intuition behind these methods to better understand
their applications and highlight their main limitations. Then, we argue that
the current methods simply target estimands that are scaled ATT (and thus move
the goalpost to a different target of interest), where we specify the scale and
the target populations. We further propose a PS weight-based alternative for
the average causal effect on the treated, called overlap weighted average
treatment effect on the treated (OWATT). The appeal of our proposed method lies
in its ability to obtain similar or even better results than trimming and
truncation while relaxing the constraint to choose a priori a threshold (or
even specify a smoothing parameter). The performance of the proposed method is
illustrated via a series of Monte Carlo simulations and a data analysis on
racial disparities in health care expenditures.
