Overlap, matching, or entropy weights: what are we weighting for?

There has been a recent surge in statistical methods for handling the lack of adequate positivity when using inverse probability weighting. Alongside these nascent developments, a number of questions have been posed about the goals and intent of these methods: to infer causality, what are they really estimating and what are their target populations? Because causal inference is inherently a missing data problem, the assignment mechanism -- how participants are represented in their respective treatment groups and how they receive their treatments -- plays an important role in assessing causality. In this paper, we contribute to the discussion by highlighting specific characteristics of the equipoise estimators, i.e., overlap weights (OW) matching weights (MW) and entropy weights (EW) methods, which help answer these questions and contrast them with the behavior of the inverse probability weights (IPW) method. We discuss three distinct potential motives for weighting under the lack of adequate positivity when estimating causal effects: (1) What separates OW, MW, and EW from IPW trimming or truncation? (2) What fundamentally distinguishes the estimand of the IPW, i.e., average treatment effect (ATE) from the OW, MW, and EW estimands (resp. average treatment effect on the overlap (ATO), the matching (ATM), and entropy (ATEN))? (3) When should we expect similar results for these estimands, even if the treatment effect is heterogeneous? Our findings are illustrated through a number of Monte-Carlo simulation studies and a data example on healthcare expenditure.