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The goals of this paper were to provide a set of conditions for inverse probability weights (IPW) to target a subpopulation of the patients who have clinical equipose, establish a relationship between matching weights and overlap weight estimators, and use the beta distribution to help determine the appropriate weights.  Whenever distributions of covariates between two treatment groups were very different, this can result in limited overlap of distribution of the propensity scores and violations of the positivity assumptions which they called lack of adequate positivity.  Difference propensity score methods have shown to be affected differently by insufficient overlap of distribution of the propensity scores or by the lack of the adequate positivity.

They discussed per creation of propensity score weights, creating balancing weight estimators. The authors then went into an incredible amount of detail regarding their estimators extending into all possible realms until they finally got to their simulations section.   They found that overall, OW (overlap weight), MW (matching weight), and EW (entropy weight) were more efficient than IPW, despite the fact that the latter is doubly robust. Out of the three, OW, MW, and EW, the OW estimator had the best RMSE in most of the scenarios we considered under the homogeneous treatment effect, while MW is the best under the heterogeneous treatment effect.

They demonstrated that the beta family can also approximate the matching weights and entropy weights very well. This is for those who desire to expand their statistical toolsets as the equipoise estimators (OW, MW, EW, and BW) to explore complex studies where the lack of adequate positivity is not just by happenstance. Although they mainly focused on continuous outcomes, the methods presented in this paper can be extended to other types of outcomes and can easily be implemented using the PSweight R-package (Zhou, Tong, et al., 2020).


Written by,


Usha Govindarajulu


Keywords:  causal inference, positivity, IPW, balancing weights



Matsouaka RA and Zhou Y (2024). “Causal inference in the absence of positivity: The role of overlap weights.” Biometrical Journal.

Zhou, T., Tong, G., Li, F., & Thomas, L. E. (2020). Psweight: An r package for propensity score weighting analysis. arXiv.