Something fun (Słoczyński et al., 2025, p. 3, footnote 1):
“IPT [inverse probability tilting] coincides with Hainmueller’s (2012) entropy balancing estimator when the propensity score is estimated with the logit model”.
(It just holds for ATT.) The picture below illustrates the correspondence, using the Lalonde (1986) dataset (in the cobalt package in R). I computed (knitted R here) the weights using the WeightIt package (Greifer, 2025). Since we’re aiming for ATT, the intervention group weights are all equal to 1, so I’ve omitted them from the graph.
The weights are perfectly correlated and the effective sample sizes are the same. They are scaled differently, though (in the WeightIt implementation at least – need to wade into the original papers): entropy weights sum to the comparison group sample size and IPT weights sum to the intervention group sample size.
References
Greifer, N. (2025). WeightIt: Weighting for Covariate Balance in Observational Studies (version 1.5.0) [R Package].
Słoczyński, T., Uysal, S. D., & Wooldridge, J. M. (2025). Covariate Balancing and the Equivalence of Weighting and Doubly Robust Estimators of Average Treatment Effects. IZA Institute of Labour Economics Discussion Paper, 18147.