Equivalence of entropy balancing and the method of moments for matching-adjusted indirect comparison

Indirect comparisons are used to obtain estimates of relative effectiveness between two treatments that have not been compared in the same randomized controlled trial, but have instead been compared against a common comparator in separate trials. Standard indirect comparisons use only aggregate data, under the assumption that there are no differences in effect-modifying variables between the trial populations. Population-adjusted indirect comparisons aim to relax this assumption by using individual patient data (IPD) from one trial to adjust for differences in effect modifiers between populations. At present, the most commonly used approach is matching-adjusted indirect comparison (MAIC), where weights are estimated that match the covariate distributions of the reweighted IPD to the aggregate trial. MAIC was originally proposed using the method of moments to estimate the weights, but more recently entropy balancing has been proposed as an alternative. Entropy balancing has an additional "optimality" property ensuring that the weights are as uniform as possible, reducing the standard error of the estimates. In this brief method note, we show that MAIC weights are mathematically identical whether estimated using entropy balancing or the method of moments. Importantly, this means that the standard MAIC (based on the method of moments) also enjoys the "optimality" property. Moreover, the additional flexibility of entropy balancing suggests several interesting avenues for further research, such as combining population adjustment via MAIC with adjustments for treatment switching or nonparametric covariate adjustment.