This paper proposes a~simple, yet powerful, method for balancing distributions of covariates for causal inference based on observational studies. The method makes it possible to balance an arbitrary number of quantiles (e.g., medians, quartiles, or deciles) together with means if necessary. The proposed approach is based on the theory of calibration estimators (Deville and S\"arndal 1992), in particular, calibration estimators for quantiles, proposed by Harms and Duchesne (2006). The method does not require numerical integration, kernel density estimation or assumptions about the distributions. Valid estimates can be obtained by drawing on existing asymptotic theory. An~illustrative example of the proposed approach is presented for the entropy balancing method and the covariate balancing propensity score method. Results of a~simulation study indicate that the method efficiently estimates average treatment effects on the treated (ATT), the average treatment effect (ATE), the quantile treatment effect on the treated (QTT) and the quantile treatment effect (QTE), especially in the presence of non-linearity and mis-specification of the models. The proposed approach can be further generalized to other designs (e.g. multi-category, continuous) or methods (e.g. synthetic control method). An open source software implementing proposed methods is available.

翻译：本文提出了一种简单而有效的方法，用于在基于观察性研究的因果推断中平衡协变量的分布。该方法能够在必要时平衡任意数量的分位数（例如中位数、四分位数或十分位数）以及均值。所提出的方法基于校准估计理论（Deville and Särndal 1992），特别采用了Harms和Duchesne（2006）提出的分位数校准估计量。该方法无需数值积分、核密度估计或关于分布的假设，可借助现有渐近理论获得有效估计。本文以熵平衡方法和协变量平衡倾向得分方法为例展示了该方法的实用性。模拟研究结果表明，该方法能高效估计处理组平均处理效应（ATT）、平均处理效应（ATE）、处理组分位数处理效应（QTT）以及分位数处理效应（QTE），尤其在模型存在非线性和设定错误的情况下表现优异。该框架可进一步推广至其他设计（如多类别、连续型）或方法（如合成控制法）。本文同时提供了实现所提方法的开源软件。