In many scientific fields such as biology, psychology and sociology, there is an increasing interest in estimating the causal effect of a matrix exposure on an outcome. Covariate balancing is crucial in causal inference and both exact balancing and approximate balancing methods have been proposed in the past decades. However, due to the large number of constraints, it is difficult to achieve exact balance or to select the threshold parameters for approximate balancing methods when the treatment is a matrix. To meet these challenges, we propose the weighted Euclidean balancing method, which approximately balance covariates from an overall perspective. This method is also applicable to high-dimensional covariates scenario. Both parametric and nonparametric methods are proposed to estimate the causal effect of matrix treatment and theoretical properties of the two estimations are provided. Furthermore, the simulation results show that the proposed method outperforms other methods in various cases. Finally, the method is applied to investigating the causal relationship between children's participation in various training courses and their IQ. The results show that the duration of attending hands-on practice courses for children at 6-9 years old has a siginificantly positive impact on children's IQ.

翻译：在生物学、心理学和社会学等许多科学领域，人们对估计矩阵暴露对结果的因果效应日益关注。协变量平衡在因果推断中至关重要，过去几十年中已提出精确平衡和近似平衡两类方法。然而，当处理变量为矩阵时，由于约束数量庞大，难以实现精确平衡或为近似平衡方法选择阈值参数。为应对这些挑战，我们提出加权欧几里得平衡方法，该方法从全局角度近似平衡协变量，并适用于高维协变量场景。我们分别提出参数和非参数方法估计矩阵处理的因果效应，并给出了两种估计的理论性质。此外，模拟结果表明，所提方法在多种情形下优于其他方法。最后，将该方法应用于探究儿童参与各类培训课程与其智商之间的因果关系。结果表明，6-9岁儿童参加动手实践课程的时长对其智商具有显著正向影响。