Functional data analysis, which handles data arising from curves, surfaces, volumes, manifolds and beyond in a variety of scientific fields, is a rapidly developing area in modern statistics and data science in the recent decades. The effect of a functional variable on an outcome is an essential theme in functional data analysis, but a majority of related studies are restricted to correlational effects rather than causal effects. As the first attempt in the literature, the causal effect is studied for a functional variable as a treatment in cross-sectional observational studies. Despite the lack of a probability density function for the functional treatment, the propensity score is properly defined in terms of its top functional principal component scores which can represent the functional treatment approximately. Two covariate balancing methods are proposed to estimate the propensity score, which minimize the correlation between the treatment and covariates. The appealing performance of the proposed method in both covariate balance and causal effect estimation is demonstrated by a simulation study. The proposed method is applied to study the causal effect of body shape on human visceral adipose tissue.

功能数据分析处理来自各种科学领域的曲线、曲面、体积、流形等数据，是近几十年来现代统计学和数据科学中快速发展的领域。功能变量对结果的影响是功能数据分析的一个基本主题，但大多数相关研究仅限于相关效应而不是因果效应。作为文献中的第一次尝试，在横断面观察性研究中研究了作为治疗的功能变量的因果效应。尽管缺乏功能性治疗的概率密度函数，但倾向得分是根据其最高功能主成分得分正确定义的，该得分可以近似代表功能性治疗。提出了两种协变量平衡方法来估计倾向评分，从而最小化治疗和协变量之间的相关性。模拟研究证明了所提出的方法在协变量平衡和因果效应估计方面的吸引人的性能。所提出的方法用于研究体型对人体内脏脂肪组织的因果影响。