We define the notions of absolute average and median treatment effects as causal estimands on general metric spaces such as Riemannian manifolds, propose estimators using stratification, and prove several properties, including strong consistency. In the process, we also demonstrate the strong consistency of the weighted sample Fr\'echet means and geometric medians. Stratification allows these estimators to be utilized beyond the narrow constraints of a completely randomized experiment. After constructing confidence intervals using bootstrapping, we outline how to use the proposed estimates to test Fisher's sharp null hypothesis that the absolute average or median treatment effect is zero. Empirical evidence for the strong consistency of the estimators and the reasonable asymptotic coverage of the confidence intervals is provided through simulations in both randomized experiments and observational study settings. We also apply our methods to real data from an observational study to investigate the causal relationship between Alzheimer's disease and the shape of the corpus callosum, rejecting the aforementioned null hypotheses in cases where conventional Euclidean methods fail to do so. Our proposed methods are more generally applicable than past studies in dealing with general metric spaces.

翻译：本文在黎曼流形等一般度量空间上定义了绝对平均处理效应与中位数处理效应的概念，提出了基于分层法的估计量，并证明了包括强一致性在内的若干性质。在此过程中，我们还证明了加权样本Fr\'echet均值与几何中位数的强一致性。分层法使得这些估计量能够超越完全随机实验的严格限制。通过自助法构建置信区间后，我们阐述了如何利用所提出的估计量来检验费希尔尖锐零假设（即绝对平均或中位数处理效应为零）。通过在随机实验与观察性研究设定中的模拟实验，我们为估计量的强一致性及置信区间的合理渐近覆盖提供了实证依据。我们还将所提方法应用于观察性研究的真实数据，以探究阿尔茨海默病与胼胝体形态之间的因果关系，在传统欧几里得方法无法拒绝零假设的情况下，我们的方法成功拒绝了上述零假设。相较于以往研究，本文提出的方法在处理一般度量空间问题时具有更广泛的适用性。