Absolute average and median treatment effects as causal estimands on metric spaces

TL;DR

This paper introduces new causal estimands, absolute average and median treatment effects, on metric spaces, along with estimators, confidence intervals, and applications demonstrating their effectiveness beyond Euclidean settings.

Contribution

It defines and estimates absolute average and median treatment effects on general metric spaces, extending causal inference tools to non-Euclidean geometries with proven consistency.

Findings

Estimators are strongly consistent in metric spaces.

Confidence intervals have reasonable asymptotic coverage.

Applied to real data, methods detect effects missed by Euclidean approaches.

Abstract

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.