Geodesic Causal Inference

Adjusting for confounding and imbalance when establishing statistical relationships is an increasingly important task, and causal inference methods have emerged as the most popular tool to achieve this. Existing methodology has been developed primarily for outcomes that lie in Euclidean spaces. We introduce here a general framework for causal inference when outcomes reside in general geodesic metric spaces, where we draw on a novel geodesic calculus that facilitates scalar multiplication for geodesics and the quantification of treatment effects through the concept of geodesic average treatment effect. Using ideas from Fr'echet regression, we obtain a doubly robust estimation of the geodesic average treatment effect and results on consistency and rates of convergence for the proposed estimators. We also develop an intrinsic uncertainty quantification framework for the treatment effect based on Fr'echet objective functions. The proposed framework is illustrated through simulations and real data applications, including network-valued outcomes from New York City taxi trips to assess the impact of the COVID-19 pandemic, and compositional data on U.S. state-level energy sources to study the effect of coal mining.