Daisuke Kurisu, Hans-Georg Mueller, Taisuke Otsu and Yidong Zhou

Published 23 June 2025

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Regression discontinuity designs, quasi-experimental designs for observational studies to estimate causal effects of an intervention or treatment at a cutoff point, have been widely applied in various fields of data science, such as economics, education, environmental studies, epidemiology, and political science. We propose an extension of regression discontinuity designs, where we extend the customary scenario of scalar responses and aim at causal inference for a general class of complex non-Euclidean outcomes. Specifically, the outcomes considered extend beyond scalars or vectors and include networks, compositional data and functional data, as well as other types of random objects situated in geodesic metric spaces that may be subject to causal effects. For this extension a major challenge is how to quantify a treatment effect, since algebraic operations are not available and taking differences is thus not feasible. To overcome this challenge, we express the causal effect at the cutoff point as a geodesic from the local Fr ́echet mean of the untreated outcome to the treated outcome, which reduces to the ordinary average treatment effect and thus to the well-established approach for regression discontinuity designs in the special case of scalar or vector outcomes. Our estimation method is based on local Fr ́echet regression, a regression method for non-Euclidean responses that corresponds to local linear regression for the special case of scalar responses. We equip local Fr ́echet regression with a novel bandwidth selection that is specifically aimed at regression discontinuity designs. The proposed bandwidth selector is shown to be competitive even in the well-established case of regression discontinuity designs for scalar outcomes. The proposed approach is supported by theory and we establish the convergence rate of the geodesic regression discontinuity design estimator. We demonstrate the relevance of this general approach for specific practical applications, where one seeks to assess causal effects in the framework of policy interventions and natural experiments that give rise to regression discontinuity designs. These applications include an assessment of the changes caused in daily CO concentration curves by the introduction of a metro system in Taipei and in UK voting behavior as quantified by compositional data after wins by the conservative party. We also develop an extension for fuzzy regression discontinuity designs with non-Euclidean outcomes, broadening the scope of causal inference to settings that allow for imperfect compliance with the assignment rule.