Shape-Preserving Treatment Effects for Functional Outcomes
By recording health measurements daily rather than at a few clinical visits, connected devices pave the way for more precise real-world evidence that measures how the effect of an intervention unfolds over time: does the response rebound, peak late, or oscillate? Yet causal methods for such functional outcomes remain scarce, and those that exist target the Functional Average Treatment Effect (FATE). Its pointwise averaging flattens the features patients share whenever their curves are misaligned in time, as is common with biosignals, where patients respond with their own timing. We develop a causal inference framework that captures the shape of the effect of a binary intervention, based on the phase-centered elastic mean from elastic shape analysis. We lift this mean from a sample statistic to a functional of the data-generating law and prove that it is well defined under explicit geometric assumptions. Because it is not linear, it yields three causal estimands instead of one: populational, individual and local. We identify the populational estimand by inverse-probability weighting and the local estimand through the conditional outcome laws, estimate them by weighting and by matching, and show that the individual one is not identifiable. On simulations with timing variability or heterogeneous effect shapes, our estimators recover the shape that the FATE loses. On real weight trajectories following GLP-1 receptor agonist initiation, they reveal a rebound that the FATE averages away. This framework thus gives clinicians a treatment-effect curve whose shape and timing reflect genuine patient trajectories rather than a misalignment artifact.