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Reasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyMachine Learning: Calibration & Uncertainty QuantificationKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
Hyperbolic statistical inference for Treatment Effects with Circular biomarker of astigmatism
🤖 AI Summary
This study addresses the challenge of comparing treatment effects on circular biomarkers, such as corneal astigmatism, whose directional data structure is poorly handled by conventional methods. The authors propose a novel two-sample hypothesis testing framework that embeds parameters of von Mises–distributed circular data into the Poincaré disk, leveraging hyperbolic geometry to construct an interpretable test statistic. Permutation and bootstrap procedures are employed to accommodate scenarios with equal or unequal concentration parameters, respectively. This work represents the first application of hyperbolic geometry to inference on circular biomarkers, preserving the intrinsic structure of directional data. Simulations demonstrate that the method achieves accurate type I error control, strong consistency, and superior asymptotic power compared to existing approaches. Its clinical utility is further validated through real-world data from cataract surgery outcomes.
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Application Category
📝 Abstract
Circular biomarkers arise naturally in many biomedical applications, particularly in ophthalmology, where angular measurements such as astigmatism are routinely recorded. Similar directional variables also occur in the study of human body rotations, including movements of the hand, waist, neck, and lower limbs. Motivated by a clinical dataset comprising angular measurements of astigmatism induced by two cataract surgery procedures, we propose a novel two-sample testing framework for circular data grounded in hyperbolic geometry. Assuming von Mises distributions with either common or group-specific concentration parameters, we embed the corresponding parameter spaces into the Poincar\'e disk, an open unit disk endowed with the Poincar\'e metric.Under this construction, each von Mises distribution is mapped uniquely to a point in the Poincar\'e disk, yielding a continuous geometric representation that preserves the intrinsic structure of the parameter space. This embedding enables direct comparison of group distributions via hyperbolic distances, leading to natural and interpretable test statistics. We develop permutation-based tests for the common concentration case and bootstrap-based procedures for unequal concentrations. Extensive simulation studies demonstrate stable empirical size, strong consistency, and superior asymptotic power compared with existing competing methods. The proposed methodology is illustrated through a detailed analysis of the cataract surgery dataset, including a clinically informed restructuring of the original observations. The results highlight the practical advantages of incorporating hyperbolic geometry into the analysis of circular biomedical data and underscore the potential of geometry-aware inference for directional biomarkers.
Problem
Research questions and friction points this paper is trying to address.
circular biomarker
treatment effect
astigmatism
statistical inference
directional data
Innovation
Methods, ideas, or system contributions that make the work stand out.
hyperbolic geometry
circular data
von Mises distribution
Poincaré disk
treatment effect inference
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