Hyperbolic statistical inference for Treatment Effects with Circular biomarker of astigmatism

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 Poincaré disk, an open unit disk endowed with the Poincaré metric.Under this construction, each von Mises distribution is mapped uniquely to a point in the Poincaré 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.

💡 Research Summary

This paper introduces a novel two‑sample testing framework for circular biomedical data, motivated by angular measurements of surgically induced astigmatism after two cataract surgery techniques. The authors model each treatment group’s observations as von Mises random variables Θ ∼ VM(μ, κ), where μ is the mean direction and κ the concentration. Rather than comparing mean directions directly, the clinical goal is to assess how closely each treatment aligns with a pre‑specified target direction (μ₀, taken as 0°).

The methodological breakthrough is the embedding of the von Mises parameter pair (μ, κ) into the Poincaré disk D = {z∈ℂ:|z| < 1}, a model of two‑dimensional hyperbolic geometry. The bijective mapping ξ = r(κ) e^{iμ} with r(κ)=κ/(1+κ) sends κ = 0 (the uniform circular distribution) to the origin and κ → ∞ (highly concentrated distributions) to the boundary of the disk, while preserving the angular information in the complex argument. This representation captures both location and concentration in a single geometric object.

Within the hyperbolic space, the target direction μ₀ corresponds to the radius R_{μ₀} = {t e^{iμ₀}: 0 ≤ t < 1}. For any embedded point ξ, the distance to the target radius is defined as d_R(ξ) = min_{0≤t<1} d_H(ξ, t e^{iμ₀}), where d_H is the Poincaré distance
d_H(w₁,w₂)=cosh⁻¹