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.

翻译：圆形生物标志物在众多生物医学应用中自然产生，尤其在眼科学领域，诸如散光等角度测量被常规记录。类似的方向性变量也出现在人体旋转研究中，包括手部、腰部、颈部和下肢的运动。受一份包含两种白内障手术方式所致散光角度测量的临床数据集启发，我们提出了一种基于双曲几何的圆形数据新型双样本检验框架。假设数据服从具有公共或组别特定集中参数的冯·米塞斯分布，我们将相应的参数空间嵌入到庞加莱圆盘（一个赋予庞加莱度量的开单位圆盘）中。在此构造下，每个冯·米塞斯分布被唯一映射到庞加莱圆盘中的一个点，从而产生一个保持参数空间内在结构的连续几何表示。这种嵌入使得能够通过双曲距离直接比较组间分布，进而得到自然且可解释的检验统计量。我们针对公共集中参数情形开发了基于置换的检验方法，并针对不等集中参数情形提出了基于自助法的检验流程。大量的模拟研究显示，与现有竞争方法相比，所提方法具有稳定的经验水平、强一致性以及更优的渐近功效。通过对白内障手术数据集的详细分析（包括基于临床知识对原始观测值的重构），我们展示了所提方法的应用。结果凸显了将双曲几何纳入圆形生物医学数据分析的实践优势，并强调了基于几何感知的推断在方向性生物标志物研究中的潜力。