Modeling the ratio of two dependent components as a function of covariates is a frequently pursued objective in observational research. Despite the high relevance of this topic in medical studies, where biomarker ratios are often used as surrogate endpoints for specific diseases, existing models are based on oversimplified assumptions, assuming e.g.\@ independence or strictly positive associations between the components. In this paper, we close this gap in the literature and propose a regression model where the marginal distributions of the two components are linked by Frank copula. A key feature of our model is that it allows for both positive and negative correlations between the components, with one of the model parameters being directly interpretable in terms of Kendall's rank correlation coefficient. We study our method theoretically, evaluate finite sample properties in a simulation study and demonstrate its efficacy in an application to diagnosis of Alzheimer's disease via ratios of amyloid-beta and total tau protein biomarkers.

翻译：对两个依赖组分之比进行协变量函数的建模，是观察性研究中经常追求的目标。尽管这一主题在医学研究中具有高度相关性（生物标志物比率常被用作特定疾病的替代终点），但现有模型基于过度简化的假设，例如假设组分间独立或呈严格正相关。本文弥补了这一文献空白，提出了一种回归模型，其中两个组分的边缘分布通过Frank copula连接。该模型的关键特性在于允许组分间存在正相关或负相关，且其中一个模型参数可直接解释为Kendall秩相关系数。我们从理论上研究了该方法，通过模拟实验评估了有限样本性质，并在阿尔茨海默病诊断应用中（通过β-淀粉样蛋白与总tau蛋白生物标志物的比率）证明了其有效性。