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 commonly based on oversimplified assumptions, assuming e.g. independence or strictly positive associations between the components. In this paper, we overcome such limitations and propose a regression model where the marginal distributions of the two components are linked by a copula. A key feature of our model is that it allows for both positive and negative associations 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.