# On the Optimal Combination of Elliptically Distributed Biomarkers to Improve Diagnostic Accuracy

## TL;DR

This paper introduces a new method for combining biomarkers that follow elliptical distributions to improve diagnostic accuracy, especially when biomarkers are not normally distributed.

## Contribution

The novel contribution is the derivation of an ROC-based method for optimally combining elliptically distributed biomarkers using a nonparametric maximum likelihood estimate.

## Key findings

- The proposed elliptical combination method outperformed traditional normality-based methods in simulation studies.
- The method improved AUC values in real-world applications for autism spectrum disorder and neural tube defects.
- The nonparametric maximum likelihood estimation technique proved effective for empirical estimation of optimal biomarker combinations.

## Abstract

Diagnostic biomarkers play a critical role in biomedical research, particularly for the diagnosis and prediction of diseases, etc. To enhance diagnostic accuracy, extensive research about combining multiple biomarkers has been developed based on the multivariate normality, which is often not true in practice, as most biomarkers follow distributions that deviate from normality. While the likelihood ratio combination is recognized to be the optimal approach, it is complicated to calculate. To achieve a more accurate and effective combination of biomarkers, especially when these biomarkers deviate from normality, we propose using a receiver operating characteristic (ROC) curve methodology based on the optimal combination of elliptically distributed biomarkers. In this paper, we derive the ROC curve function for the elliptical likelihood ratio combination. Further, proceeding from the derived best combinations of biomarkers, we propose an efficient technique via nonparametric maximum likelihood estimate (NPMLE) to build empirical estimation. Simulation results show that the proposed elliptical combination method consistently provided better performance, demonstrating its robustness in handling various distribution types of biomarkers. We apply the proposed method to two real datasets: Autism/autism spectrum disorder (ASD) and neural tube defects (NTD). In both applications, the elliptical likelihood ratio combination improves the AUC value compared to the multivariate normal likelihood ratio combination and the best linear combination.