Marginal, conditional, and pseudo likelihood ratio approaches for biomarker combination to predict a binary disease outcome

It is a common practice in public health research that multiple biomarkers are collected to diagnose or predict a disease outcome. A natural question is how to combine multiple biomarkers to improve the diagnostic accuracy. It has been shown by Neyman-Pearson lemma that the likelihood ratio statistic achieves the optimal AUC in theory. However, practical difficulty often lies in the estimation of the multivariate density functions. We propose three novel methods for the biomarker combination, with the idea of breaking down the joint densities to a series of univariate densities. The marginal likelihood ratio approach only assumes the marginal distribution of each biomarker. While the conditional likelihood ratio (CLR) and pseudo likelihood ratio (PLR) approaches assume the conditional distributions of a marker given others, and hence make use of the correlation structure to estimate the combination rules. The proposed methods make it much easier to assume and validate the univariate distributions of a biomarker than making multivariate distributional assumptions. Extensive simulation studies demonstrate that the CLR and the PLR approaches outperform many existing methods, and are therefore recommended for practical use. The proposed methods are motivated by and applied to a biomarker study to diagnose childhood autism/autism spectrum disorder.