## Mathematics Dissertations

#### Title

Novel Nonparametric Methods For ROC Curves

12-15-2016

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

Yichuan Zhao

Xin Qi

Jing Zhang

Yanqing Zhang

#### Abstract

The receiver operating characteristic (ROC) curve is a widely used graphical method for evaluating the discriminating power of a diagnostic test or a statistical model in various areas such as epidemiology, industrial quality control and material testing, etc. One important quantitative measure summarizing the ROC curve is the area under the ROC curve (AUC). The accuracy of two diagnostic tests with right censored data can be compared using the difference of two ROC curves and the difference of two AUC's. Moreover, the difference of two volumes under surfaces (VUS) is investigated to compare two treatments for the discrimination of three-category classification data, extending the ROC curve to the ROC surface in the three-dimensional case.

A few scientific progresses have been achieved in ROC curves and its related fields over the past decades. In this dissertation, we propose a plug-in empirical likelihood (EL) procedure combining placement values and weighting of inverse probability techniques, to construct stable and precise confidence intervals of the ROC curves, the difference of two ROC curves, the AUC's and the difference of two AUC's with right censoring. We proved that the limiting distribution of the EL ratio is a weighted $\chi^2$ distribution. Furthermore, we introduce a jackknife empirical likelihood (JEL) procedure to explore the difference of two correlated VUS's with complete data. We proved that the limiting distribution of the proposed JEL ratio is a $\chi^2$ distribution, i.e., the Wilk's theorem holds. Extensive simulation studies demonstrate that the new methods have better performance than the existing methods in terms of coverage probability of confidence intervals in most cases. Finally, the proposed methods are applied to analyze data sets of Primary Biliary Cirrhosis (PBC), Alzheimer's disease, etc.

#### DOI

https://doi.org/10.57709/9461609

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