# Some Novel Interval Estimation Methods

8-11-2020

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Mathematics and Statistics

Gengsheng Qin

Jun Kong

Jing Zhang

Ruiyan Luo

## Abstract

In medical diagnostic studies, the Youden index is a summary measure widely used in the evaluation of the diagnostic accuracy of a medical test. When covariates are not considered, the diagnostic accuracy of the test can be biased or misleading. By incorporating information from covariates, we propose and compare various confidence intervals for the covariate adjusted Youden index and its optimal cut-off point. In ROC analysis, the area under the ROC curve (AUC) is a popular one number summary index of the discriminatory accuracy of a diagnostic test. Adjustment for covariate effects can greatly improve the diagnostic accuracy of a test for individual patient. AUC Regression is widely used to evaluate the effects of the covariates on the diagnostic accuracy. Using side information provided by the influence function, empirical likelihood methods are proposed for inferences of AUC in the presence of covariates. For parameters in the AUC regression model, it is shown that the asymptotic distribution of the influence function-based empirical log-likelihood ratio statistic is a standard chi-square distribution. Hence, confidence regions for the regression parameters can be easily obtained without any variance estimates.

The latter half of this dissertation focuses on empirical likelihood (EL) based interval estimation methods for correlation coefficient (CC) and coefficient of variation (CV). Under normal distribution assumptions, there are many types of confident intervals for CC or CV, such as the GPQ-based ‘exact’ interval, the Z transformation-based interval, and maximum likelihood-based intervals. However, the exact method is computationally cumbersome, and approximation methods can't be applied when the underlying distribution is unknown. Therefore, we propose influence function-based empirical likelihood intervals for CC and CV. Extensive simulation studies are conducted to evaluate the finite sample performances of the proposed EL-based intervals in terms of coverage probability. Finally, we illustrate the proposed methods with real examples.

## DOI

https://doi.org/10.57709/18614951