## Mathematics Ph.D. Dissertations

#### Title

Likelihood-Based Confidence Bands for a ROC Curve

2006

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Mathematics/Mathematical Statistics

#### Abstract

In the area of ROC analysis, confidence intervals and the corresponding confidence bands are usually constructed by way of first oder Normal approximations based on the popular delta method. However, it has been always known that all the information in the data is contained in the corresponding likelihood function. On another note there has recently been a surge in research in empirical data analysis. Of note is the work of Art Owen on empirical likelihood methods. This research endeavors to explore the use of likelihood-based methods in statistical analysis of the ROC curve, specifically in the construction of confidence bands, and compare them with ROC confidence bands based on Normal approximation resultant from the delta method.

In the first part we explore the theory behind the construction of likelihood-based and Normal approximation based confidence bands for the ROC curve, under the assumption that the data are sampled from parametric models. In the second part we explore the theory behind the construction of confidence bands for the ROC curve using the Normal approximation for empirical estimators and empirical likelihood-based confidence bands for the ROC curve. The basis for both of the likelihood-based methods is the chi-squared approximation, popularly known as Wilks’ method. It has been observed that the chi-squared approximation to the likelihood function may not be a very good approximation. As a result a Bartlett correction to the chi-squared approximation can be used to adjust the approximation. The Bartlett correction has been known to be plausible for parametric models. However, it has only been recently known that the empirical likelihood function is also Bartlett correctable, under some conditions. This research goes further to explore Bartlett correction of empirical likelihood confidence bands for the ROC curve.

Simulation studies were carried out to demonstrate the improvement of the likelihood-based approach over the Normal approximation approach as well as the further improvement of the Bartlett correction over empirical likelihood method.

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