A Study on the Bi-Rayleigh ROC Curve Model

Abstract

Receiver Operating Characteristic (ROC) curves are used to describe and compare the accuracy of diagnostic test or the ability of a continuous biomarker in discriminating between the subjects into healthy or diseased cases in medical field. The most familiar form of ROC curve is Bi-normal (Gaussian) ROC curve model, which assumes that the test scores or a monotone transformation of the test scores are from two normal populations (i.e. healthy and diseased). It may not be true all the time, it may violate the assumptions of normal distribution in some situations and also we cannot adopt the model as it is when the sample size is small. In this paper, we have proposed ROC curve model for Rayleigh distribution which can be used even when sample size is small. The properties of Bi-Rayleigh ROC model are studied and Area Under the ROC Curve (AUC) are derived. The proposed model is supported by real life example as well as simulation studies. The confidence interval for the population parameter is studied with simulation studies of varying sample sizes. It is found that Bi-Rayleigh ROC model provides better accuracy of classification than the conventional bi-normal ROC model. In the ROC model section, Bi-Rayleigh ROC model and its AUC are derived and its properties are discussed. In application section, two real life example viz. Head trauma data and Multiple Sclerosis data were used for the proposed model and then these are compared with the conventional bi- normal model. In the simulation section, simulation studies have been carried out for different values of parameters to support the Bi-Rayleigh ROC model. In the last section, the confidence intervals for the parameter σi, i=0,1 are discussed for real life example and simulated data sets. II. ROC MODEL AND AREA UNDER THE CURVE Let the scores (S) be a random variable in healthy (H) and diseased (D) populations respectively. The ROC curve is plotted over values of threshold value (t) and is given by y(t) = h (x(t)) where x(t) is the False positive rate which is given by x(t) = P(S>t|H) and y(t) is True positive rate which is given by y(t) = P (S>t|D) The general functional form of ROC can be represented as