The Bi-Gamma ROC Curve in a Straightforward Manner

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Abstract
In biomedical research, biomarkers (diagnostic tests) are used in distinguishing healthy and diseased populations. The effectiveness and accuracy of a biomarker generally assessed through the use of a Receiver Operating Characteristic (ROC) curve model, and its functional such as area under the curve (AUC). The parametric (smooth) ROC curves are obtained under the specific distributions assumptions. A resulting ROC curve model is the plot of sensitivity versus 1-specificity for all possible threshold values. Most popular and widely used ROC curve model is bi-normal ROC curve model under the assumptions of normality. When the biomarker results are continuous and positively skewed (non-normal). The gamma distribution is supposed to a flexible model for positively skewed measurements. In practice use of bi-gamma ROC curve model is hindered by the fact that ROC function cannot be written in closed-form. The solution of the problem is to use transformed invariance property of ROC curve model. Which assumes that the test results of both diseased and healthy are normally distributed after some monotone transformation [1]. In this paper we propose a simple approximation solution for the problem mentioned in above lines using a normal approximation due to Wilson and Hilfertys [2]. Which is useful to approximate gamma distribution results with classical normal distribution based results.

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