Evaluation of Area under the Constant Shape Bi-Weibull ROC Curve

The Receiver Operating Characteristic (ROC) curve generated based on assuming a constant shape Bi-Weibull distribution is studied. In the context of ROC curve analysis, it is assumed that biomarker values from controls and cases follow some specific distribution and the accuracy is evaluated by using the ROC model developed from that specified distribution. This article assumes that the biomarker values from the two groups follow Weibull distributions with equal shape parameter and different scale parameters. The ROC model, area under the ROC curve (AUC), asymptotic and bootstrap confidence intervals for the AUC are derived. Theoretical results are validated by simulation studies. Notations and Terminologies X Random variable representing controls t Cutoff point of classification, t x y  Y Random variable representing cases I(θ) Fisher Information matrix m Number of controls y(x) ROC model n Number of cases MLE Maximum Likelihood Estimate f(x) Probability Density Function (PDF) of X x(t) False Positive Rate (FPR) at cutoff t g(y) PDF of Y y(t) True Positive Rate (TPR) at cutoff t F(x) Distribution function of X TPR Probability that cases are correctly identified (Sensitivity) G(y) Distribution function of Y FPR Probability that controls are wrongly identified as cases (1-Specificity)