Confidence Intervals Estimation for ROC Curve, AUC and Brier Score under the Constant Shape Bi-Weibull Distribution

The accuracy of diagnostic test is typically evaluated by sensitivity and specificity. Receiver Operating Characteristic (ROC) curve analysis is one of the most familiar techniques and it will provide accuracy for the extent of correct classification of a test and it is a graphical representation of the relationship between sensitivity and specificity. The conventional way of expressing the true accuracy of test is by using its summary measures Area Under the Curve (AUC) and Brier Score (B). Hence the main issue in assessing the accuracy of a diagnostic test is to estimate the ROC curve and its AUC and Brier Score. The ROC curve generated based on assuming a Constant Shape Bi-Weibull 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, AUC, MLE, asymptotic, bootstrap confidence intervals for the AUC, asymptotic confidence intervals for the ROC curve and Brier Score are derived. However, the accuracy of a test is to be explained by involving the scale and shape parameters. Theoretical results are validated by simulation studies. An illustrative example is also provided to explain the concepts.