ROC Curve and AUC for a Left-Truncated Sample from Rayleigh Distribution

摘要

In a diagnostic process, biomarker plays an important role in predicting the presence or absence of a disease. Biomarkers are biological properties that can be measured from blood, tissue etc. They are continuous in nature and can be molecules, cells, genes or gene derivatives, enzymes, or hormones. Examples of such measures are blood pressure; cholesterol etc and their presence above a level indicate the presence of a disease. Biomarkers can be used to classify the receivers as healthy (H) or diseased (D). Application of receiver operating curve (ROC) and the area under the ROC curve can indicate the accuracy of the biomarker in predicting the presence of the disease. Sensitivity of the biomarker can be defined as the probability that the diseased individuals are correctly predicted and are also called as true positive rate (TPR). Specificity can be defined as the probability that the healthy individuals are correctly identified. I- specificity is defined as the probability that healthy individuals are wrongly specified as diseased and is also called as false positive rate (FPR). ROC curve is the graphical representation of FPR versus TPR. The accuracy of the biomarker can be got from the AUC defined by Bamber (Ref. 1). Parametric modeling a parametric distribution for ROC curves deriving the sensitivity and the specificity. For the left-truncated samples a left-truncated bi-Rayleigh (LTR2) ROC model is proposed here. This curve evaluates the accuracy of the biomarker since due to some technical problems, some of the measurements may be missing and hence the traditional approach cannot be used. For LTR distributions, MLE parameters are not explicitly existent. Wu et al. (Ref. 2) have derived an approximate MLE of scale parameters for right and doubly truncated Rayleigh distribution under failure censored sampling plan. This study is an extension of the above work to obtain approximate MLE of the scale parameter from left-truncated sample. (21 refs.)