The total operating characteristic to measure diagnostic ability for multiple thresholds

The relative operating characteristic (ROC) is a popular statistical method to measure the association between observed and diagnosed presence of a characteristic. The diagnosis of presence or absence depends on whether the value of an index variable is above a threshold. ROC considers multiple possible thresholds. Each threshold generates a two-by-two contingency table, which contains four central entries: hits, misses, false alarms, and correct rejections. ROC reveals for each threshold only two ratios, hits/(hits + misses) and false alarms/(false alarms + correct rejections). This article introduces the total operating characteristic (TOC), which shows the total information in the contingency table for each threshold. TOC maintains desirable properties of ROC, while TOC reveals strictly more information than ROC in a manner that makes TOC more useful than ROC. We illustrate the concepts with an application to land change science.