Abstract
In a multivariate setup, the classification techniques have its significance in identifying the exact status of the individual/observer along with accuracy of the test. One such classification technique is the Multivariate Receiver Operating Characteristic (MROC) Curve. This technique is well known to explain the extent of correct classification with the curve above the random classifier (guessing line) when it satisfies all of its properties especially the property of increasing likelihood ratio function. However, there are circumstances where the curve violates the above property. Such a curve is termed as improper curve. This paper demonstrates the methodology of improperness of the MROC Curve and ways of measuring it. The methodology is explained using real data sets.
Highlights
In classification, there are plenty of techniques to accommodate the need for identifying an individual/observer’s status in a wide variety of fields like Psychology, Banking, Forensic, Medicine, etcetera [12, 3]
Since the slope of a Multivariate Receiver Operating Characteristic (MROC) Curve for a continuous decision variable is equal to the likelihood ratio at the corresponding threshold, it follows that the slope of a MROC Curve decreases as the false positive rate (FPR)increases, that is, a MROC Curve will be concave everywhere (0 ≤ F P R ≤ 1)
Since φ √(c−b′μ0) > 0, it follows from Equation (9) that the second derivative of the MROC Curve and the (b′ Σ0 b) derivative of the likelihood ratio have opposites signs when evaluated at t and c, respectively
Summary
There are plenty of techniques to accommodate the need for identifying an individual/observer’s status in a wide variety of fields like Psychology, Banking, Forensic, Medicine, etcetera [12, 3]. Since the slope of a MROC Curve for a continuous decision variable is equal to the likelihood ratio at the corresponding threshold, it follows that the slope of a MROC Curve decreases as the false positive rate (FPR)increases, that is, a MROC Curve will be concave everywhere (0 ≤ F P R ≤ 1). If the decision variable is not an increasing function of the likelihood function, its model and corresponding MROC Curve are said to be improper. This MROC Curve is concave for F P R < 0.76, but is convex for F P R > 0.76.
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