Parametric estimates for the receiver operating characteristic curve generalization for non-monotone relationships

Abstract

Diagnostic procedures are based on establishing certain conditions and then checking if those conditions are satisfied by a given individual. When the diagnostic procedure is based on a continuous marker, this is equivalent to fix a region or classification subset and then check if the observed value of the marker belongs to that region. Receiver operating characteristic curve is a valuable and popular tool to study and compare the diagnostic ability of a given marker. Besides, the area under the receiver operating characteristic curve is frequently used as an index of the global discrimination ability. This paper revises and widens the scope of the receiver operating characteristic curve definition by setting the classification subsets in which the final decision is based in the spotlight of the analysis. We revise the definition of the receiver operating characteristic curve in terms of particular classes of classification subsets and then focus on a receiver operating characteristic curve generalization for situations in which both low and high values of the marker are associated with more probability of having the studied characteristic. Parametric and non-parametric estimators of the receiver operating characteristic curve generalization are investigated. Monte Carlo studies and real data examples illustrate their practical performance.

Improving the biomarker diagnostic capacity via functional transformations

“…A rapid overview suggests that the Mode frequency in male is more likely to take extreme values of the spectrum. Considering classification subsets in the form (a, b) (a ≤ b ∈ R) according to the so-called gROC curve approach [11,12] improves the global diagnostic capacity and gets an AUC of 0.741 [0.726, 0.758]. McIntosh and Pepe [15] proved that the optimal classification criterion among those based on one particular biomarker can be computed from a binary regression with an adequate functional transformation of the biomarker.…”

“…Martínez-Camblor and Pardo-Fernández [12] proved that the AUC under continuous ROC curves defined as R(t) = P(ξ ∈ b(t)) (0 ≤ t ≤ 1) is the probability of selecting randomly and independently two subjects, one negative and one positive, for which there exists one classification subset such that both subjects are correctly classified. In any case, the AUC can always be interpreted as the average sensitivity (specificity) for all specificity (sensitivity) values and beyond the determined classification rules, it can always be used to compare the global classification accuracy of two different biomarkers.…”

“…Martínez-Camblor and Pardo-Fernández [12] proved that, assuming that extreme values of the biomarker are associated with having more probability of being positive (that implies b ≥ 1), for each t ∈ [0, 1], the gROC curve is based on the subsets…”

“…Since m/n is only a positive constant, the order induced by n l /m l is the one leading to optimize equation (11). In order to satisfy (C 1 ) and definition in (10) we can define h op (e l ) = n l n l + m l for 1 ≤ l ≤ k (12) because the order of induced by n l /(n l + m l ) is that induced by n l /m l but the values are in the unit interval. Notice that, since (C 2 ) does not affect (C 1 ) those functions which satisfy both simultaneously maximize S 1 , S 2 , S 3 and S 4 .…”

“…They considered the problem in which both the highest and the lowest biomarker values are associated with having more probability of being positive. The parametric estimator of the resulting ROC curve (called generalized ROC curve, gROC) was considered by Martínez-Camblor and Pardo-Fernández [12]. For particular real problems, more customized criteria have been proposed.…”