Relationship between Brier score and area under the binormal ROC curve

Abstract

If we consider the Brier score (B) in the context of the signal detection theory and assume that it makes sense to consider the existence of B as a parameter for the population (let B̄ be this B), and if we assume that the calibration in the observer's probability estimate is perfect, we find that there is a theoretical relationship between B̄ and the area under the binormal receiver operating characteristic (ROC) curve, AZ. We have derived this theoretical functional relationship between B and AZ, by using the parameter a and b in the binormal ROC model and the prior probability of signal events (α); here, the two underlying normal distributions are Nμs,σs and Nμn,σn; and, a = (μs - μn)/σs and b = σn/σs. We empirically found that, if parameters b and α are constant, B̄ values in relation to given AZ values monotonically decrease as AZ values increase, and these relationship curves have monotonically decreasing slopes. © 2002 Elsevier Science Ireland Ltd. All rights reserved.