Relationship between Brier score and area under the binormal ROC curve

  • Ikeda M
  • Ishigaki T
  • Yamauchi K
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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,σsand Nμn,σn; and, a = (μs- μn)/σsand b = σn/σs. We empirically found that, if parameters b and α are constant, B̄ values in relation to given AZvalues monotonically decrease as AZvalues increase, and these relationship curves have monotonically decreasing slopes. © 2002 Elsevier Science Ireland Ltd. All rights reserved.

Author-supplied keywords

  • Brier score
  • Medical decision making
  • Probabilistic judgments
  • Receiver operating characteristic (ROC) curve

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