Standard errors and confidence intervals for measures of sensitivity

Abstract

It is quite simple to obtain maximum-likelihood estimates of the slope (b0) and intercept (a0) of the linearized ROC curve under the unequal-variance Gaussian assumption. If a measure of sensitivity is a function (f) of (a, b) and f is differentiable at (a0, b0), then it is not hard to show that f(â, b̂) is asymptotically normally distributed with mean f(a0, b0) and variance Var(f) = [σa2(∂f/∂a) + σb2(∂f/∂b) + 2σab(∂f/∂a)(∂f/∂b)]. This result was used to obtain large-sample variances for estimates of Δm, d′e, and A, the area under the ROC curve, based on maximum likelihood. Confidence intervals for these estimates are obtained in the usual manner. The maximum-likelihood method for estimating area is compared with standard non-parametric methods. The maximum-likelihood approach is recommended for a variety of good reasons.