When I asserted a few minutes ago that:
> In such non-normal situations, the indices are *not*
> defined in terms of means and standard deviations, but
> instead in terms of the straight-line ROC curve
... I should have warned that some computer programs may (incorrectly)
estimate the index values from the decision-variable means and standard
deviations rather than from the ROC curve. The former approach is
incorrect because ROC curves (and therefore detectability, in a
fundamental sense) are invariant under monotonic transformations of the
decision variable, whereas means and standard deviations aren't
invariant in this way.
For example, consider the situation in which a decision variable x
arises from a pair of normal densities, and a monotonically-related
decision variable y = exp(x) arises from a pair of log-normal
densities. Both ROC curves will be the same. However, only the value
of d_a computed from the means and standard deviations of x will agree
with the common value of d_a computed from both ROC curves, whereas the
value of d_a computed from the means and standard deviations of y will
be wrong.
Charles Metz
