Victor A. Gombos wrote:
> I am using Systat 9.0 for my master's thesis data--the
> nature of my analyses depend heavily on Signal Detection
> Theory. Therefore, of course I am using the Signal
> Detection Analysis program in Systat. Systat reports a
> number of things, including the ROC, d-prime, D-sub-a,
> and Sakitt's D.
>
> It seems to me that using d' is not as good as D-sub-a or
> Sakitt's D -- the subjects' responses I am analyzing are
> confidence ratings, on a scale from one-to-eight -- as
> there tends to be greater fluctuations/variability of d-
> prime as opposed to the other measures. In this way,
> I'm committed to D-sub-a thinking it is more robust and
> consistent.
>
> But I need to know how D-sub-A and Sakitt's D differ from
> d'. I haven't been able to find information on this
> from any other source thus far.
>
> Can anyone tell me what these measures are and how they
> differ from d'?
The distinction is explained very nicely in:
Simpson AJ, Fitter MJ. What is the best index of detectability?
Psych Bull 1973; 80:481-488.
I strongly recommend that you read this paper, but I'll try to summarize
its essence (and add a point or two) here.
For two states of truth (1 and 2) and decision variables densities that
are normally distributed with generally different means (mu_1 and mu_2)
and standard deviations (sigma_1 and sigma_2):
d_a = (mu_1 - mu_2)/SQRT((sigma_1**2 + sigma_2**2)/2)
whereas
Sakitt's D = (mu_1 - mu_2)/SQRT(sigma_1 * sigma_2)/2).
In the special case where sigma_1 = sigma_2 = sigma, both d_a and
Sakitt's D reduce to:
d' = (mu_1 - mu_2)/sigma .
All three of these indices also apply rigorously to non-normal
decision-variable densities as long as the resulting ROC curve plots as
a straight line on "normal deviate axes" (e.g., see Metz CE. ROC
methodology in radiologic imaging. Investigative Radiology 1986; 21:
720), in which case some (usually unknown) monotonic transformation of
the decision variable must yield normal densities. 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. If the
"y intercept" and "slope" of such an ROC are given by "a" and "b",
respectively, then:
d_a = a/SQRT((1 + b**2)/2)
whereas
Sakitt's D = a/SQRT(b) ,
both of which approach
d' = a
for the special case where b = 1.
When any ROC curve plots as a straight line on normal-deviate axes, its
value of d_a also equals the normal deviate which corresponds to the
area under the ROC when that curve is plotted on *conventional* (i.e.,
probability, rather than normal-deviate) axes. The latter
interpretation of d_a is sometimes used for other ROC curve forms as
well, which isn't strictly "legal" but, from a practical standpoint, is
rarely misleading.
Charles Metz
