Psychophysical Methods 105(We need not assume that observers actually use Bayes’s rule,
only that they have a sense of the prior odds and the likeli-
hood ratios, and that they do something akin to multiplying
them.)
Once the observers have calculated the posterior probabil-
ity or odds, they need a rule for saying “Yes” or “No.” For ex-
ample, they could choose to say “Yes” if p(SN)≥.5. This
strategy is by and large equivalent to choosing a value of
below which they would say “No,” and otherwise they would
say “Yes.” This value of ,c, is called the criterion.
We have already seen how we can generate an ROC curve
by inducing observers to vary their guessing rates. These
procedures—manipulating prior probabilities and payoffs—
induce the observers to vary their criteria (Figures 4.8C and
4.8D) from lax (cis low, hit rate and false-alarm rate are
high) to strict (cis high, hit rate and false-alarm rate are
low), and produce the ROC curve shown in Figure 4.8F.
Different signal energies (Figure 4.8G) produce different
ROC curves. The higher d, the further the ROC curve is from
the positive diagonal.
The ROC Curve; Estimating d
The easiest way to look at signal detection theory data is to
transform the hit rate and false-alarm rate into log odds. To
dothis,wecalculateH=kln 1 p(hp)(h)andF=kln 1 p(fpa()fa),
wherek=^ 3 =0.55133(whichisbasedonalogisticapprox-
imation to the normal). The ROC curve will often be linear
after this transformation. We have done this transformation
with the data of Cook and Wixted (1997; see Figure 4.9).
If we fit a linear function, H=b+mF, to the data, we
can estimate d=mband (^) SN=m^1 , the standard deviation of
theSNdistribution (assuming (^) N=1). Figure 4.9 shows
these computations. (This analysis is not a substitute for more
detailed and precise ones, such as Eng, 2001; Kestler, 2001;
Metz, 1998; Stanislaw & Todorov, 1999.)
Energy Thresholds and Observer Thresholds
It is easy to misinterpret the signal detection theory’s as-
sumption that there are no observer thresholds (a potential
misunderstanding detected and dispelled by Krantz, 1969).
The assumption that there are no observer thresholds means
that observers base their decisions on evidence (the likeli-
hood ratio) that can vary continuously from 0 to infinity. It
need not imply that observers are sensitive to all signal ener-
gies. To see how such a misunderstanding may arise, consider
Figures 4.8A and 4.8B. Because the abscissas are labeled
“energy,” the panels appear to be representations of the input
to a sensory system. Under such an interpretation, any signal
whatsoeverwouldgiverisetoasignal+noisedensitythat
differs from the noise density, and therefore to an ROC curve
that rises above the positive diagonal.
To avoid the misunderstanding, we must add another layer
to the theory, which is shown in Figure 4.10. Rows (a) and (c)
are the same as rows (a) and (b) in Figure 4.8. The abscissas
in rows (b) and (d) in Figure 4.10 are labeled “phenomenal
evidence” because we have added the important but plausible
assumption that the distribution of the evidence experienced
by an observer may not be the same as the distribution of
the signals presented to the observer’s sensory system (e.g.,
because sensory systems add noise to the input, as Gorea &
Sagi, 2001, showed). Thus in row (b) we show a case where
the signal is not strong enough to cause a response in the ob-
server: the signal is below this observer’s energy threshold.
In row (d) we show a case of a signal that is above the energy
threshold.
Some Methods for Threshold Determination
Method of Limits
Terman and Terman (1999) wanted to find out whether retinal
sensitivity has an effect on seasonal affective disorder (SAD;
–1
1
0
-1 0 1
k = 0.55133
H = k ln 1– hrhr
H = 0.92 + 0.52 F
–2 0 2 4 6
0.1
0.2
0.3
0.4
d ́=0.920.52 = 1.77
σN = 1
σSN = 0.52^1 = 1.92
F = k ln1 – farfar
Figure 4.9 Simple analysis of the Cook and Wixted (1997) data.