100 Foundations of Visual Perception
Panel 5: The ROC Diagram. This panel differs from
the corresponding one in Figure 4.1A because the false-alarm
rate has changed. When the observer is in a Dstate,p(fa)=
p(h)=g(represented by the unfilled dot); when the observer
is in a Dstate,p(fa)=gandp(h)=1.0.
Variable Energy Threshold—Guessing Observer
The preceding versions of threshold theory are idealizations;
step functions are nowhere to be found in psychophysical
data. In response to this realization Jastrow (1888) assumed
that the threshold varies from moment to moment. This con-
ception is depicted in Figure 4.1C.
Panel 1: Threshold Location. The idea that the loca-
tion of a momentary threshold follows a normal density func-
tion comes from Boring (1917). Subsequently, other density
functions were proposed: the lognormal (Gaddum, Allen, &
Pearce, 1945; Thurstone, 1928), the logistic (which is a par-
ticularly useful approximation to the normal; Bush, 1963;
Jeffress, 1973), and the Weibull (Marley, 1989a, 1989b;
Quick, 1974), to mention only three.
Panel 2: Detection Probability. This panel shows that
when the threshold is normally distributed, the probability of
being in a Dstate follows the cumulative distribution that
corresponds to the density function of the momentary thresh-
old. When that density is normal, this cumulative is some-
times called a normal ogive.
Panel 3: Catch Trials. Unchanged from the corre-
sponding panel in Figure 4.1B.
Panel 4: Signal Trials. The psychometric function
shown in this panel takes on the same shape as the function
that describes the growth of detection probability (Fig-
ure 4.1C, second panel).
Panel 5: The ROC Diagram. Instead of observing two
points in the diagram, as we did when we assumed that the
threshold was fixed, the continuous variation of the psycho-
metric function gives rise to a continuous variation in the hit
rate, while the false-alarm rate remains constant.
Variable Energy Threshold—Variable Guessing Rate
It was a major breakthrough in the study of thresholds when,
in 1953–1954, psychophysicists induced their observers to
vary their guessing rate (Swets, 1996, p. 15). This was a de-
parture from the spirit of early psychophysics, which implic-
itly assumed that observers did not develop strategies. This
manipulation was crucial in revealing the weaknesses of
threshold theory. We see in a moment how this manipulation
is done.
The effect of manipulating the observer’s guessing rate is
shown in Figure 4.1D.Panel 1: Threshold Location. Unchanged from the cor-
responding panel in Figure 4.1C.Panel 2: Detection Probability. In this panel the energy
of the stimulus is indicated by a downward-pointing arrow.
The corresponding p(D) is indicated on the ordinate. For rea-
sons we explain in a moment, this value is connected to a
point on the ordinate of the ROC diagram.Panel 3: Catch Trials. In this panel we assume that we
have persuaded the observer to adopt four different guessing
rates (g 1 ,..., g 4 ) during different blocks of the experiment.
The corresponding values are marked on the abscissa (the
false-alarm rate) of the ROC diagram (Figure 4.1D, fifth
panel).Panel 4: Signal Trials. The general structure of a detec-
tion experiment, assuming two observer states, detect and D,
is shown in Figure 4.2. In this figure (which is an augmented
version of Figure 4.3) we show how to calculate the hit rate
and the false-alarm rate, as well as which parts of this model
are observable and which are hidden.Panel 5: The ROC Diagram. In this panel hit rate and
false-alarm rate covary and follow a linear function. In the
note that follows we give the equation of this line and show
that it allows us to estimate p(Dsignal), which is the measure
of the signal’s detectability.
Note:If in Equation 1 we let b=p(Dsignal) and m=
1 – p(Dsignal), and we recall that g=p(“Yes”D)=p(fa),
then the equation of the ROC is p(hit)=b+mp(fa), a
straight line. The intercept bgives the measure of the signal’s
detectability:p(Dsignal).
We can now understand the importance of the ROC dia-
gram. Regardless of the detection theory we hold, it allows us
to separate two aspects of the observer’s performance: stimu-
lus detectability(or equivalently observer sensitivity) and ob-
server bias. In high-threshold theory these measures are
p(Dsignal) and g.