98 Foundations of Visual Perception
statesDorD, and (b) on the scale of stimulus energy there is
a fixed value below which observers can never detect a stim-
ulus. Above this threshold, observers always detect it.
Panel 2: Detection Probability. This panel represents
the same idea in more modern terms. This graph represents the
probability of an observer being above the observer thresh-
old—in stateD—as a function of stimulus energy. As you
would expect, this probability is 0 below the energy threshold
and 1 above it. The multiple arrows represent stimuli; they are
unfilled if they are at an energy that is below the energy thresh-
old and filled if they are above it.
Panel 3: False Alarm Rate (Catch Trials). Suppose
that the psychophysical experiment we are doing requires the
observers to respond “Yes” or “No” depending on whether
they detected a stimulus or not. This is called a yes-no task.
This panel represents what would happen if on some trials,
calledcatch trials,we withheld the stimulus without inform-
ing the observers. In Table 4.1 we show the nomenclature of
the four possible outcomes in such an experiment, generated
by two possible responses to two types of stimuli.
According to a naive conception of detection, observers
are “honest”; they would never respond “Yes” on catch trials.
In other words, they would never produce false alarms.That
is why p(fa)=0 for all values of stimulus energy.
Panel 4: Hit Rate (Signal Trials). This panel shows the
probability that the observer says “Yes” when the signal was
presented (a hit) depends on whether the stimulus energy is
above or below threshold. If it is above threshold, then
p(h)=1, otherwise p(h)=0. The function that relates
p(“Yes”), or hit rate, to stimulus energy is called the psycho-
metric function.
Panel 5: The ROC Space. This panel is a plot of the hit
rate as a function of the false-alarm rate. Here, where ob-
servers always respond “No” when in a Dstate, there is little
point in such a diagram. Its value will become clear as we
proceed.
Fixed Energy Threshold—Guessing ObserverInstead of assuming naive observers, we next assume sophis-
ticated ones who know that some of the trials are catch trials,
so some of the time they choose to guess. Let us compare this
threshold theory, called high-threshold theory,with the most
unrestricted form of two-state threshold theory (Figure 4.2).
The general theory is unrestricted because it (a) allows ob-
servers to be in either a Dstate or a Dstate on both signal and
catch trials and (b) imposes no restrictions on when guessing
occurs.
In contrast, high-threshold theory (Figure 4.3) has three
constraints: (a) During a catch trial, the observer is always in
aDstate:p(Dcatch)=0. (b)When in a Dstate, the observer
always says “Yes”: p(“Yes”D)=1. (c) When in a Dstate,
the observer guesses “Yes” (emits a false alarm) at a rate
p(fa)=g, and “No” at a rate 1–g. So p(“Yes”D)=g. The
theory is represented in Figure 4.1B, whose panels we dis-
cuss one by one.Panel 1: Threshold Location. Unchanged from the
corresponding panel in Figure 4.1A.Panel 2: Detection Probability. Unchanged from the
corresponding panel in Figure 4.1A.Panel 3: Catch Trials. In this panel we show that ob-
servers, realizing that some stimuli are below threshold and
wishing to be right as often as possible, may guess when in a
Dstate. This increases the false-alarm rate.Panel 4: Signal Trials. The strategy depicted here does
not involve guessing when the observers are in a Dstate. This
panel shows that this strategy increases the observers’ hit
rates when they are in a DDstate, that is, below the energy
threshold. Note that the psychometric function does not rise
from 0 but from g.
Note:With the help of Figure 4.3 we can see that when
signal energy is 0, the hit rate is equal to the false-alarm rate,
g. We begin by writing down the hit rate as a function of
the probability of the observer being in a Dstate when a
signal is presented, p(Dsignal), and the probability of the
observer guessing, that is, saying “Yes” when in a Dstate,
p(“Yes”D)=g:p(h)p(Dsignal)[1p(Dsignal)]g
p(Dsignal)(1g)g. (1)When signal energy is 0, p(Dsignal)0, and therefore
p(h)=g.TABLE 4.1 Outcomes in a Yes-No Experiment with Signal and
Catch Trials (and their abbreviations)
ResponseStimulus Class Yes No
Signal Hit (h) Miss (m)
Catch False alarm (fa) Correct rejection (cr)