Handbook of Psychology, Volume 4: Experimental Psychology

Psychophysical Methods 97

Herbart (1824/1890) may have been the first to use the Latin
term for threshold, limen, and its German equivalent,
schwelle,to refer to the limit below which a given stimulus
ceases to be perceptible. Although the idea of threshold ap-
pears straightforward, it turns out to be complex. We now ex-
plore the original idea of threshold and show its limitations.
After that we present it in its current form: signal detection
theory.

Threshold Theories

We begin with the simplest threshold theory, to which we will
gradually add elements until it can be confronted with data.

By that point we will have introduced two fundamental ideas:

the receiver operating characteristic (ROC) and the possibil-

ity of disentangling the sensitivity of observers from their

response bias.

Fixed Energy Threshold—Naive Observer

The simplest threshold theory is depicted in Figure 4.1. We

look at the panels from left to right.

Panel 1: Threshold Location. This panel represents

two fundamental ideas: (a) The observer can be in one of two

(3) FALSE ALARM RATE

catch trials

(1)

THRESHOLD LOCATION

(2)

DETECTION PROBABILITY

(5)

ROC CURVE

(4) HIT RATEsignal trials

p(false alarm)

1

energy^0 energy

energy threshold

1

(^0) energy
hit rate
false alarm ratebelow
p(hit)
1
(^0) energy
A. FIXED ENERGY
THRESHOLD—
NAIVE OBSERVER
p(false alarm)
1
(^0) energy
g
perceptible
(=D-state)
imperceptible
(=D-state)
energy
p(D)
p(D)
1
(^0) energy
false alarm rate
g
hit rate
below
above
STIMULUS
p(hit) ENERGY
1
(^0) energy
B. FIXED ENERGY
THRESHOLD—
GUESSING
OBSERVER g
p(false alarm)
1
(^0) energy
g
energy
mean energy threshold p(D)
1
(^0) energy false alarm rate
hit rate
low
high
STIMULUS
p(hit) ENERGY
1
(^0) energy
C. VARIABLE ENERGY
THRESHOLD—
GUESSING
OBSERVER
p(false alarm)
1
(^0) energy
g 2
g 4
g 1
g 3
p(false alarm)
1
(^0) energy
g 2
g 4
g 1
g 3
energy
p(D)
1
(^0) energy
false alarm rate
hit rate
g 1 g 2 g 3 g 4
energy
p(D)
1
(^0) energy
false alarm rate
hit rate
g 1 g 2 g 3 g 4
.14
.65
p(hit)
1
(^0) energy
p(hit)
1
0
.28
.71
(i) low
energy
(ii) higher
energy
D. SINGLE
STIMULUS—
GUESSING RATE
MANIPULATED
perceptible
(=D-state)
imperceptible
(=D-state)
above
STIMULUS
ENERGY
Figure 4.1 High-threshold theories: (A) Fixed energy threshold—naive observer; (B) Fixed energy threshold—sophisticated observer; (C) Variable
energy threshold—sophisticated observer; (D) The effect of manipulating guessing rate: (i) Low-energy stimulus, (ii) high-energy stimulus.