Thresholds, Weber’s law,
Fechner’s three methods
Research Methods
Fall 2010
Tamás Bőhm
Introspection
• Method applied by early psychologists (Wundt)
• Self-observation of conscious thoughts and
sensations
– Do I perceive this light brighter than that? Why?
• Inherently subjective
 rejected as a scientific method
• But it provides useful intuitions when designing
experiments (helps forming hypotheses)
– I realized that it is hard for me to hear very high and
very low tones. Thus the frequency of a tone may be
a factor in its perceived loudness/audibility. Let’s test
this in an experiment!
Formal methods
• Quantifying the sensations
evoked by physical stimuli
• Gustav Fechner (1860):
established 3 formal
psychophysical methods
• Fechner’s methods have
been (and are being) widely
used
• More reliable and accurate
than introspection
• Results from different
experiments are comparable
Sensation magnitude
Thresholds
If a linear relationship is
assumed, two values
determine the function:
Linear
psychophysical
equation
X-intercept
slope
Stimulus intensity
– X-intercept: minimum
stimulus value that evoked
any sensation;
absolute threshold
– Slope: the rate at which
sensation grows as we
increase intensity;
difference threshold
(inversely proportional to
slope)
Thresholds
General definitions (not assuming linearity):
– Absolute threshold:
intensity that the observer can just barely detect
• Intensities below absolute threshold: undetectable
• Intensities above absolute threshold: detectable
– Difference threshold (aka. just noticeable difference
/JND/ and difference limen):
minimum intensity difference that is noticeable to the
observer
• A change in intensity that is smaller than the difference
threshold: undetectable
• A change in intensity that is larger than the difference
threshold: detectable
Difference thresholds
Sensation magnitude
•
Linear
psychophysical
equation
•
constant
slope
Stimulus intensity
•
•
•
•
Linear function 
difference threshold (slope) is
constant
An observer able to detect the
difference between intensities
100 and 110 should also be
able to detect the difference
between 1000 and 1010. This
is not the case: the observer is
able to detect the difference
only between 1000 and 1100
500 & 550 Hz tones
5000 & 5050 Hz tones
5000 & 5500 Hz tones
Difference threshold is not
constant!
Difference thresholds
Sensation magnitude
•
Nonlinear
psychophysical
equation
slope changes
with intensity
•
•
•
Stimulus intensity
Difference threshold is not
constant (changes with
intensity)
 function is nonlinear
Weber’s law: difference
threshold is a constant
proportion of the initial
stimulus value
ΔI / I = c
Previous
examples:
c=10%
Weber’s law
holds only
approximately!
Absolute thresholds
• Even in the absence of stimulation, there is some
random firing on sensory nerves
• This inner noise can even vary from moment to moment
• Observers cannot distinguish inner noise from the effect
of a weak stimulus
• Even when there is no light (perfect darkness),
observers may experience a dim light (dark light, intrinsic
light)
• Observers in an anechoic chamber often report hearing
a whistling sound
 Measuring truly „absolute” thresholds is problematic:
observers may confuse inner noise with the real thing
Psychophysical methods
•
Threshold measurements: detection of small intensities (absolute thr.) and
discrimination of small intensity differences (difference thr.)
Is it intense enough to see? How small a difference can you see?
– Fechner’s 3 methods
• Method of constant stimuli
• Method of limits
• Method of adjustment
– Modifications of Fechner’s methods
• Staircase method
• Modifications of the method of constant stimuli (adaptive, no standard)
– Forced choice, objective methods
– Sensory decision theory (SDT)
– Psychophysical functions from psychometric data
•
Direct scaling: growth of sensation with intensity
How bright do you see a light?
– Magnitude estimation and the power law
•
Multidimensional scaling: degree to which stimuli are comparable along
some dimensions
Along which dimensions do you judge the similarity of two stimuli?
Fechner’s three methods
•
•
•
Absolute threshold
Difference threshold
Method of constant stimuli


Method of limits


Method of adjustment

not used
•
Presenting one stimulus at a time
The stimulus is very weak
Possible responses:
“Yes, I see it.” /
“No, I don’t see it.”
•
Presenting two stimuli at a time:
– Standard: fixed, easily
detectable
– Comparison: either more or
less intense than the standard
Possible responses:
“Comparison is stronger.” /
“Comparison is weaker.”
Method of constant stimuli for
measuring absolute thresholds
Light intensity
Weak
Strong
1. Select a range of light intensities from certainly
invisible to certainly visible
2. Pick a few (4-7) points uniformly in this
intensity range; this will be the constant
stimulus set
Method of constant stimuli for
measuring absolute thresholds
…
3. Test each stimulus many times (20-25) in
random order
Method of constant stimuli for
measuring absolute thresholds
Visible?
YES
NO
4. Present the stimuli one at a time and ask the
observer if it was visible or not
Click to start
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Could you see the spot of light?
Method of constant stimuli for
measuring absolute thresholds
0%
+
-
5%
+ +
+
20%
-
-
50%
-
+
80%
+
+
95%
100%
-
-
-
-
5. Calculate the proportion of “yes” and “no”
responses at each light level
+
Percentage “seen”
Method of constant stimuli for
measuring absolute thresholds
100%
75%
50%
25%
0%
Stimulus intensity
6. Plot the percentages against stimulus intensity
 psychometric function
Psychometric function for absolute
thresholds
Ideal
FIG (Sekuler)
• Fixed absolute threshold
• Step function
Actual
sigmoid function
FIG (Sekuler)
• Absolute threshold varies
somewhat from trial to trial
(due to constant fluctuations in
sensitivity)
• Conventionally, the intensity
corresponding to 50% is
considered to be the threshold
Method of constant stimuli for
measuring difference thresholds
Standard stimulus:
Comparison stimuli:
Light intensity
1. Standard stimulus has a fixed intensity
2. The intensities of comparison stimuli bracket
the standard
Method of constant stimuli for
measuring difference thresholds
3. All pairs of standard and comparison stimuli
are tested many times
Method of constant stimuli for
measuring difference thresholds
STRONGER
WEAKER
4. For each pair, the observer judges if the
comparison stimulus was stronger or weaker
than the standard
http://www.yorku.ca/psycho
Percentage “stronger”
Method of constant stimuli for
measuring difference thresholds
100%
75%
50%
25%
0%
Light intensity of comparison stimuli
5. For each comparison level, the percentage of
“stronger” responses is calculated and results
are plotted as a psychometric function
Percentage “stronger”
Psychometric function for
difference thresholds
100%
75%
50%
25%
0%
PSE
Light intensity of comparison stimuli
•
When the observer cannot see a difference, he/she
chooses randomly between “stronger” and “weaker”;
this corresponds to 50% on the psychometric function
 point of subjective equivalence (PSE)
Percentage “stronger”
Psychometric function for
difference thresholds
100%
75%
50%
25%
0%
DW
DS
Light intensity of comparison stimuli
•
•
•
By convention, the intensity at 75% is considered to be just
noticeably stronger than the standard  DS
A comparison intensity at 25% is just noticeably weaker than the
standard  DW
Difference threshold = the average of DS and DW
Method of limits for measuring
absolute thresholds
1.
+
Light intensity
+
+
2.
+
+
+
-
threshold estimate -
3.
4.
Trials
On each trial, the observer
reports if she/he could see
the light or not.
Start with presenting a light
intensity well above the
expected threshold (the
observer can certainly see it)
Decrease the intensity until
the observer cannot see it
Threshold estimate: the
intensity at which the
response changes
• Descending series: start from above the expected threshold and
decrease intensity
• Ascending series: start from below the expected threshold and
increase intensity
Click to start
Method of limits for measuring
absolute thresholds
+
+
Light intensity
•
+
+
+
+
+
+
+
-
-
+ •
+
-
-
-
-
-
-
-
Trials
•
Ascending and descending
series may yield different
results
 use both
Even in the same direction,
there is variability in the
threshold (inner noise, etc)
 average many
measurements
Measured threshold
corresponds to 50% point in
a psychometric function
(method of constant stimuli)
pure-tone audiometry
Light intensity of comparison stimulus
Method of limits for measuring
difference thresholds
+
+
+
+
+
+
+
threshold
+
estimate
-
+
+
+
+
-
-
-
Trials
+ comparison brighter
- comparison weaker
• Intensity of the
comparison stimulus is
decreased
(descending) or
increased (ascending)
until the response
changes
• Threshold estimate:
intensity difference
between the standard
and comparison stimuli
where the response
changes
• Average results from
multiple series in both
directions
Method of adjustment for
measuring absolute thresholds
• Observer can vary the
stimulus intensity
• Instructed to adjust it
so that it is just visible
or just invisible
• Initial intensity is set
to be far from the
expected threshold
value
adjustment
device
stimulus with
adjusted intensity
observer
Method of adjustment for
measuring absolute thresholds
threshold
estimate
descending
ascending
• Adjustment: by a real or a software
device (e.g. knob, slider)
• Threshold estimate: final
intensity value
• Descending: initial
intensity is well above
expected threshold;
adjusted to just visible
• Ascending: initially well
below threshold;
adjusted to just invisible
• Ascending and
descending task repeated
several times and results
averaged
• Similar to method of limits
but observers find it
easier