Part 1
Psychometric Functions
Psychometric Functions
• A function is a rule for turning one number into
another number.
• In a psychometric function, we take one number
(e.g. a quantified stimulus) and turn it into
another number (e.g. the probability of a
behavioral response).
• By convention, the physical quantity is
represented on the abscissa, and the behavioral
response is represented on the ordinate.
Proportion of Chosen Response
(Psycho)
The Axes of a Psychometric Function
1
0.75
0.5
0.25
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Relative Stimulus Value
(Physics)
Part 4: Psychometric Functions
Linear Function =
Sigmoidal Function =
(Slope * X) + “Y-Intercept”
1_________________
1 + {( exp^ - Slope )^ - ( X - “X-Intercept”)}
Psychometric Functions
About Slope
About Slope
• Psychometric functions vary from each other in slope.
• Steeper slopes, better discrimination, lower thresholds:
Shallower slopes, worse discrimination, higher thresholds.
• If your slope is infinite (i.e., a step function), you have a
“ceiling effect”. Your task is too easy for the subject.
• If your slope is zero (i.e., a flat function), you have a “floor
effect”. Your task is too difficult for the subject.
• Intermediate slopes are desirable, and allow you to dismiss
objections that your subjects didn’t understand the task.
(Perceptual limits, not “Conceptual” limits)
Slope Indicates Discriminability
Proportion of Chosen Response
Step Function
1
0.75
0.5
0.25
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Relative Stimulus Value
Slope Indicates Discriminability
Step Function
Proportion of Chosen Response
Flat Function
1
0.75
0.5
0.25
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Relative Stimulus Value
Slope Indicates Discriminability
Step Function
Flat Function
Proportion of Chosen Response
Intermediate Slope
1
0.75
0.5
0.25
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Relative Stimulus Value
Psychometric Functions
About X-Intercept
About X-Intercept
• Psychometric functions vary from each other in X-intercept.
• The X-intercept is an index of bias, and an index of the
Point-of-Subjective-Equality (PSE).
• To the extent that the X-intercept departs from the center
of the abscissa (i.e., the center of the range of stimuli being
tested), there is bias.
• The PSE is equal to the abscissal value (i.e., the stimulus
quantity) that is associated with the 50% ordinal value (the
50% response rate).
Mid-Point Indicates Bias (or PSE)
Proportion of Chosen Response
No Bias
1
0.75
0.5
0.25
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Relative Stimulus Value
Mid-Point Indicates Bias (or PSE)
No Bias
Proportion of Chosen Response
Liberal Bias
1
0.75
0.5
0.25
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Relative Stimulus Value
Mid-Point Indicates Bias (or PSE)
No Bias
Liberal Bias
Proportion of Chosen Response
Conservative Bias
1
0.75
0.5
0.25
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Relative Stimulus Value
Psychometric Functions
About Goodness-of-Fit
About Goodness-of-Fit
• Psychometric functions vary from each other in
“goodness of fit”.
• To the extent data points (or their error bars) fall on or
near the psychometric function, the fit is good.
• The goodness of fit can be indexed by the correlation ( “r”
statistic) between the data and the function.
• If the fit (that is, the “r” statistic) is statistically greater
than the would be expected by chance ( p < 0.05 ), we can
be confident in estimating thresholds and P.S.E.’s from
them.
"r" Statistic Indicates Goodness-of-Fit
Proportion of Chosen Response
Perfect Fit ( r(8) = 1, p < 0.01 :
r^2 = 1 )
1
0.75
0.5
0.25
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Relative Stimulus Value
"r" Statistic Indicates Goodness-of-Fit
Proportion of Chosen Response
Acceptable Fit ( r(8) = 0.72, p < 0.05 :
1
0.75
0.5
0.25
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Relative Stimulus Value
r^2 = 0.53 )
"r" Statistic Indicates Goodness-of-Fit
Proportion of Chosen Response
Unacceptable Fit ( r(8) = 0.57, n.s. :
1
0.75
0.5
0.25
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Relative Stimulus Value
r^2 = 0.33 )
Class Data From A Lab Exercise
Proportion "Longer" Responses
1
When in doubt, say “Longer”:
slope = 1.8 arbitrary units
mid-point (PSE) = -0.23 secs
r statistic = 0.99
0.75
0.5
When in doubt, say “Shorter”:
slope = 2.4 arbitrary units
mid-point (PSE) = +0.13 secs
r statistic = 0.99
0.25
0
-1.5 -1 -0.5
0
0.5
1
1.5
Relative Duration (Seconds)
Learning Check
• For next class….
• On one plot, draw two psychometric functions that differ
from each other only in slope (i.e., discriminability).
• On another plot, draw two psychometric functions that
differ from each other only in mid-point (i.e., PSE).
• On a third plot, draw two psychometric functions that
differ from each other only in ‘goodness of fit” (r stat).