1. Psychometric Functions
Part 1
Psychometric Functions

2. 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.

4. Part 4: Psychometric Functions
Linear Function =
(Slope * X) + “Y-Intercept”
1_________________
1 + {( exp^ - Slope )^ - ( X - “X-Intercept”)}
Sigmoidal Function =

5. Psychometric Functions
About Slope

6. 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)

10. Psychometric Functions
About X-Intercept

11. 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).

15. Psychometric Functions
About Goodness-of-Fit

16. 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.

20. Class Data From A Lab Exercise
When in doubt, say “Longer”:
slope = 1.8 arbitrary units
mid-point (PSE) = secs
r statistic = 0.99
When in doubt, say “Shorter”:
slope = 2.4 arbitrary units
mid-point (PSE) = secs
r statistic = 0.99

21. 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).