Understanding Psychometric Functions: Slope, X-Intercept, and Goodness of Fit

1. 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) = -0.23 secs r statistic = 0.99 When in doubt, say “Shorter”: slope = 2.4 arbitrary units mid-point (PSE) = +0.13 secs r statistic = 0.99

21. Learning Check • 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).