[From Rick Marken (2007.09.06.1030)]

It just struck me that one of the most mathematically precise

demonstrations of the effect of a person's purpose on the results of

an experiment is the difference between the Fechner's and Steven's

laws of psychophysics. According to Fechner, the relationship between

stimulus intensity (I) and sensation magnitude (R) is R = k log I.

According to Steven's the relationship is R = I^k. I refer to

sensation as R because the measure of sensation is based on the

subject's response. In Steven's case R is an actual numerical response

to a stimulus presentation; in Fechner's case R is derived from the

proportion of times the subject correctly reports the more intense

stimulus as being more intense.

What's interesting is that these two different laws -- which can be

viewer as disturbance-output functions from a control theory

perspective -- are obtained when subjects have different purposes

(different variables they control). In the Fechner studies what is

controlled is the relationship between a binary response ("more" vs

"less" intense) and the relative intensity of pairs of stimuli

presented on each trial. In the Stevens studies what is controlled is

the relationship between a numerical response and the intensity of

single stimuli presented on each trial.

Since, according to control theory, the functional relationship

between disturbance and output is the inverse of the feedback function

connecting output to controlled variable, the difference in the

Fechner and Stevens laws is a reflection of the difference in the

feedback functions in these two cases. And the feedback function is

different in the two cases because the controlled variables and the

outputs that affect the states of these variables are different in the

two cases.

I would love to be able to describe, quantitatively, the controlled

variables and the feedback functions in the Fechner and Stevens

experiments so I could show how control theory explains the difference

in the relationships found between input (I, which is actually a

disturbance variable) and output (R) found in the two cases. But I'm

finding this a difficult task. So that's why I'm posting this. I'd

love to get some help on this. I guess the first thing is to get

confirmation that my basic premise is true -- that Fechner's log law

and Steven;s power law are just the inverse of the different feedback

functions that exist in Fechner's discrimination approach to measuring

R versus Stevens' numerical judgment (he called it magnitude

estimation) approach.

Best

Rick

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Richard S. Marken PhD

rsmarken@gmail.com