[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
···
--
Richard S. Marken PhD
rsmarken@gmail.com