# About Narens' paper on ratio estimation

I believe I now understand the basic point of the paper. Stevens' experiments asked subjects to make ratio judgements, i.e. given a stimulus x, to be deemed to have one unit of intensity, say for various stimuli y how many times y is greater than x. Narens' notation for this is (x,p,y), where p is the number named by the subject.

Stevens assumed (according to Narens) that the words uttered by the subject could unproblematically be understood as numerical estimates of the actual ratio of subjective intensities. Narens points out that this is an assumption that might not hold. In general, there will be some function f mapping a verbally expressed ratio p to the actual (but unobserved) number f(p) relating the subjective intensities.

Narens calls the scale implied by a subject's judgements multiplicative if (x,p,y) and (y,q,z) imply (x,pq,z). This is a property of the subject's verbal responses, and therefore can be observed to hold or fail to hold in the experimental data. But the unknown function f implies that even if the experimental data are multiplicative, that may not be true of the underlying objective relationships between intensities f(p), f(q), and f(pq). If f(p) = log(p), then instead we would have f(pq) = f(p) + f(q), giving an additive objective scale underlying the verbal scale. Narens only mentions multiplicative and additive scales (and unless I've overlooked it, manages not to mention the log function at all, although it must be lurking offstage). However, there are an unlimited number of other possibilities. Provided only that f is monotonic, we could define an operation * on real numbers by f(p)*f(q) = f(pq). When f is the identity, * is multiplication. When f is log, * is addition. If f(p) = tanh(p), then * would be the formula for relativistic addition of velocities.

Narens explores various assumptions that might be made to ensure that f(pq) = f(p)f(q), although the derivation is not deep. There seems to be an assumption that the objective relationships ought to form a multiplicative scale. That seems to me to be rather undercut by the paper of Torgensen that Narens cites. It reports that when subjects are asked to give differences instead of ratios, they generate additive scales, i.e. (x,p,y) and (y,q,z) imply (x,p+q,z).

···

--
Richard Kennaway, jrk@cmp.uea.ac.uk, http://www.cmp.uea.ac.uk/~jrk/
School of Computing Sciences,
University of East Anglia, Norwich NR4 7TJ, U.K.

Richard Kennaway
(2007.09.13)
[From Bill Powers (2007.09.13.1230 MDT)]

Thank you extensively for your reading of the Narens paper. In my
rejected Science paper so long ago, I mentioned the same possibility, of
a transformation that was undetectable because it applied to all
perceptions. But of course I hadn’t the math to make the idea rigorous.
This also applies to the log hypothesis of Weber-Fechner.

It’s a shame that people who publish long involved mathematical
treatments don’t make any effort to clarify what they’re talking about
for the sake of non-mathematicians. You have shown that it can be done,
if I may say, brilliantly. The adjective applies to both
“shown” and “done.”

I think you have cleared the way for the paper Rick and I are now just
about finished with. How about posting a version of your analysis on your
web page so we can cite it in the paper?

Best,

Bill P.

[From Bruce Abbott (2007.09.13.1535 EDT)]

In reviewing the literature for your
paper, I would suggest you have a look at Norman H. Anderson’s work on
subjective estimation (“Integration Theory”). A physicist
turned experimental psychologist, Anderson
provided some elegant demonstrations of both multiplicative and additive
relationships resulting from subjective estimation. I’m not sure, but
this might relate to the problem you have under consideration. A quick
Google search turned up a number of links.

Bruce

[From Bill Powers (2007.09.13.1230 MDT)]

Richard Kennaway (2007.09.13)

Thank you extensively for your reading of the Narens paper. In my rejected
Science paper so long ago, I mentioned the same possibility, of a
transformation that was undetectable because it applied to all perceptions. But
of course I hadn’t the math to make the idea rigorous. This also applies to the
log hypothesis of Weber-Fechner.

It’s a shame that people who publish long involved mathematical treatments don’t
make any effort to clarify what they’re talking about for the sake of
non-mathematicians. You have shown that it can be done, if I may say,
brilliantly. The adjective applies to both “shown” and
“done.”

I think you have cleared the way for the paper Rick and I are now just about
finished with. How about posting a version of your analysis on your web page so
we can cite it in the paper?

In reviewing the literature for
your paper, I would suggest you have a look at Norman H. Andersons work
on subjective estimation (Integration Theory). A physicist turned
experimental psychologist, Anderson provided some elegant demonstrations
of both multiplicative and additive relationships resulting from
subjective estimation. Im not sure, but this might relate to the problem
you have under consideration. A quick Google search turned up a
[From Bill Powers (2007.09.13.1406 MDT)]

Bruce Abbott (2007.09.13.1535
EDT)

Too hard to get full text. You have to be a member of something, not
free.

I found this diagram in a Wiki:

Doesn’t look very relevant to our paper.

Best.

Bill P.

[From Rick Marken (2007.09.13.1345)]

Stevens assumed (according to Narens) that the words uttered by the
subject could unproblematically be understood as numerical estimates
of the actual ratio of subjective intensities. Narens points out
that this is an assumption that might not hold. In general, there
will be some function f mapping a verbally expressed ratio p to the
actual (but unobserved) number f(p) relating the subjective
intensities.

It sounds to me like this mapping could be described as "the way the
subjects uses numbers". The function mapping p to the ratio of
intensities, f(p), describes how subjects assign numbers (in terms of
their ratio) to subjective intensities, f(p). Does that sound right to
you, Richard?

Best

Rick

···

On 9/13/07, Richard Kennaway <jrk@cmp.uea.ac.uk> wrote:
--
Richard S. Marken PhD
rsmarken@gmail.com