# Stevens' Power Law

[From Bill Powers (990928.0250 MDT)

This is for Mark (Isaac) Kurtzer, but I thought the whole group might be
interested.

Isaac, I'm finally getting around to reconstructing that argument with
Stevens' Power Law of perception. This isn't the one I was told about, but
an earlier one that I actually sent to Science years ago when Stevens
published his article there. The letter was rejected, on the basis of
Stevens' comment to the editors, relayed to me, that "logarithms have
nothing to do with it." Here, as best as I can recreate it, is my argument.

Let perception of any objective (physical) stimulus measure S be
proportional to the logarithm of S.

P = s*log(S)
where s represents the coefficient appropriate to the stimulus S.

One of the perceptions against which the apparent magnitude of stimuli is
measured is the perception of the magnitude of a _number_, N.

Let Pn = n*log(N), where
Pn = perception of magnitude of a number.
n = coefficient
N = number

For some other stimulus A, let

Pa = a*log(A)

The subject will generate a number such that its perceived magnitude is the
same as the perceived magnitude of the stimulus in question. If perception
Pa is judged equal to perception Pn, then

a*log(A) = n*log(N), or

a n
log(A ) = log(N ), or

a n
A = N , or

a/n
N = A

Thus the number in which the subjective magnitude of the stimulus is
expressed will be some power of the stimulus magnitude. This will apply to
changes as well.

Note that the _absolute_ perceptual law cannot be obtained this way.
Suppose that for all perceptions, there is a common function f applied to
the expression x*log(X) before perceptual comparison occurs. Thus we would
have

Pn = f(n*log(N)), and

Pa = f(a*log(A)).

If Pn = Pa, then

f(n*log(N)) = f(a*log(A)),

a/n
N = A.

QED.

Best,

Bill P.

[From Bill Powers (990928.1140 MDT)]
[For Isaac Kurtzer but copied for general interest]

Hi, Isaac --

Me:

Let perception of any objective (physical) stimulus measure S be
proportional to the logarithm of S.

P = s*log(S)
where s represents the coefficient appropriate to the stimulus S.

You:

i don't understand the first statement. It it worded the way you
intended.."stimulus measure S proportional to the logrithm of S"?
should'nt it be P is proportional to the log of S?

The sentence is "Let perception .... be proportional to the logarithm of
S." The prepositional phrase "of any objective physical stimulus measure"
refers to the subject, perception. You missed the "be" in the sentence, or
connected it to the wrong grammatical element.

after having read the following, which i am happy so say i followed
mathematically, i am not sure if i get the concepetual point. Is it if
two
signals are made to match that have similar input functions, logrithm vs
exponential, but different mulitpliers, then any common
relation between the two drops out and there are no terms that correspond
actual value of the signal P in expressing N.

That is it. Stevens thought he was measuring the true relationship between
the objective stimulus and the subjective perception, which is what
psychophysics is about. But he was only getting people to compare one
perception with another. This, of course, has also been a problem in all
other psychophysics experiments.

So if you are to lift one load 1 and then we'll call that one "UGA" heavy
then at first you can give very reliable assessments of other weights
according to a power law.

Let me reprise a bit and then carry the analysis a little further.

In the following, I'm going to assume that the magnitude of a quantitative
perception of any kind is equal to some constant times the log of the
actual magnitude of the stimulus. I will show that from this basic law,
which is like the Weber-Fechner law, we can show that the reported
magnitude of a stimulus, in numbers, will be proportional to a power of the
actual stimulus magnitude, the power being the constant just mentioned.

Let's say that perception of weight W is Pw, following the law

Pw = w*log(W).

Estimates of weight are given in numbers, N; the subjective magnitude of a
number N is, according to the same assumed law,

Pn = n*log(N)

Following the derivation in my previous post, when the subject vocalizes a
number N whose magnitude is perceived as the same as the magnitude of the
weight perception, we have Pw = Pn and therefore

w*log(W) = n*log(N), or

N = W^(w/n) (using ^ to indicate that the following expression is an
exponent).

This is the observed power law relating the reported numbers and the
perceived magnitude of the numbers Pn. The power in the exponent is w/n,
although we can't measure n and w separately.

At the same time, let Pe be the perceived effort that is related to the
signals entering the muscles (whether derived from reafferent signals or
from other sensory data). The new equation is

Pe = e*log(E),
where E is the actual effort.

Note that if (for example) effort were sensed from Golgi tendon organs and
weight were sensed from skin-pressure sensors, the physical stimuli for Pu
would be different from those for Pe, and we would expect different
calibration factors to apply. The measurement of E in particular entails an
unknown calibration factor.

To continue, we now have

Pe = e*log(E) and
Pn = n*log(N).

When the subject vocalizes a number N whose magnitude is perceived as the
same as the magnitude of the effort perception, we have Pe = Pn and therefore

n*log(N) = e*log(E), or

N = E^(e/n).

Note that log(E) is a constant times log(N): therefore a log-log plot of E
versus N will be a straight line of slope e/n. Likewise, a log-log plot of
W versus N will be a straight line with a slope of w/n. You mention that
these plots, under certain conditions, will be parallel to each other,
implying that the two straight lines are separated from each other.

If they are separated but parallel, we can say two things: first, the
parallelism implies that (e/n) = (w/n) and hence that e = w, and second,
that one of the plots must have a constant added or subtracted to make it
coincide with the other plot. Let's see how this would be done.

n*log(N) = e*log(E)

Construct a constant C such that e times the log of the constant is equal
to the separation of the two parallel lines, and add this quantity to the
right side:

n*log(N) = e*log(E) + e*log(C), or

n*log(N) = e*[(log(E) + log(C)], or

n*log(N) = e*log(C*E)], or (to skip a couple of steps)

N = (C*E)^(e/n).

Let's call the two reported numbers Ne and Nw, for effort and weight. They are

Nw = W^(w/n), and
Ne = (C*E)^(e/n).

The constant C, if calculated as above, makes Ne equal to Nw, so

W^(w/n) = (C*E)^(e/n)

The exponent 1/n drops out, so we have

W^w = (C*E)^e, or

W = (C*E)^(w/e)

Note that perceived weight and perceived effort are also related by a power
law. However, if E is calibrated as suggested, w = e, and we get

W = C*E

The weight is some constant times the effort.

This would make sense if the perceived effort were calibrated so it is
equivalent to the upward force being applied at the load. Being sensed at
some place other than at the load, it would not be objectively of the same
8:1, an actual effort sensed at the tendon would be 8 times the force
applied at the load, but of course the magnitude of the aggregate sensory
signal would depend on the number of sensors and their individual
sensitivities. There is no direct way to determine C.

Now to the final problem: what happens after exercise fatigues the system.
What you report as being observed is that the ratio of perceived Effort to
perceived Weight increases, while both Effort and Weight perceptions increase.

Fatigue, it is known, reduces the response of muscles to driving signals.
If actual tension in a tendon were being sensed, there would be no
difference in sensed effort, because actual muscle forces are simply
increased as required to hold the load in position. The driving signals,
however, would be larger because of the reduced muscle response. Thus this
increase in the ratio of perceived effort to perceived weight might be
explained if effort is sensed in the imagination mode -- that is, if the
driving signals are sensed via what have been called re-afferents,
perceptual signals derived directly from motor output signals rather than
from their effects. I believe you said that this is the currently accepted
explanation.

The other possibility is that sensed Weight might be decreased due to
habituation of skin-pressure sensors (or whatever sensors are involved).
This would not, however, affect muscle tension because the load is still
maintained in the required position. Thus it appears that the present
analysis supports the explanation that effort is perceived via
re-afferents, or some equivalent path.

How does this square with the concurrent increase in sensed weight? The
implication is that sensors directly affected by the weight of the load
become _more_ sensitive after repeated lifts. But this is not the usual
effect of habituation; the usual effect is in the opposite direction.

One answer might come from the process you call "hefting" a load:
accelerating it up and down as a way of measuring its mass (as opposed to
its weight). If the muscles fatigue, they will be less effective in
accelerating the load, a phenomenon which would also result if the muscles
remained as sensitive as before but the load's mass increased. If the brain
assumes constant muscle strength, it will perceive an increase in mass and
hence infer an increase in weight.

if the procedure is not telling you the absolute relation than what is it
conveying?

It's telling you about a relationship between two perceptions, rather than
a relationship between an objective stimulus and a perception. You start by
estimating the number of EFFs per UGA. Then you fatigue the system, and
estimate the number again. The ratio has changed. But you still don't know
how big an EFF is in subjective units. You're saying that one perception
has changed relative to another one, or perhaps relative to its previous
value. That's all you can say without physically measuring impulses per
second over all pathways affected by a given stimulus. Even that would
leave open the question of just how big X impulses per second appears to
the rest of the brain.

Best,

Bill P.

[Martin Taylor 990929 17:49]

Bill Powers (990928.0250 MDT

I've just downloaded four months of CSG messages, which is why you may
have thought you lost me--no such luck (for you). I've been away a lot
and over-busy when not away. And that is likely to continue. But I've
been trying to carry the PCT torch, and indeed there are now two separate
research groups now who are including PCT as a core aspect of their work.
One is concerned with developing intelligent aircraft cockpits,
and the other with team behaviour. So I'm not lost--just incommunicado
for a while.

Sorry, but I may not see any comments on this message for a while. I'm
just away again for a couple of weeks on Saturday.

···

-----------------

In two messages, Bill talks about the power law as matching one perception
against another. 40 years ago I did an experiment that showed (at least to
my satisfaction) that this is a bit too simple. What I did was to take a
whole host of types of perception--colour, grey level, time interval,
placement of a nonsense syllable in a partly learned list, angle of a
pointer... and ask people to set an item in one dimension to match a
presented item in another. The core question is "does it matter which
dimension the subject controls?"

It does matter which the subject controls. If I ask the subject to set
an item in dimension B to equal an item in dimension A, and do this a lot
of times, I can get a mapping of A values onto B values. That's what
Stevens did, for lots of such pairs of dimensions. But if I then ask
the subject to set an item in dimension A equal to a presented item in
dimension B, I can get another mapping. If the subject is simply equating
two perceptions whose values are set by the "stimulus" the two mappings
whould be identical. They aren't. There's a characteristic difference
between A->B and B->A. The magnitudes on the ->X side seem to be a bit
more linear than on the X-> side.

Now, if we look at another set of mappings: A->B->X and A->C->X, we can
map A onto X by two routes, through the mappings of A onto B and C, and
of B and C onto X. These mappings of A->X look alike. So do the mappings
X->B->A and X->C->A. Both ways, A and X have the same property of being
presented or being controlled in the two mappings.

The bottom line for me, then and now, is that how you perceive something
(quantitatively) depends on whether you are controlling its value.

Unfortunately, before you ask, this stuff was never published. It was a
summer student project and was not finished by the end of the summer.
However, the result that the magnitude you assign to a perception is
influenced by whether you are controlling that magnitude is, I think,
pretty solid.

Martin

[From Rick Marken (990930.1450)]

[Martin Taylor (990929 17:49) --

If I ask the subject to set an item in dimension B to equal
an item in dimension A, and do this a lot of times, I can get
a mapping of A values onto B values...But if I then ask
the subject to set an item in dimension A equal to a presented
item in dimension B, I can get another mapping.

This is a surprising result! What you are saying is that if
I adjust, say, perceived light intensity (Pa) to match perceived
tone intensity (Pb) I get a different relationship between
A and B (physical light and tone intensity, respectively)
than I would if I adjust perceived tone intensity to match
perceived light intensity. In both cases, the expected (and
observed, I thought) relationship is:

a/b
B = A

How different is the relationship when A is adjusted to make
Pa = Pb versus when B is adjusted to make Pa = Pb?

The bottom line for me, then and now, is that how you
perceive something (quantitatively) depends on whether
you are controlling its value.

If this is a real result (and I'm prepared to believe that
it is) then yours is a possible explanation of the result.
But it seems implausible to me; you are saying that we
perceive, say, light intensity differently depending on
whether we are controlling it's value (by adjusting A)
or not. There must be some more plausible possibilities.

Best

Rick

···

---
Richard S. Marken Phone or Fax: 310 474-0313

[Martin Taylor 9901001 08:16]

[From Rick Marken (990930.1450)]

[Martin Taylor (990929 17:49) --

If I ask the subject to set an item in dimension B to equal
an item in dimension A, and do this a lot of times, I can get
a mapping of A values onto B values...But if I then ask
the subject to set an item in dimension A equal to a presented
item in dimension B, I can get another mapping.

This is a surprising result! What you are saying is that if
I adjust, say, perceived light intensity (Pa) to match perceived
tone intensity (Pb) I get a different relationship between
A and B (physical light and tone intensity, respectively)
than I would if I adjust perceived tone intensity to match
perceived light intensity.

Yes. That's what I am saying.

In both cases, the expected (and
observed, I thought) relationship is:

a/b
B = A

How different is the relationship when A is adjusted to make
Pa = Pb versus when B is adjusted to make Pa = Pb?

In most of my dimensions (not all) I used interpolated distance between
two end points, rather than from one end point only, as is the case with
unbounded estimates of tone intensity or light intensity. What I was
interested in at the time was the effect of anchor points on the
perception of magnitude. Near an anchor point distances (magnitudes of
perceived distance/size/colour difference/time... are expanded--perhaps
logarithmically). I wanted to test whether the unanchored perception of
the controlled comparison "stimulus" was the same as for that "stimulus"
when it was fixed outside of the subject's ability to control.

What I got can't therefore be quoted directly in the form you ask. The
effect, which I studied further in my thesis for geometric distance
(Perceptual and Motor Skills 1961 or 62--you could probably find it in
a library), is that the controlled value acts as if it is in a more
curved perceptual space than the comparison value. That jibes with
what I got with cross-dimension measures such as "tell me the nonsense
syllable that was as far along the list you learned as the middle
grey was between the black and the white" compared with "show me the
grey that is as far between black and white as JEZ was along the list
you learned."

I might hazard a guess at an answer to your direct question, and it is
only a guess--there would be a power law relationship in each direction,
but the exponents would be slightly different, perhaps slightly enough
that if you were not expecting it you would not test to see whether it
was a reliable difference.

The bottom line for me, then and now, is that how you
perceive something (quantitatively) depends on whether
you are controlling its value.

If this is a real result (and I'm prepared to believe that
it is) then yours is a possible explanation of the result.
But it seems implausible to me; you are saying that we
perceive, say, light intensity differently depending on
whether we are controlling it's value (by adjusting A)
or not. There must be some more plausible possibilities.

The possibility is that the value of a perception is affected by its
local context, and that when you are manipulating a value it does not
serve as a fixed reference magnitude within the context.

In the later thesis work, I asked people to put a dot on a blank 3x5 index
card just where a dot had been on one shown either immediately previously
or even simultaneously. There was a very consistent pattern of bias
in the placements. In one of the experiments, the "blank" card had a
dot already on it somewhere in the central region. On those cards, the
replaced dot position showed little or no bias--a dramatic difference.
The argument was that the fixed dot served as an anchor point, but the
pencil tip under the control of the subject did not, until the "response
dot" had been placed and was no longer under the subject's control. At
that point, many subjects almost immediately said "that's wrong", and
I allowed them a second try--which resulted in a placement biased
exactly like the first (I mean the scatter of second dot placements
was centred around the first dot placement). The results of several
separate experiments could all be accounted for by the assumption of
a curved relationship between physical and perceived distance from
any available anchor point--if and only if the controlled pencil point
did not serve as an anchor point itself.

During the thesis work, I also used tactile distance, but that wasn't
published then. It became part of a study with Susan Lederman published
in Perception and Psychophysics probably around 1967-73 time frame
called "Perception of interpolated position by active touch" or something
like that. In that, I had a 5 inch brass rod on which a small solder
lump had been placed. The subject had to put a dot on an index card
at a location corresponding to the location of the lump on the rod
(which the subject couldn't see). The result was much the same, perhaps
a bit exaggerated if I remember aright.

One way of looking at it may be that a controlled variable is perceived
as less embedded within its context than is a variable that is simply
presented (as a disturbance). I know that is a very imprecise--one might
say waffled--way of saying something that isn't an explanation. It's
a way of putting words to a subjective impression.

Martin

[From Bruce Nevin (991001.1103 EDT)]

Martin Taylor 9901001 08:16 --

One way of looking at it may be that a controlled variable is perceived
as less embedded within its context than is a variable that is simply
presented (as a disturbance).

This is reminding me (maybe wrongly) of this exchange from last April:

............................. begin quote .............................
Dick Robertson (980416.1319CDT)--

Subject: example of conflict
Bruce Nevin (980415.1517 EDT)--

I experienced an amusing example of conflict this morning.

I've been finishing the shingling on the front of my house. The scaffolding
is supported on pump jacks attached to 4x4 columns that are set off from
the roof overhang at the gable end so they stand about 4 feet away from the
front wall of the house. I had a cup of liquid that I was throwing out into
the bushes beyond this. In the direction I was throwing were three objects
I did not want to hit, the column, the edge of a trash can beyond the
corner of the house, and the edge of a ladder leaning up against the scaffold:

Trash

can
\ Column
* |
> * /
> * I was standing in the doorway and had a choice
> to pitch it between the column and the trash
> can, or between the column and the ladder
me/ leaning up against the scaffold. The three
> objects were about equidistant by line of sight.
> I was looking for the widest aperture, couldn't
decide, and threw the liquid with most exquisite
accuracy onto the 4x4 column!

This conflict would have been resolved, I feel, if I were not under time
pressure. The telephone was ringing. This raised the gain on "get rid of
what's in that cup" at the expense of "decide which side of the column to
throw it".

Urgency is involved in many a conflict.

Bruce Nevin

This is a neat story, and I think it is illustrative of how many accidents
occur. It reminds me of an incident many years back when Bob McFarland and I
were at a AAAS convention in Denver and it started to snow heavily in Aspen for
the first time that season. We ditched the convention and hiked out there to
discover hardly anyone there. That afternoon on Big Burn (a slope a couple
miles long and maybe almost a mile wide, for you non-skiers) we stopped half
way down to rest and marveled to note that we were the only people there. Just
then I noticed a single skier top the rise way above us. We continued talking
and a couple minutes later he wiped Bob out. He got up awkwardly, apologized
and slunk off.

I tried to understand with PCT how a guy could do that with a couple square
miles of empty space around him, and came up with a thought and an observation.

The thought was that, as a novice, which he clearly seemed to be, he might have
been saying to himself, "I have to stear clear of those guys," all the while
keeping his eye on us. And his feet went where his eye was aiming. The
observation I have made on myself many times, that suggested this
interpretation, is that when (e.g.) I hit the ball out of the court, it ends up
where I have been looking.

I wonder, Bruce, if you can recall where you were looking as you hastened to
depart?

Best, Dick Robertson

.............................. end quote ..............................

BTW, for the way I found that msg, check out this cool tool. I've been
trying out the free download. If you ever say "let me see, where did I put
that message where so-and-so said XYZ?", this is the ticket. Pretty cool.

Bruce Nevin

............................. begin quote .............................

···

At 08:48 AM 10/01/1999 -0400, Martin Taylor wrote:
Date: Tue, 21 Sep 1999 09:52:43 -0600 (MDT)
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.............................. end quote ..............................

[From Bill Powers (991001.1415 MDT)]

Martin Taylor 990929 17:49--

In two messages, Bill talks about the power law as matching one perception
against another. 40 years ago I did an experiment that showed (at least to
my satisfaction) that this is a bit too simple. What I did was to take a
whole host of types of perception--colour, grey level, time interval,
placement of a nonsense syllable in a partly learned list, angle of a
pointer... and ask people to set an item in one dimension to match a
presented item in another. The core question is "does it matter which
dimension the subject controls?"

It does matter which the subject controls. If I ask the subject to set
an item in dimension B to equal an item in dimension A, and do this a lot
of times, I can get a mapping of A values onto B values. That's what
Stevens did, for lots of such pairs of dimensions. But if I then ask
the subject to set an item in dimension A equal to a presented item in
dimension B, I can get another mapping. If the subject is simply equating
two perceptions whose values are set by the "stimulus" the two mappings
would be identical. They aren't. There's a characteristic difference
between A->B and B->A. The magnitudes on the ->X side seem to be a bit
more linear than on the X-> side.

Fascinating. Of course it's hard to tell which side is changing -- when you
say one direction gives a somewhat more "linear" mapping, this could be due
to a change in the linearity of either perception or both perceptions
relative to the objective measures of the stimuli. Can you give an estimate
of the size of this effect?

Also, if one perception is being used as the standard and the other is
being matched to it, the relation between the two perceptions has different
exponents, doesn't it? That is, suppose

p1 = a1*log(S1) and

p2 = a2*log(S2).

Then, when p1 is judged equal to p2, we have

S1^a1 = S2^a2, or

S1 = S2^(a2/a1).

when S2 is used as the standard and S1 is varied to match it. However, we have

S2 = S1^(a1/a2)

when S1 is used as the standard and S2 is varied to match it.

I obviously haven't worked any of this out in detail, but there may be some
effect like that of calculating percentages of changes. A measurement of 3
units is 60% larger than a measurement of 2 units, but a measurement of 2
units is only 33.3% percent smaller than a measurement of 3 units. The
apparent relative size of the change in the controlled variable depends on
which variable it is compared with.

It's hard to guess offhand the effect of averaging nonlinear measures. I
presume that in the experiments in which one perception was controlled
while the other became, in effect, the reference value, you started the
variable perception both above and below the value of the other, to
randomize effects of the direction of change. If the initial "high" values
were symmetrical with the initial "low" values about the value of the
stationary variable, but the subjective judgement depended on the relative
amounts of change, delta-v/v, there would be a systematic difference of
judgements. I wouldn't want to guess off the top of my head how this
would affect the comparisons of which you speak, but if you're still
interested after 40 years you might want to give the analysis here another
look with this in mind.

Best,

Bill P.