# Perceptual distance measures

[Martin Taylor 2007.09.21.17.23]

In light of the recent discussion about power-laws and such, I
thought you might be interested in the attachment. I mentioned it in
one of my posts, to argue that no perceptual distance measure could
properly be obtained, since additivity simply doesn't apply.

The demonstration in the attached paper is due to W. P. Tanner, who
never published it, but who encouraged me to do the experiment and
report the results. It's easy to set up, and if you doubt the results
you can set it up for yourself.

The one-page paper sets up a condition in which two physically equal
distances AA' and BB' are perceived as different, AA' being seen as
longer than BB' (they are the distances between the arrow tips of the
Mueller-Lyer illusion in its outward and inward pointing forms, but
the intervening line is not drawn).

AA' and BB' are subdivided, physically identically, so that AA1 =
BB1, A1A2 = B1B2, ... AnA' = BnB'. Now we ask which is seen as
larger, AjAk or BjBk.

The result is counterintuitive (to me, at least). If Abs(j-k) = 1,
then BjBk is seen as longer than AjAk, for all j, k > 0. The
exception is for distances AA1 vs BB1 and AnA' vs BnB', where the "A"
versions are longer (in agreement with the distances AA' vs BB'). But
for Abs(j-k) > 1, the greater Abs(j-k), the less the inverse
illusion, and the more likely it is that the A form will be seen as
longer.

Visually (I'm not goint to draw the arrows, just the layout of the
dots, which I'll call X, X1...Xn, X'. You can substitute A or B.
Physically they are the same distances. X and X' are the arrow tips
of the illusion figure.

. . . . . . .
X X1 X2 X3 X4 X5 X'

Physically, X1X5 = X1X2 + X2X3 + X3X4 + X4X5

Perceptually, A1A2 < B1B2, ...., A3A4 < B3B4, so A1A5 should be seen
as shorter than B1B5, but the reverse is true. The sum of the shorter
distnaces is perceived as longer than the sum of the lonfer distances.

If you don't believe it, give it a try. I can't guarantee it will
work for you since you've been warned and some people can overcome
such illusions by using an analytic approach. But it worked for my
naive subjects, and it probably will work for you.

Anyway, the point is that for perceived distances at least, it is
impossible to infer anything about the perceptions of large
magnitudes from the combination of perceptions of small
magnitudes...which means that work such as that of Narens that
Richard explained [Thu, 13 Sep 2007 18:15:58] may b mathematically
interesting, but is psychologically pointless.

Martin

[From Rick Marken (2007.09.21.1925)]

Martin Taylor (2007.09.21.17.23)--

In light of the recent discussion about power-laws and such, I
thought you might be interested in the attachment.

For some reason I wasn't able to look at at the jpg's. But it sounds
very interesting. But it doesn't seem to have anything to do with what
we were talking about in terms of power laws. If I understand you
correctly, what you find is that you are showing that the Muller-Lyer
illusion seems to work the opposite way on the lines connecting the
"wings" or "feathers" than on subsegments of the lines. That actually
doesn't seem that surprising to me if what the subject is judging when
comparing the relative lengths of subsegments is the ratio of the
length of the subsegment to the apparent length of the whole line. The
subsegments in the lines that are perceptually lengthened by the
illusion will thus appear slightly shorter than the subsegments in the
lines that a perceptually shortened by the illusion.

Maybe you could try sending the jpgs again? I'd like to see it. Though
I guess I could draw it pretty easily. Actually, I just drew it and
didn't see what you describe. Maybe I'm not getting it right. If
anything, the subsegment in the "long" line seems longer than the
equivalent lenfgth subsegment in the "short" line.

Best

Rick

···

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

Re: Perceptual distance
measures
[Martin Taylor 2007.09.22.00.16]

[From Rick Marken (2007.09.21.1925)]

Martin Taylor (2007.09.21.17.23)–

In light of the recent discussion about power-laws and such,
I

thought you might be interested in the attachment.

For some reason I wasn’t able to look at at the jpg’s. But it
sounds

very interesting. But it doesn’t seem to have anything to do with
what

we were talking about in terms of power laws.

I think it does. It says that NO laws of that kind can be internally
consistent. I don’t know why you couldn’t see the jpg. It’s simply a
scan of the text of the paper (one page). I append it to the bottom of
this message rather than attaching it.

If I understand you

correctly, what you find is that you are showing that the
Muller-Lyer

illusion seems to work the opposite way on the lines connecting
the

“wings” or “feathers” than on subsegments of the
lines.

I suppose that’s one way of putting it. But that’s not the point.
The point is that if you have perceptual extents a1, a2, …, an and
b1, b2, … bn, in which aj > bj for all j, then a1+a2_…+an
should be greater than b1+b2+…bn, if the idea of a scale relating
perceptual length to physical length has any meaning. But in fact
a1+a2+…+an < b1+b2+…+bn.

That actually

doesn’t seem that surprising to me if what the subject is judging
when

comparing the relative lengths of subsegments is the ratio of the

length of the subsegment to the apparent length of the whole line.
The

subsegments in the lines that are perceptually lengthened by the

illusion will thus appear slightly shorter than the subsegments in
the

lines that a perceptually shortened by the illusion.

Yes, that’s my interpretation of why it happens, too. But I must
say that I did find it surprising when Spike Tanner showed it to me. I
didn’t expect teh effect a priori, and whether our interpretation is
right or not, it still demonstrates that the idea of a physical to
perceptual scale of length is itself an illusion.

In the experiment, I didn’t have the viewer look at both
Mueller-Lyer configurations at the same time. The task was to adjust a
line to appear the same length as the interval in the illusion figure.
Afterward we looked to see which match (to what should have been the
same distance) was longer.

Maybe you could try sending the jpgs
again? I’d like to see it. Though

I guess I could draw it pretty easily. Actually, I just drew it
and

didn’t see what you describe. Maybe I’m not getting it right. If

anything, the subsegment in the “long” line seems longer
than the
equivalent lenfgth subsegment in the
“short” line.

That’s another interesting point about a lot of these illusions,
which I mentioned in my first posting – when you are aware of what to
expect, the illusion goes away, or at least that’s what I find for
myself. I’m not sure where this fits with PCT, but it does seem that
different perceptual attitudes (for want of a better word) can change
perception in some conditions. However, that problem doesn’t arise in
the experiment, the way the figures were separately presented.

I hope you can read the paper this time.

Martin

[From Rick Marken (2007.09.22.0940)]

Martin Taylor (2007.09.22.00.16) --

It says that NO laws of that kind can be internally consistent. I don't know why you
couldn't see the jpg.

Actually, I did see the jpg of the paper. There were two jpgs attached
so I thought that one was an image of the illusion itself. I read the
paper but I can't seem to get my mind around what was done, what the
data mean or why you conclude what you conclude. Maybe if you could
explain what is in the table. You mention seven consecutive distances
in the paper, for example, but then the table has only 5 dot to dot
intervals in the table.

I also don't understand what you mean by an "internally consistent"
law relating stimulus to perceptual magnitude or why you conclude that
there can be no such law based on your results. I see the power law as
one possible mapping of stimulus magnitude to perceptual magnitude
(perceptual magnitude being the intensity level of the perceptual
hierarchy in PCT), the other main contender being the log law. Bill
Powers noted that the apparent power law could be explained by
assuming that all perceptual magnitudes, including the perceptual
magnitude of the numbers used in magnitude estimation tasks, are a log
function of stimulus magnitude. I realized that this was a nice
illustration of the behavioral illusion with power law being the
inverse of the log perceptual function of number, which is the
feedback function in the magnitude estimation experiment.

Does your experiment relate to that observation in some way.

Best regards

Rick

···

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

Bill Powers noted that the
apparent power law could be explained by

assuming that all perceptual magnitudes, including the perceptual

magnitude of the numbers used in magnitude estimation tasks, are a
log

function of stimulus magnitude.
[From Bill Powers (2007.09.22.1120 MDT)]

Rick Marken (2007.09.22.0940) –

A slight correction: I showed that IF perception follows a log law, that
is consistent with Stevens’ power law given that the sizes of numbers are
also perceived as logs. Whether any of these laws actually exists depends
on how well the data support the mathematical forms proposed. I have no
idea what the quality of these purported facts is.

Best,

Bill P.

[From Rick Marken (2007.09.22.1100)]

Bill Powers (2007.09.22.1120 MDT)

>Rick Marken (2007.09.22.0940) --

> Bill Powers noted that the apparent power law could be explained by
> assuming that all perceptual magnitudes, including the perceptual
> magnitude of the numbers used in magnitude estimation tasks, are a log
>function of stimulus magnitude.

A slight correction: I showed that IF perception follows a log law, that is
consistent with Stevens' power law given that the sizes of numbers are also
perceived as logs.

Right. That's what I meant when I said that it could be explained by
_assuming_ a log perceptual function: _If_ the perceptual function is
log then the power function is predicted.

By the way, your posts on libertarianism have been wonderful (that is,
_assuming_ -- that is, IF -- one is in the in an ideological position
to read them without treating them as a disturbance;-). Thanks!

Very best

Rick

···

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

[From Rick Marken (2007.09.22.0940)]

Martin Taylor (2007.09.22.00.16) --

It says that NO laws of that kind can be internally consistent. I don't know why you

> couldn't see the jpg.

I also don't understand what you mean by an "internally consistent"
law relating stimulus to perceptual magnitude or why you conclude that
there can be no such law based on your results. I see the power law as
one possible mapping of stimulus magnitude to perceptual magnitude
(perceptual magnitude being the intensity level of the perceptual
hierarchy in PCT), the other main contender being the log law. Bill
Powers noted that the apparent power law could be explained by
assuming that all perceptual magnitudes, including the perceptual
magnitude of the numbers used in magnitude estimation tasks, are a log
function of stimulus magnitude.

I know. That was the point of bringing this old experiment into play. It says that NEITHER of these, nor any other of a similar kind, can be correct. Both say that the sum of items of type A, all of which are greater than the corresponding items of type B, should be greater than the sum of the items of type B. That isn't true in this situation, so both laws fail, as do all others that suggest that perceptual magnitude is a function of stimulus magnitude.

I realized that this was a nice
illustration of the behavioral illusion with power law being the
inverse of the log perceptual function of number, which is the
feedback function in the magnitude estimation experiment.

Does your experiment relate to that observation in some way.

Only in that the "observation" is of a universal mathematical truth, whereas the experiment relates either side of the equation to observations of what people do. You have no evidence at all that the perceived magnitude of a number is a strict log function of its numerical value. I suspect that it isn't, and that if there is such a function relating perceived magnitude to number value, it may well not even be monotonic.

(As for the two-jpg problem, soorry about that, but I know of nothing I can do about it. I've noticed that a second file that is nonsense to PCs is sometimes attached to my mails. I suspect it may be the "rsrc" part of the Mac file. It never seems to matter, apart from confusing PC users who think it's something they should be able to see. From my end, I can't see when that happens. I know about it only when someone tells me there was an extra attachment they couldn't open, as you did.)

Martin

[From Rick Marken (2007.09.22.1140)]

I know. That was the point of bringing this old experiment into play.
It says that NEITHER of these, nor any other of a similar kind, can
be correct. Both say that the sum of items of type A, all of which
are greater than the corresponding items of type B, should be greater
than the sum of the items of type B. That isn't true in this
situation, so both laws fail, as do all others that suggest that
perceptual magnitude is a function of stimulus magnitude.

Then how would you explain the power law relationship that is observed
between stimulus magnitude and magnitude estimates?

I think the finding in your experiment is very interesting but I don't
see that it implies that there is no orderly functional relationship
between physical stimulus magnitude and the perception of that
magnitude. The violations of additivity could be the result of the
fact that the subjects were controlling different types of
perceptions.

Maybe you could give an explanation of what actually happened in your
experiment another try.

Best

Rick

···

On 9/22/07, Martin Taylor <mmt-csg@rogers.com> wrote:
--
Richard S. Marken PhD
rsmarken@gmail.com

## Martin Taylor (2007.09.22)

That isn’t true in this
situation, so both laws fail, as do all others that suggest that
perceptual magnitude is a function of stimulus
magnitude.
[From Bill Powers (2007.09.22.1230MDT)]

I suspected the same thing long ago, but this doesn’t mean that the
actual perceptual magnitude is not any reliable function of stimulus
magnitude. It just says that we can’t measure the function by any of the
means so far tried (including yours of adjusting a line length to match
the perceived length of the intervals).

What I didn’t get from your paper was any indication of how repeatable
the measures were. With only two trials per subject, I don’t see how you
could get p < 0.005 for the first interval, unless you were averaging
the interval across different subjects. If you were averaging across
subjects, you weren’t getting the right measurement for ANY subject, and
of course you have no way of knowing how much variability there was in a
single subject’s estimates. Maybe the comparison you were looking for is
difficult to make accurately.

I don’t suppose that you allowed the subjects to adjust the line used for
measurement by holding it up to the figure with the dots on it. Doing it
that way would have resulted in no errors at all to speak of. So how did
they make the comparison?

The reason I am a little reluctant to accept the results at face value is
that in modeling tracking data, I assume a linear proportionality between
the actual and perceived positions of a target and ditto for the
perceived position of the cursor, and I get fits within 5% rms (sometimes
2%) for the whole minute’s run. Perhaps because the reference condition
is “zero distance” between target and cursor, the scaling is
rendered unimportant, but it seems to me that the loop gain would be
varying a lot (if your data are correct), which would lead us to expect a
rather poor fit of the model to the data. One thing I am sure of – the
slopes are monotonic. If they ever reversed there would be amplitudes
where positive feedback existed, and that is not observed.

You wouldn’t happen to still have the raw data, would you?

Best,

Bill P.

Then how would you explain the
power law relationship that is observed

between stimulus magnitude and magnitude
estimates?
[From Bill Powers (2007.09.22.1300 MDT)]
Rick Marken (2007.09.22.1140) –
One explanation might be that there is no power law or log law
“observed” between stimulus and magnitude estimates. Instead,
what we may have is an arbitrary fit of a curve to a plot with a very
large scatter. We need to see the original raw data that Fechner/Weber
used, and also that Stevens used. Just how constant was the JND in the
original finding that led to a log relationship? Could it be that, as
Martin is implying, the aggregate data include lots of different
laws which, when averaged together, look a little like a log or a power
relationship?

I agree with you that there is probably an orderly relationship between
perception and the outside world. But measuring it may not be as simple
as some people have assumed it would be.

Best,

Bill P.

## Martin Taylor (2007.09.22.00.16)

[From Bill Powers (2007.09.22.1310 MDT)]

Here is a figure from an experiment with vision:

A pure Weber-law response would be a straight horizontal line: constant
JND. Obviously, this is approximated only in the center part of the plots
(right-hand curve). I don’t have the original data – it would cost me
\$15 to see the original paper, which says I value knowing the result at a
smaller figure.

Best,

Bill P.

[Martin Taylor 2007.09.22.16.22]

[From Rick Marken (2007.09.22.0940)]

paper but I can't seem to get my mind around what was done, what the
data mean or why you conclude what you conclude. Maybe if you could
explain what is in the table. You mention seven consecutive distances
in the paper, for example, but then the table has only 5 dot to dot
intervals in the table.

There were six dots between the arrowtips, which makes seven intervals. But there are five intervals between the outermost dots

X . . . . . . X

I considered intervals that included an arrowtip separately from intervals that didn't, so there are only five possible dot-to-dot intervals (if you ignore the fact that the intervals weren't all the same length). The intervals from an arrowtip to a dot or the other arrowtip are called "Intervals from a point" in the table. Anyway the message to take from the table is that the greater the inter-dot interval, the more likely it is that the match to the "Feather" interval will be greater than the match to the corresponding "Arrowhead" interval.

[From Bill Powers (2007.09.22.1230MDT)]

Martin Taylor (2007.09.22) --

That isn't true in this situation, so both laws fail, as do all others that suggest that perceptual magnitude is a function of stimulus magnitude.

I suspected the same thing long ago, but this doesn't mean that the actual perceptual magnitude is not any reliable function of stimulus magnitude. It just says that we can't measure the function by any of the means so far tried (including yours of adjusting a line length to match the perceived length of the intervals).

That's a red herring. Nothing in my experiment has any bearing on what form a functional relationship between stimulus and perception might take, nor did I suggest it did. The experiment says there IS no such functional reslationship, at least not one of the form p = f(s). If at any moment a functional relationship between perceived magnitude and stimulus magnitude exists, it depends on the configuration in which the stimulus exists. In other words p = f(s, C) if there's any function at all.

In the HPCT structure, it implies either that there must be some feedback between the configuration level and the intensity level of perception or that there is no functional relation between stimulus magnitude and perceptual magnitude.

What I didn't get from your paper was any indication of how repeatable the measures were. With only two trials per subject, I don't see how you could get p < 0.005 for the first interval, unless you were averaging the interval across different subjects. If you were averaging across subjects, you weren't getting the right measurement for ANY subject,

I disagree. There are situations in which your objection to averaging is absolutely correct, and others in which it's not. What you can always say is that if you don't measure indivdual subjects, there's no way of knowing where in the distribution any one subject's data may lie. That's what makes the combination of subjects in correlation so subversively dangerous. In one-dimensional data, it's always possible that there are different classes of subjects, but what you can say is that if you choose a person at random from the same population, the likelihood is greatest that the person's data will be near the mode of the distribution.

I don't say that everyne will see the inverse Mueller-Lyer illusion the same way -- almost every illusion will show up people who say "I don't see it". What I observe is that most of the responses given by subjects in the experiment suggest that more people see the inverse illusion than don't see it.

As for the p < .005 measure, I should have stated in the paper what it is in reference to, but I think it's from the binomial probability distribution -- how likely is it that you would get so much divergence from 50-50 from tossing a fair coin.

and of course you have no way of knowing how much variability there was in a single subject's estimates.

Actually, since each subject made two estimates for each figure, I would have done at the time. I don't, from the published paper. It's possible the "p-value" is based on the inter-estimate variation within subjects, but I doubt it.

Maybe the comparison you were looking for is difficult to make accurately.

All I looked for what whether measure A was greater or less than measure B. That's not hard to make accurately.

I don't suppose that you allowed the subjects to adjust the line used for measurement by holding it up to the figure with the dots on it. Doing it that way would have resulted in no errors at all to speak of. So how did they make the comparison?

I don't remember the physical setup, but I can imagine how I probably would have done it at the time. It's only guessing, but the sort of thing I would have done would have been to draw a solid line (or maybe a dot) on a card, and have some arrangement that allowed the card to slide out from under some cover, probably also made of card to minimize the edge contrast. If I were to do the experiment now, I would arrange for the Mï¿½ller-Lyer figure to appear horizontally on one part of the screen and have the mouse control a vertical line length or inter-dot distance.

The reason I am a little reluctant to accept the results at face value is that in modeling tracking data, I assume a linear proportionality between the actual and perceived positions of a target and ditto for the perceived position of the cursor, and I get fits within 5% rms (sometimes 2%) for the whole minute's run. Perhaps because the reference condition is "zero distance" between target and cursor, the scaling is rendered unimportant, but it seems to me that the loop gain would be varying a lot (if your data are correct), which would lead us to expect a rather poor fit of the model to the data.

Early on in my modelling of the sleep study tracking data, I used an exponential relation between the actual error and the perceived error. I found that it was very hard to optimize this variable because it's effects could be closely mimicked by a change in overall gain. However, I did find that setting a tolerance region in which a small error was taken to be zero error, but outside that range taking the error to be just the difference between the reference and the perception gave better fits than taking a linear relation all the way.

One thing I am sure of -- the slopes are monotonic. If they ever reversed there would be amplitudes where positive feedback existed, and that is not observed.

You wouldn't observe it, would you, if you were doing either a compensatory or a pursuit tracking task. It's possible you might if you were doing a tracking equivalent of a cross-modal magnitude matching study. It would be difficult to detect in the results, though, because although what you would observe in theory would be a rapid transition across the non-monotonic region, the jump that you would observe in practice wouldn't be instantaneous, but would be limited by the response bandwidth, and would be easily lost in the normal tracking noise.

We aren't talking about big deviations from non-monotonicity, but little wiggles on a generally monotonic function. The only reason I suggested the possibility is from little hints in peripheral results over the years in studies that had nothing to do with magnitude estimation itself (inter-aural pitch matching as a function of frequency is one; number factorization and association value is another, neither published because they were preliminaries for other stuff that was interesting at the time). I just think it's a possibilty that should be kept in mind when one is dealing with a condition in which it might matter.

You wouldn't happen to still have the raw data, would you?

I very much doubt it, but it is remotely possible. I think a rerun of the experiment on a computer screen would give much better data. Particularly for the shorter intervals, the published data are likely to be noisy simply because of the need to measure accurately the distance produced by the subjects. That shouldn't be an issue in a computerized experiment.

Martin

PS. I'm puzzled as to what the plot of the Weber fraction you posted a few minutes later is supposed to imply.

[From Rick Marken (2007.09.22.1430)]

Bill Powers (2007.09.22.1300 MDT)

One explanation might be that there is no power law or log law "observed"
between stimulus and magnitude estimates.

Of course. But that isn't what Martin is saying. He's saying that
there is no single functional relationship between stimulus and
perceptual magnitude because any such functional relationship "must
include terms involving configurational variables". This conclusion,
which basically assumes that there is only a configuration level of
perception, is based on some questionable data; basically the
proportion of times a group of individuals adjusted a line to be
longer (with no indication of how much longer) for segments of one
component of the Muller Lyer relative to another. I'm going to spend
some time on this beautiful afternoon trying to figure out what the
subjects in Martin's experiment were up to.

I agree with you that there is probably an orderly relationship between
perception and the outside world. But measuring it may not be as simple as
some people have assumed it would be.

Of course!

Best

Rick

···

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

[Martin Taylor 2007.09.22.17.49]

[From Rick Marken (2007.09.22.1430)]

Bill Powers (2007.09.22.1300 MDT)

One explanation might be that there is no power law or log law "observed"
between stimulus and magnitude estimates.

Of course. But that isn't what Martin is saying. He's saying that
there is no single functional relationship between stimulus and
perceptual magnitude because any such functional relationship "must
include terms involving configurational variables". This conclusion,
which basically assumes that there is only a configuration level of
perception,

No it doesn't. However, basic physiological data at the level of the retina tell you that what gets up the optic nerve does depend on spatio-temporal configuration, so from that point of view you could be right. But neither my expriment nor my interpretation force that interpretarion.

is based on some questionable data;... I'm going to spend
some time on this beautiful afternoon trying to figure out what the
subjects in Martin's experiment were up to.

You are a good and quick programmer, and you have access to students. Why not program the experiment and set it up so that you can judge for yourself as best you can what they are doing, rather than making abstract guesses at what might have been going on over 40 years ago in an experiment that is only sketchily described (and remembered).

I agree with you that there is probably an orderly relationship between
perception and the outside world. But measuring it may not be as simple as
some people have assumed it would be.

Of course!

I agree, but I think this orderly relationship is based largely on the learned ability to control perceptions through actions on this outer world, not primarily on perceptual transformations enforced by neural sensitivities, though those do matter (of course).

Martin

[From Rick Marken (2007.09.23.1130)]

Martin Taylor (2007.09.22.17.49)

> Rick Marken (2007.09.22.1430)

> I'm going to spend
>some time on this beautiful afternoon trying to figure out what the
>subjects in Martin's experiment were up to.

You are a good and quick programmer, and you have access to students.
Why not program the experiment and set it up so that you can judge
for yourself as best you can what they are doing,

Before I can program it I have to know what was actually done. From
what I can tell from reading the article it seems that eight subjects
made two line length estimates of all inter-dot distances for both
figures. So there were 8X2X2 = 32 line length judgments of each
inter-dot distance. This suggests that the sum of the number of
"Arrowhead greater" and "Feather greater" judgments should be a
multiple of and greater than 32 . But this is not true for _any_ of
the "dot to dot intervals" in the table. So I don't know what the
data is that's reported. Were trials eliminated when the length
estimates for the ""Arrowhead" and "Feather" versions of the
Mueller-Lyer were exactly equal?

And just taking the data as reported I don't see why you conclude that
they imply that there can be no single functional relationship between
perceived and physical distance. On what do you base that conclusion?
On the fact that with zero interpolated dots the distance between the
dots is more often judged greater for the Arrowheads, which make the
overall distance between "points" look greater? I don't see why. The
change in the rate of "Arrowhead greater" vs "Feathers greater"
estimates as a function of increasing dot to dot interval can be
accounted for by context effects. The 0 and 1 dot to dot interval
length is judged smaller for the Feather than for the Arrowheads
because these judgments are made relative to the apparently longer
point to point distance with Feathers. This context effect diminishes
as the dot to dot distance approaches the point to point distance.

The point is that these results can be explained if there is a single
perceptual function relating physical to perceived distance. All you
have to assume is that the perception of dot to dot distance = f(dot
to dot distance) and that this perceived distance is judged relative
to the perceived distance of the context distance, which is longer for
the feathers than for the arrowhead.

Best

Rick

···

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

[Martin Taylor 2007.09.23.14.55]

[From Rick Marken (2007.09.23.1130)]

Martin Taylor (2007.09.22.17.49)

> Rick Marken (2007.09.22.1430)

> I'm going to spend
>some time on this beautiful afternoon trying to figure out what the
>subjects in Martin's experiment were up to.

You are a good and quick programmer, and you have access to students.
Why not program the experiment and set it up so that you can judge
for yourself as best you can what they are doing,

Before I can program it I have to know what was actually done.

I don't think you have to know that, though it would be much better if you did. You ask a question that I can't answer, off the top of my head.

From
what I can tell from reading the article it seems that eight subjects
made two line length estimates of all inter-dot distances for both
figures. So there were 8X2X2 = 32 line length judgments of each
inter-dot distance. This suggests that the sum of the number of
"Arrowhead greater" and "Feather greater" judgments should be a
multiple of and greater than 32 . But this is not true for _any_ of
the "dot to dot intervals" in the table. So I don't know what the
data is that's reported.

You are absolutely right, and I don't know the answer. There must be a reason, but I doubt that it would be omission of "equal" judgments. If you look at just one column of the table, the "4 interval" distance between the outermost dots, there are only 14 recorded judgments. Since that would be measuring a match to a physical interval of 2.82 inches, to have 18 settings exactly equal to within .01 inch seems most unlikely. I don't have another suggestion at the moment.

But that shouldn't prevent you from testing the proposition that short inner distances are larger in the "wrong" figure. You could do it for a variety of different illusions with the same programmatic core, too. The results would be rather more reliable than my old ones.

And just taking the data as reported I don't see why you conclude that
they imply that there can be no single functional relationship between
perceived and physical distance. On what do you base that conclusion?... The
change in the rate of "Arrowhead greater" vs "Feathers greater"
estimates as a function of increasing dot to dot interval can be
accounted for by context effects.

Yes, that is why!

The point is that these results can be explained if there is a single
perceptual function relating physical to perceived distance. All you
have to assume is that the perception of dot to dot distance = f(dot
to dot distance) and that this perceived distance is judged relative
to the perceived distance of the context distance, which is longer for
the feathers than for the arrowhead.

Precisely. We agree completely on the probable reason for the effect, just not on how it applies. I say that p = f(s) is not the same as p = f(s, context), whereas you claim that the two functions are identical. That's the only disagreemment I can see between us. To me, your way of looking at it is tantamount to saying that

x = ky for a given value of k

is the same function as

x = ky*z

which is true if z is fixed and k has the appropriately changed value. But that doesn't make the two fucntions the same, does it?

Psychologically, both our ways of looking at it say the same thing: that perception at the intensity level depends on configurations that include that intensity.

Martin

PS. I haven't left the "libertarian" discussion. Just trying to understand the implications in PCT terms. It's not easy.

[Martin Taylor 2007.09.23.15.55]

[From Rick Marken (2007.09.23.1130)]

Martin Taylor (2007.09.22.17.49)

> Rick Marken (2007.09.22.1430)

> I'm going to spend
>some time on this beautiful afternoon trying to figure out what the
>subjects in Martin's experiment were up to.

You are a good and quick programmer, and you have access to students.
Why not program the experiment and set it up so that you can judge
for yourself as best you can what they are doing,

Before I can program it I have to know what was actually done. From
what I can tell from reading the article it seems that eight subjects
made two line length estimates of all inter-dot distances for both
figures. So there were 8X2X2 = 32 line length judgments of each
inter-dot distance. This suggests that the sum of the number of
"Arrowhead greater" and "Feather greater" judgments should be a
multiple of and greater than 32 . But this is not true for _any_ of
the "dot to dot intervals" in the table. So I don't know what the
data is that's reported. Were trials eliminated when the length
estimates for the ""Arrowhead" and "Feather" versions of the
Mueller-Lyer were exactly equal?

I said I didn't know the answer, and I still don't. But here's a plausible interpretation: the two measures for one subject's matches to one interval in each configuration were averaged, and the average of the two "Feathers" matches compared to the average of the two "Arrowhead" matches. Equal values were omitted from the table. That makes one comparison per subject.

This works, because there are 16 subjects, now each yielding one comparison for each inter-dot interval. There is only one interdot interval of 4 (the interval between the outermost dots), two of 3, 3 of 2, 4 of 1 and 5 of zero. Seven of the "point-to-dot intervals were tested. That gives respectively 16, 32, 48, 64, 80, and 108 actual responses. The table shows 14, 31, 45, 59, 68, and 94 which would mean that the number of "equals" per 16 comparisons would be 2, 0.5, 1, 0.8, 2.4, and 2.1 which doesn't sound out of line. So that's what I guess the table entries mean.

Including the "equals" values in a new table, using this assumption, we get, now entering to the nearest integer percent rather than number:

intervening dots 0 1 2 3 4 point-to-dot
Arrowhead Greater 50 49 35 28 12 4
Equal 12 3 6 7 15 13
Feathers Greater 35 37 58 69 75 83

Martin

[From Rick Marken (2007.09.23.2115)]

Martin Taylor (2007.09.23.14.55) --

Precisely. We agree completely on the probable reason for the effect,
just not on how it applies. I say that p = f(s) is not the same as p
= f(s, context), whereas you claim that the two functions are
identical. That's the only disagreemment I can see between us.

I agree. That's where we disagree. I think of f() as the function
mapping physical stimulus magnitude (s) into perceptual magnitude (p).
I assume that f() is of the same form regardless of the value of
context. If f() is the inverse of the feedback function in a
magnitude estimation task, then I'm sure you would find that f() is
approximately a log function (inverse of power) in a distance
estimation task when all the distances are enclosed by "feathers" or
when all the distances are enclosed by "arrowheads" or when all the
distances are not enclosed at all. It would be pretty easy to do the
experiment to find out but I'm pretty sure it has already been done.
I'll try to find one. If I can't I'll do the magnitude estimation
experiment myself.

Best

Rick

···

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

[Martin Taylor 2007.09.24.00.47]

[From Rick Marken (2007.09.23.2115)]

Martin Taylor (2007.09.23.14.55) --

Precisely. We agree completely on the probable reason for the effect,
just not on how it applies. I say that p = f(s) is not the same as p
= f(s, context), whereas you claim that the two functions are
identical. That's the only disagreemment I can see between us.

I agree. That's where we disagree. I think of f() as the function
mapping physical stimulus magnitude (s) into perceptual magnitude (p).
I assume that f() is of the same form regardless of the value of
context.

But given these data, it can't be, can it, at least not if by "form" you mean visual shape when graphed? I show below that what you say can actually be true if you mean a parameterized mathematical form, but that it's irrelevant to the point under discussion.

If f() is the inverse of the feedback function in a
magnitude estimation task, then I'm sure you would find that f() is
approximately a log function (inverse of power) in a distance
estimation task when all the distances are enclosed by "feathers" or
when all the distances are enclosed by "arrowheads" or when all the
distances are not enclosed at all.

Let's see. If I undersand you correctly, you assert that in all cases p = k*log(s) approximately, where p is the perceptual value and s is the stimulus magnitude, in this case physical distance. I take it that this is what you mean by a log function, and that changing k is the way to get a function of the same form that is different between the A (Arrow) and F (Feathers) context configurations. It's the only way, actually. The question is whether by changing the value of k from kA to kF you get a situation in which pA1 > pF1 for s1 small, but pA2 < pF2 for s2 large. Let's work with these equations.

pA1 = kA*log(s1), pA2 = kA*log(s2)
PF1 = kF*log(s1), pF2 = kF*log(s2)

log(s1) = pA1/kA = pF1/kF
log(s2) = pA2/kA = pF2/kF

pA1/PF1 = kA/KF
pA2/PF2 = kA/kF

But pA1/pB1 > 1.0, pA2/pB2 < 1.0

Therefore either I misunderstand your claim, or it's an impossible claim.

So, let's consider "of the same form" to not mean simply being a log function, but being of some form with a single parameter. Can we make an analogous calculation? We can, and it turns out that with appropriate parameterization the claim is feasible. Label the parameterized form of the function fA and fF

pA1 = fA(s1) , pA2 = fA(s2)
pF1 = fF(s1) , pF2 = fF(s2)

pA1/pF1 = fA(s1)/fF(s1) > 1.0
pA2/pF2 = fA(s2)/fF(s2) < 1.0

This is feasible, provided the parameterization changes the curvature of the function f so that fA increases relatively more slowly than fF (as would, for example, fA linear and fF quadratic -- the parameterization being in the exponent).

As an example, try fA(s) = s, fF(s) = s^2. Then for s<1 fA(s) > fF(s), but for s>1, fA(s) < fF(s).

I think this is actually what is happening (not the example, but the principle). The context changes the shape, but perhaps not the mathematical form, of the function relating perception to stimulus.

It's kind of irrelevant to the main point, though, since even if your logarithmic assumption had been feasible, the result would still have been that the configuration influences the relation between stimulus magnitude and perception magnitude.

It would be pretty easy to do the
experiment to find out but I'm pretty sure it has already been done.

Interesting to know if it has. More interesting to know if anyone has ever cited my paper

I'll try to find one. If I can't I'll do the magnitude estimation
experiment myself.

I wish you would. As I said, my data are old, and done with techniques less accurate than are now available. Anyway, my sense is that you don't very much trust experiments done by people who didn't have PCT in mind.

Martin

[From Bill Powers (2007.09.24.0409 MDT)]

Martin Taylor 2007.09.24.00.47] –

Rick Marken (2007.09.23.2115)] –

I’m copying here a few paragraphs I recently send to Warren Mansell. They
may have a bearing on the Lyell Illusion that’s being discussed.

···

=======================================================================================

I once worked with a graduate student who was a former student of Dick
Robertson’s, Dick having been an intern in Carl Rogers’ Counselling
Center at the U. of Chicago along with Mary, even before I met her. Just
after Mary and I married, I gave a seminar along with Clark and
MacFarland at the counselling center (1957), and Dick has been aboard
ever since then. He recently retired from Northeastern Illinois
University where he pioneered the teaching of PCT for many
years.

The graduate student suffered from motion sickness to a debilitating
degree, even on subway (Underground) trains and on buses. She wanted to
study it, so I wrote a program to use a (sideways) waterfall illusion to
measure after-“images” of motion (a transition-level
experiment). A computer screen was filled with moving dots, which ceased
to move when a beep sounded. At that point, the participant could affect
the rate of motion of the dots by using a mouse, the instructions being
to use the mouse to keep the dots stationary after the beep. Of course
that means making the dots move oppositely to the direction of the
illusion. The result was a very nice negative exponential decay of the
participant-caused motion, dying out to zero in twenty or thirty seconds.
This made the magnitude and time-course of the illusion directly
measurable. We were probably measuring the motion of fluid in the
semicircular canals!

In refining the experiment and data-taking, this student and I passed
many trial versions of the experiment back and forth, and after a while I
began to have trouble with the program – some bug had crept into it so
the data were not being correctly reported – so I thought. Eventually,
had it any more. We had reorganized it away. Not only that, but the
graduate student found that she no longer suffered from motion sickness.
Since that had been her main motivation for doing this thesis, she said
thank you very much and she was changing her thesis to some other
subject, since the old thesis had evaporated. (I’m copying this post to
Dick, and also to Rick Marken who is looking for PCT experiments for his
students at UCLA).

I have felt for a long time that therapy takes precedence over studying
abnormalities. We may as well admit that psychological testing has
effects on the people we’re testing, so we might as well study changes
along with whatever else we think we’re measuring. And as long as we’re
doing that, we might as well select “experiments” that also do
people some good rather than harm.

=======================================================================================

What occurred to me, of course, was that while doing Martin’s
experiments, the participants might have been reorganizing, so that
gradually, over time, the distortions would tend to disappear. Perhaps
the course of the experiments was short enough that this effect would not
appear, but if the comparisons were made in the same order every time,
there could have been some confusing adaptational effects – or even if
they were made in randomized order.

It seems to me that this experiment calls for a pct-style version, in
which the participants simply remove what they see as distortions until
the dots all appear equally spaced. If there really are context effects
of the kind Martin tried to measure, this should show them very clearly,
as my experiment did for the waterfall illusion. And if adaptation really
does take place, the illusion should disappear with extended practice
(though if it doesn’t, that would be informative, too).

I think we’re always reorganizing to maintain the mutual calibration
among sensory modalities, particularly kinesthesia, touch, and vision.
With practice, the cross-calibration seems to get better, and it seems to
deteriorate otherwise. PCT experiments would seem to suit this sort of
experimentation very nicely.

And my answer to Martin’s comment is that no, I don’t trust anyone’s
experiments that were done without using PCT techniques. There are too
many assumptions about things that PCT makes measurable. Some older
experiments were done by adjustment of objects and relationships and
might be PCT-certifiable, more or less, but most weren’t. I don’t know
how the line-length comparisons to dot intervals were made, so I can’t
judge Martin’s experiment on that basis. It seems that so many details
have been lost that the results can no longer be interpreted. If
reorganization were involved, it would not be surprising to find
temporary inconsistencies in the judgements, but I would expect them to
disappear with practice.

Best.

Bill P.