# Measurement of functional components (was Controlled Variables (was ...)

[Martin Taylor 2009.04.09.00.33]

[From Bill Powers (2009.04.06.1547 MDT)]

Martin Taylor 2009.04.06.14.11] –

Sorry, but you’ve left
out a
vital function, the environmental feedback function in that path that
loops from Output O to S2. S2 is not equal to O, but to some function
g(O).

Could that be why I said: “S2 is a function of O”? I
admit I called the function “f” rather than
g”, but I didn’t think that distinction was as
important as you seem to make it. “A rose by any other name would
smell as sweet.”

You said that, but you didn’t include it in your equations.

I don’t understand why you say that. I went back to my original message
to check, and there it was.

S2 depends on
O, so everywhere you see S2 you need to substitute f(O) (or g(O) if you
like). This emphasizes the point that you don’t know the form of that
function, either, so you’re stymied.

I think you are missing the point more than a little. The point is that
the perception p is being controlled. The variable p is a function of
two inputs, which we call S1 and S2, so if control were perfect, the
trace of S2 vs S1 would follow a simple curve of constant p in the
S1-S2 space, a different curve for each different reference value for
p. Since S2 is a single-valued function of O, the same is true for the
trace of S1 versus D in the S1-D space under the condition of constant
p. We may not know the form of any of these functions, but we can
determine quite a bit about them, especially when we set things up so
we can analyse differences. Everything you say, above or below, applies
equally when you are modelling a pursuit tracking experiment, because
the formal setup is the same.

When I first learned golf, one important trick everyone had to know was
how to get over a stymie without damaging the green. So many greens
were damaged by people who didn’t do it very well that the rule was
changed, and now golf has no stymies. Maybe we could change the rules
here, so that they apply equally to all experiments we might consider,
rather than damaging the “green” of the discussion group by being
applied selectively only to experiments I mention as being interesting.

The observed
relationship of O
to S1, therefore, will be determined by the reference level R and the
inverse of the environmental feedback function.

Yes. Do you think that might possibly be why I said: “The form of
the function relating S2 to S1 is controlled by the form of p(S1, S2).
For example, if p == (S1 * S2), the answer would be quite different
than
it would if p == (S1-S2). Under some circumstances, we may be able to
assume the form of p, or test it by modelling the control
loop.

I do admit that I did not mention the possibility of varying reference
levels.

But when you say “is controlled by the form” you’re alluding to
a relationship without spelling it out. The result is that you’re
talking
about a mathematical derivation without actually producing it. I want
to
see the derivation itself, not statements about what it might
demonstrate. I have no idea how you arrive at your conclusions, and
showing the derivation might show that.

I did explain what I meant by “the form of p(S1,S2)”. Since that could
be anything at all, it’s quite impossible to specify. “The Test” can
approximate what that form might be in any particular case (I’ll have
things to say about “The Test” at some later time, when this is all
sorted out). “The Test” is one element of modelling the control loop,
anyway. So I think that in the part you quoted I covered your
objection. I specifically mentioned two possibilities p == (S1-S2) (the
pursuit tracking and the answer-matching cases) and p == (S1*S2)
(controlling an area). I could add the possibilities Rick tested in his
size study, and a myriad of others, but how would that affect anything

It will not indicate a
property
of the forward path through the organism.

Which is not under examination. What is under examination is S1 =
f(D).

That is part of the forward path through the organism. You can’t
“examine” that path because you can observe only one end of
it.

Aaargh!!! Isn’t the whole point of the demonstration to show how you
can? I really don’t care for critiques of a proof that X = Y that
consist simply of the assertion “you can’t say X = Y”. That’s rather
like Rick’s dismissal of all methods of studying the human other than
“The Test” for the controlled variable. You are saying that you can’t
tell whether something is hot or cold because you can only see a
reading on a thermometer and that’s not a feeling of warmth or cold.
Well, if you want to make that kind of argument, there’s little more to
be said, I think.

Because this is a
control loop
for which we assume an invariant reference value set by the
experimenter,
if control is perfect, then

p(S1.0, S2.0) = p(S1.1, S2.1), which is equivalent to p(S1.0, S2.0) -
p(S1.1, S2.1) = 0.

What we are interested in is not really the transfer function between
S1
and O, but the one between the disturbance D introduced by the
experimenter and S1. We want to know whether we can deduce anything
this transfer function by observing the relationship between variations
in D and variations in O. In conventional psychology, D is the
independent variable (IV) and O the dependant variable
(DV).

This is where the problem lies for me. What you deduce about the path
from D to S1 depends entirely on the kinds of functions you assume to
lie
in that pathway. You haven’t drawn any box between the disturbing
variable and S1, but that’s what we’re talking about
here.

Yes, that’s what the experiment is presumed to be wanting to determine.
It’s why I said: “if we can determine the form of p by other
experiments (such as by allowing the subject to influence both S1 and
S2,
for example, and modelling the control loop), then we could determine
the
function D -> S1
.”

I think we are using the word “determine” very differently
here. If the function S1 -> O is postulated, we can then measure the
function D -> O and by dividing the second function by the first,
arrive at an expression for the function D -> S1. But we can’t claim
to have determined the actual function

D -> S1 because that function depends entirely on what function
we assumed for S1 -> O. We can make it whatever we want it to be,
just
by changing what we assume about S1 -> O.

Is that really what you mean by “modelling the control loop”? Making an
assumption? Don’t “other experiments” and “modelling” have a little
more behind them than that?

By the way, just when did the object of the study become necessarily
“to have determined the actual function D -> S1”? I guess one could
try to do that, but I don’t remember ever suggesting it as the object
of any study I have mentioned. You and Rick did claim to have done
that, I believe, in respect of Steven’s power law, but it’s not
something I would normally want to do.

Also, notice what is
observed
after D: nothing, until we get to O. So the measurements you’re making
could include any function between D and O, and might well include more
than one function. You can imagine any number of arrangements inside
the
system, including a simple S-O-R connection – and that is clearly what
traditional psychologists often assumed. There is nothing to say that
it
is a relationship of the disturbing variable to the output that is
being
controlled. You can’t be sure of that without doing some tests
specifically to see what is being controlled, and at what reference
level. You know what you told the subject to do, but that is not
necessarily what the subject is doing. Only the proper tests can show
what the subject is actually doing.

Yes. Is that contradictory, confirmatory, or irrelevant to what I
wrote?
I think it is a restatement of: "_Knowing that O is a function of
D, what can we say about S1 as a function of D? That depends on what we
can determine about the control loop. The form of the function relating
S2 to S1 is controlled by the form of p(S1, S2). For example, if p ==
(S1

• S2), the answer would be quite different than it would if p ==
(S1-S2).
Under some circumstances, we may be able to assume the form of p, or
test
it by modelling the control loop. If we can determine the form of the
function that relates O to S1, and the function that relates O to D,
then
we can determine the function D -> S1, which is what we wanted to
know._"

You’re assuming that we can observe S1 and S2. We can’t.

I made no such assumption. I made the contrary very clear, or so I
thought.

Then how can we “determine” anything about S1 and S2? We don’t
know the form of the function relating S1 and S2, …

until we perform “The Test” to estimate it … (and “estimate” or
“approximate” is the best the Test can ever do). Since the function
relating S1 and S2 is derived from the function that creates the
controlled perception, we either have to assume it (as is reasonable
when we are doing pursuit tracking or answer matching, when p = (S1-S2)
or (S2-S1)), or Test for it. Notice that when we are doing a pursuit
tracking experiment, there is no way to distinguish between perfect
tracking of a changing reference level, imperfect tracking of a fixed
reference level, or anything between. The problems are exactly the
same, because the setup is exactly the same.

and we don’t know the
values of S1 and S2, …

until we experiment and model the experimental results …

so … well, I think we’re suffering a severe
difference in the way we use language. You speak in ways that make me
think you’re talking about verifying the model by comparing its
behavior
with real behavior, …

Yes, that’s what I’ve been talking about…

but then you claim to know things as if you could
measure the real counterparts of the modeled variables.

When?

I am beginning to think that when I think you’re talking about the real
system, the one we’re trying to model, you’re really talking only about
the model. Here it is in a nutshell: you say " If we can
determine the form of the function that relates O to S1, and the
function
that relates O to D …"
and my immediate response is to think
“But you can’t determine the function that relates O to S1 because
you don’t know the function that relates D to S1 and the only
measurement
you have is the relationship of D to O.”

Both of those imagined functions exist inside the organism and you
can’t
see either of them. The best you can do is assume a form for one of the
missing functions and see what that implies about the other one. That
means you’re working entirely with the imagined model, where you can’t
“determine” anything. All you can do is see what the logical
consequences of making assumptions are.

There are three answers to that.

(1) in the most general case, you are quite correct, and would be
equally correct when talking about a pursuit tracking study. That’s why
we see whether the logical consequences of the assumptions are
consistent with the observations – the data, in experiments. That’s
true of all modelling based on observations of what happens outside the
physical boundary of the organism. It’s one reason why I believe
Bayesian analysis is the most appropriate way to approach the relation
between theory and experiment.

(2) Very often (as is the case in the pursuit tracking study, and in
many detection and discrimination experiments) the experiment is set up
so as to make it highly likely that the function relating the sensory
input to S1 is very similar to the function relating sensory input to
S2. Furhtermore, when both S1 and S2 are category perceptions
themselves, the function generating the category is hardly relevant to
whether the categories are the same or different. In the case of a
discrimination experiment, one of them is derived from a current
sensory input, the other from a past instruction by an experimenter.
That does not affect the control of p(S1, S2), where p(x,y)==(x-y),
using a reference value of zero for the difference between the
categories X and Y.

(3) When (2) is not the case, one can often make reliable differential
estimates, without knowing the forms of the functions. From them, if
such is desired, one can determine certain aspects of the functions
themselves. That’s what you do when modelling tracking experiments. In
the case of detection and discrimination experiments the experiment is
deliberately designed so that only differential measures are available
(was the presentation class 1 or class 2?). When you and Rick did your
re-analysis of Stevens’s power functions, you had to make some
assumptions, and found that if S1= f(sensation) had a logarithmic form
regardless of the sensation, the results of the experiments would be
power functions. But the same would have been true for an infinite
number of other input functions, within experimental error. That
doesn’t in any way invalidate the results of your analysis, since it is
consistent with the data. In the case of a detection or discrimination
experiment, the differential inputs D1 and D2 allow for an estimate of
the precision of the channel D->S1. If the perception of S2 were
also uncertainly derived from the initial instructions (the subject
wasn’t quite sure how to react), the results would put a lower bound on
that precision (as indeed they do in any case, since one never knows
whether there are noise sources one has not incorporated in the
analysis, such as hitting the unintended button).

All we can observe are
D and O,
and the form of the environmental feedback function without the input
perceptual function that generates S2.

## Apart from the dubious question of whether we can observe the environmental feedback function – we may, but we may have to model it

that was my starting point.

Yes, but you didn’t seem to stick to those initial stipulations.

I thought I did. I don’t think your comments demonstrate the contrary,
though your language suggests they do.

Martin

[From Rick Marken (2009.04.13.1235)]

Martin Taylor (2009.04.09.00.33)–

Bill Powers (2009.04.06.1547 MDT)–

Your reply to this 7day old post from Bill seems to be 4 days old itself. So it looks like there’s some lag in the system. If you get a chance, what I would love to see is your reply to Bill’s most recent post: [Bill Powers (2009.04.10.2300 MDT)].

Best

Rick

···

Richard S. Marken PhD
rsmarken@gmail.com

[David Goldstein (2009.04.13.22:32 EDT)]

As a person who is trying to follow this interesting discussion, please come up with a better way of showing who is talking than using indentation.

For example, it would help if you put BP in front of words that Bill said and MT in front of words that MT said. The indentation is driving me crazy.

Thank you for considering my request.

David

···

----- Original Message -----

From:
Martin Taylor

To: CSGNET@LISTSERV.ILLINOIS.EDU

Sent: Monday, April 13, 2009 12:10 PM

Subject: Re: Measurement of functional components (was Controlled Variables (was …)

[Martin Taylor 2009.04.09.00.33]

[From Bill Powers (2009.04.06.1547 MDT)]

``````Martin Taylor 2009.04.06.14.11] --
``````
``````    Sorry, but you've left out a vital function, the environmental feedback function in that path that loops from Output O to S2. S2 is not equal to O, but to some function g(O).
``````

Could that be why I said: “* S2 is a function of O*”? I admit I called the function “f” rather than "g ", but I didn’t think that distinction was as important as you seem to make it. “A rose by any other name would smell as sweet.”

``````You said that, but you didn't include it in your equations.
``````

I don’t understand why you say that. I went back to my original message to check, and there it was.

``````S2 depends on O, so everywhere you see S2 you need to substitute f(O) (or g(O) if you like). This emphasizes the point that you don't know the form of that function, either, so you're stymied.
``````

I think you are missing the point more than a little. The point is that the perception p is being controlled. The variable p is a function of two inputs, which we call S1 and S2, so if control were perfect, the trace of S2 vs S1 would follow a simple curve of constant p in the S1-S2 space, a different curve for each different reference value for p. Since S2 is a single-valued function of O, the same is true for the trace of S1 versus D in the S1-D space under the condition of constant p. We may not know the form of any of these functions, but we can determine quite a bit about them, especially when we set things up so we can analyse differences. Everything you say, above or below, applies equally when you are modelling a pursuit tracking experiment, because the formal setup is the same.

When I first learned golf, one important trick everyone had to know was how to get over a stymie without damaging the green. So many greens were damaged by people who didn’t do it very well that the rule was changed, and now golf has no stymies. Maybe we could change the rules here, so that they apply equally to all experiments we might consider, rather than damaging the “green” of the discussion group by being applied selectively only to experiments I mention as being interesting.

``````    The observed relationship of O to S1, therefore, will be determined by the reference level R and the inverse of the environmental feedback function.
``````
``````  Yes. Do you think that might possibly be why I said: "_      The form of the function relating S2 to S1 is controlled by the form of p(S1, S2). For example, if p == (S1 * S2), the answer would be quite different than it would if p == (S1-S2). Under some circumstances, we may be able to assume the form of p, or test it by modelling the control loop._"

I do admit that I did not mention the possibility of varying reference levels.
``````
``````But when you say "is controlled by the form" you're alluding to a relationship without spelling it out. The result is that you're talking about a mathematical derivation without actually producing it. I want to see the derivation itself, not statements about what it might demonstrate. I have no idea how you arrive at your conclusions, and showing the derivation might show that.
``````

I did explain what I meant by “the form of p(S1,S2)”. Since that could be anything at all, it’s quite impossible to specify. “The Test” can approximate what that form might be in any particular case (I’ll have things to say about “The Test” at some later time, when this is all sorted out). “The Test” is one element of modelling the control loop, anyway. So I think that in the part you quoted I covered your objection. I specifically mentioned two possibilities p == (S1-S2) (the pursuit tracking and the answer-matching cases) and p == (S1*S2) (controlling an area). I could add the possibilities Rick tested in his size study, and a myriad of others, but how would that affect anything about the argument?

``````    It will not indicate a property of the forward path through the organism.
``````
``````  Which is not under examination. What is under examination is S1 = f(D).
``````
``````That is part of the forward path through the organism. You can't "examine" that path because you can observe only one end of it.
``````

Aaargh!!! Isn’t the whole point of the demonstration to show how you can? I really don’t care for critiques of a proof that X = Y that consist simply of the assertion “you can’t say X = Y”. That’s rather like Rick’s dismissal of all methods of studying the human other than “The Test” for the controlled variable. You are saying that you can’t tell whether something is hot or cold because you can only see a reading on a thermometer and that’s not a feeling of warmth or cold. Well, if you want to make that kind of argument, there’s little more to be said, I think.

``````      Because this is a control loop for which we assume an invariant reference value set by the experimenter, if control is perfect, then

p(S1.0, S2.0) = p(S1.1, S2.1), which is equivalent to p(S1.0, S2.0) - p(S1.1, S2.1) = 0.

What we are interested in is not really the transfer function between S1 and O, but the one between the disturbance D introduced by the experimenter and S1. We want to know whether we can deduce anything about this transfer function by observing the relationship between variations in D and variations in O. In conventional psychology, D is the independent variable (IV) and O the dependant variable (DV).
``````
``````    This is where the problem lies for me. What you deduce about the path from D to S1 depends entirely on the kinds of functions you assume to lie in that pathway. You haven't drawn any box between the disturbing variable and S1, but that's what we're talking about here.
``````
``````  Yes, that's what the experiment is presumed to be wanting to determine. It's why I said: "*      if we can determine the form of p by other experiments (such as by allowing the subject to influence both S1 and S2, for example, and modelling the control loop), then we could determine the function D -> S1*."
``````
``````I think we are using the word "determine" very differently here. If the function S1 -> O is postulated, we can then measure the function D -> O and by dividing the second function by the first, arrive at an expression for the function D -> S1. But we can't claim to have determined the actual function
D -> S1 because that function depends entirely on what function we assumed for S1 -> O. We can make it whatever we want it to be, just by changing what we assume about S1 -> O.
``````

Is that really what you mean by “modelling the control loop”? Making an assumption? Don’t “other experiments” and “modelling” have a little more behind them than that?

By the way, just when did the object of the study become necessarily “to have determined the actual function D -> S1”? I guess one could try to do that, but I don’t remember ever suggesting it as the object of any study I have mentioned. You and Rick did claim to have done that, I believe, in respect of Steven’s power law, but it’s not something I would normally want to do.

``````    Also, notice what is observed after D: nothing, until we get to O. So the measurements you're making could include any function between D and O, and might well include more than one function. You can imagine any number of arrangements inside the system, including a simple S-O-R connection -- and that is clearly what traditional psychologists often assumed. There is nothing to say that it is a relationship of the disturbing variable to the output that is being controlled. You can't be sure of that without doing some tests specifically to see what is being controlled, and at what reference level. You know what you told the subject to do, but that is not necessarily what the subject is doing. Only the proper tests can show what the subject is actually doing.
``````
``````  Yes. Is that contradictory, confirmatory, or irrelevant to what I wrote? I think it is a restatement of: "_      Knowing that O is a function of D, what can we say about S1 as a function of D? That depends on what we can determine about the control loop. The form of the function relating S2 to S1 is controlled by the form of p(S1, S2). For example, if p == (S1 * S2), the answer would be quite different than it would if p == (S1-S2). Under some circumstances, we may be able to assume the form of p, or test it by modelling the control loop. If we can determine the form of the function that relates O to S1, and the function that relates O to D, then we can determine the function D -> S1, which is what we wanted to know._      "
``````
``````    You're assuming that we can observe S1 and S2. We can't.
``````
``````  I made no such assumption. I made the contrary very clear, or so I thought.
``````
``````Then how can we "determine" anything about S1 and S2? We don't know the form of the function relating S1 and S2, ...
``````

until we perform “The Test” to estimate it … (and “estimate” or “approximate” is the best the Test can ever do). Since the function relating S1 and S2 is derived from the function that creates the controlled perception, we either have to assume it (as is reasonable when we are doing pursuit tracking or answer matching, when p = (S1-S2) or (S2-S1)), or Test for it. Notice that when we are doing a pursuit tracking experiment, there is no way to distinguish between perfect tracking of a changing reference level, imperfect tracking of a fixed reference level, or anything between. The problems are exactly the same, because the setup is exactly the same.

and we don’t know the values of S1 and S2, …

until we experiment and model the experimental results …

``````so .... well, I think we're suffering a severe difference in the way we use language. You speak in ways that make me think you're talking about verifying the model by comparing its behavior with real behavior, ....
``````

Yes, that’s what I’ve been talking about…

``````but then you claim to know things as if you could measure the real counterparts of the modeled variables.
``````

When?

``````I am beginning to think that when I think you're talking about the real system, the one we're trying to model, you're really talking only about the model. Here it is in a nutshell: you say " *    If we can determine the form of the function that relates O to S1, and the function that relates O to D ..."* and my immediate response is to think "But you can't determine the function that relates O to S1 because you don't know the function that relates D to S1 and the only measurement you have is the relationship of D to O."
Both of those imagined functions exist inside the organism and you can't see either of them. The best you can do is assume a form for one of the missing functions and see what that implies about the other one. That means you're working entirely with the imagined model, where you can't "determine" anything. All you can do is see what the logical consequences of making assumptions are.
``````

There are three answers to that.

(1) in the most general case, you are quite correct, and would be equally correct when talking about a pursuit tracking study. That’s why we see whether the logical consequences of the assumptions are consistent with the observations – the data, in experiments. That’s true of all modelling based on observations of what happens outside the physical boundary of the organism. It’s one reason why I believe Bayesian analysis is the most appropriate way to approach the relation between theory and experiment.

(2) Very often (as is the case in the pursuit tracking study, and in many detection and discrimination experiments) the experiment is set up so as to make it highly likely that the function relating the sensory input to S1 is very similar to the function relating sensory input to S2. Furhtermore, when both S1 and S2 are category perceptions themselves, the function generating the category is hardly relevant to whether the categories are the same or different. In the case of a discrimination experiment, one of them is derived from a current sensory input, the other from a past instruction by an experimenter. That does not affect the control of p(S1, S2), where p(x,y)==(x-y), using a reference value of zero for the difference between the categories X and Y.

(3) When (2) is not the case, one can often make reliable differential estimates, without knowing the forms of the functions. From them, if such is desired, one can determine certain aspects of the functions themselves. That’s what you do when modelling tracking experiments. In the case of detection and discrimination experiments the experiment is deliberately designed so that only differential measures are available (was the presentation class 1 or class 2?). When you and Rick did your re-analysis of Stevens’s power functions, you had to make some assumptions, and found that if S1= f(sensation) had a logarithmic form regardless of the sensation, the results of the experiments would be power functions. But the same would have been true for an infinite number of other input functions, within experimental error. That doesn’t in any way invalidate the results of your analysis, since it is consistent with the data. In the case of a detection or discrimination experiment, the differential inputs D1 and D2 allow for an estimate of the precision of the channel D->S1. If the perception of S2 were also uncertainly derived from the initial instructions (the subject wasn’t quite sure how to react), the results would put a lower bound on that precision (as indeed they do in any case, since one never knows whether there are noise sources one has not incorporated in the analysis, such as hitting the unintended button).

``````    All we can observe are D and O, and the form of the environmental feedback function without the input perceptual function that generates S2.
``````
``````  Apart from the dubious question of whether we can observe the environmental feedback function -- we may, but we may have to model it -- that was my starting point.
``````
``````Yes, but you didn't seem to stick to those initial stipulations.
``````

I thought I did. I don’t think your comments demonstrate the contrary, though your language suggests they do.

Martin