Mistaken analysis

[From Rick Marken (950730.1210)]

Bruce Abbott (950730.1130 EST) --

Bill, where _you_ go astray here is that you apparently do not realize
that

p = 1/k by definition:

But this definition assumes that the observed 1/r is k*m +c. Bill's point
was that we don't know that the intercept of the regression equation is
the actual collection interval (c); all we know is:

m/p + c = k*m + j

where m/p + c represents the actual pressing rate and collection
interval that produce the observed 1/r at each m. The slope and
intercept of the best fitting regression equation (k and j) are not
necessarily the 1/p and c values the rat actually produced at each m.

You are _embarrassing_ yourselves. Perhaps a vacation would help.
(;->

We're used to it. Whenever we present PCT to conventional psychologists
we know that we must seem like we're just too stupid to be believed;
heck, we're saying that everything they take for granted - the bedrock
of their discipline -- is wrong. Of course, they think we are embarassing
ourselves; they know that if they said these things, they would be embarassing
themselves in front of their colleagues. We don't mind. When they say that
we're embarassing ourselves, they are just embarassing themselves in
our eyes. But we are VERY understanding;-)

I've got a couple of exams to write today and some work to do on the
research methods text, so I may not get to that right away

No problem. Your work on the research methods text is very important and
I support all the time you put into it because we really need a book about
how to study living control systems. Since you know that organisms are
living control systems I know that you would find it too embarassing to
publish another edition of a conventional research methods text. So I
can't wait to see your new research methods text, which will be the first
text devoted to the description of methods for studying living control
systems.

Bill said:

Rick and I have both been going slightly nuts over your derivation
showing that the pressing rate is constant.

Bruce:

Why? What's to go nuts over? This is about as simple as it gets.

It's not quite as simple as it seems. And the derivation was driving me
nuts because it didn't seem to jibe with other evidence (data and models)
that suggests that reinforcement rate is controlled. Your derivation
doesn't prove that reinforcement rate is NOT controlled, but it does
suggest an interesting puzzle, viz. how one determines what the
variance of a variable (like reinforcement rate) would be if it were NOT
under control? Your analysis shows that conventional operant experiments
are even less useful as studies of control behavior than we had imagined.

Sorry to ruin your day, Rick. And it started out so promising! (;->

You never ruin my day; you MAKE it;-)

And I can hardly wait to see the new edition of your research methods
book. I was thinking of writing such a text myself -- we definitely need
such a book. But now you are writing it so I'm just going to sit back and
enjoy it when it comes out -- soon, I hope.

Best

Rick

[From Bruce Abbott (950730.1920 EST)]

Rick Marken (950730.1210) --

Bruce Abbott (950730.1130 EST) --

Bill, where _you_ go astray here is that you apparently do not realize
that

p = 1/k by definition:

But this definition assumes that the observed 1/r is k*m +c. Bill's point
was that we don't know that the intercept of the regression equation is
the actual collection interval (c); all we know is:

m/p + c = k*m + j

where m/p + c represents the actual pressing rate and collection
interval that produce the observed 1/r at each m. The slope and
intercept of the best fitting regression equation (k and j) are not
necessarily the 1/p and c values the rat actually produced at each m.

So, now p and c are to be the "actual" values rather than best estimates
from the data, as I had assumed in my previous response. Then what you are
stating is that I have erred by assuming that the actual values of 1/p and c
are identical to the values of these parameters as estimated by linear
regression (k and j, respectively). I made no such assumption.

Furthermore it makes little practical difference. Because of the extremely
high r-squares, the data severely constrain the possible values of 1/p and c
to values close to those estimated from the fit.

What I will grant you (as I have pointed out before) is that the true
relationship may not be linear, as the fits suggest, but may depart
systematically from linearity in some (small) way; my linear fit is only an
excellent empirical approximation, given the data.

As to interpretation, even p may not represent the actual rate of pressing
the lever. 1/p is the time penalty associated with each lever press, but it
need not be the time required per lever press. For example, it is possible
that the rat always presses at some constant high rate once it has returned
to the lever, and that this rate is higher than p. Assume for the moment
that the rat presses at a rate of 5 presses per second while completing the
ratio. That implies that 1/p, the time to complete a lever press, is 0.2
seconds. To complete a ratio run of, say, 8 presses, would require 8*0.2 =
1.6 seconds. Now assume that p computed from the data is 2.0 seconds. This
implies that 1/p = 0.5 seconds and that the rat will require 8*0.5 = 4.0
seconds to complete the ratio. Given that the actual time is 1.6 seconds,
there is a discrepancy of 4.0 - 1.6 = 2.4 between the observed press rate
and the rate estimated from the regression equation. Where is the "missing"
time?

The answer is that it is spent in the pause. As the number of required
responses per reinforcement increases, each additional response is
associated with a small constant increase in the prereinforcement pause
length. This adds an amount to the constant collection time proportional to
the number of responses in the upcoming ratio.

Given this possibility, it would not be surprising to find that "collection
times" vary somewhat in proportion to the ratio requirement (rather than
being strictly constant as given by c) and that response rates during the
ratio "run" are higher than expected from estimates of p. Such a result
would not be inconsistent with observed fit of the data at hand.

Regards,

Bruce