[From Bill Powers (950731.1200 MDT)]
Bruce Abbott (950730.1920 EST) --
This is getting tiresome. Bruce, you are right again. Boy, are you
right!
I started setting up cyclic3.pas. First I fitted a straight line to the
original behavioral data, to get a regression and an intercept. Since
that used up the data, I then computed p by using that straight line for
each value of m instead of the actual data. This gave me a smooth curve
for the values of p as a function of m. I then discovered that by
adjusting the collection time slightly away from your values, I could
get p to be exactly constant, as close as I could read the displays.
The values of c for rats 1 through 4 were 5.60, 5.66, 5.70, and 5.44 --
not much different from the values you got.
That sent me back to the drawing board. Knowing that it was possible for
the pressing rate p to be exactly constant, I went through the equations
again with that in mind.
First, the function describing the mean behavior rate b as a function of
the food delivery rate r:
(1) b = G*(r0 - r) where r0 is the "reference level." You'll see why I
put that in quotes shortly.
b = m/(m/p + c)
where p is pressing rate between collections, c is collection time.
r = 1/(m/p + c), so
m/(m/p + c) = G*r0 - G/(m/p + c)
Collecting terms with a common demoninator, we get
(m + G)/(m/p + c) = G*r0, or
(2) m+G = G*ro*(m/p+c)
If p is a constant the only variable is m and we can differentiate both
sides with respect to m:
d/dm:
1 = G*r0/p, or
(3) p = G*r0
Plugging that back into (2) we get
m + G = m + G*r0*c or
G = G*ro*c, or
r0 = 1/c
The so-called reference level turns out to be just the maximum rate at
which food deliveries can be obtained.
From (3), this gives us
G = pc
We can now put these derived values into the original "control system"
equation (1) and get
b = pc*(1/c - r)
For Rat 1, p = 5500 and c = 5.60/3600, so the equation is
b = 8.4*(643 - r)
This looks like a control system with a gain of 8.4 and a reference
level of 643 reinforcements per hour. The only problem is that it's not
a control system: it's a crock.
ยทยทยท
---------------------------------
In this experiment each rat is either collecting food or pressing the
bar at a constant rate. This is true for all schedules from FR-2 to FR-
64. The schedule has absolutely no effect on the behavior. All of my
elaborate modeling, and all of Staddon's, I presume, is worth exactly
nothing. The rat was either pressing the bar at a constant rate or
collecting the food, and that is all that happened in these experiments.
We aren't seeing reinforcement and we aren't seeing control. What we're
seeing is a very hungry rat pressing the bar as fast as it can, given
other limitations, and collecting the food whenever it appears.
---------------------------------
Obviously the same consideration will be seen in the Motherall data, at
least for the right limb where there is essentially a negative straight-
line relationship. So all of my modeling of those experiments and
Staddon's models as well are bull***t.
What remains now is either to get hold of the original data and verify
that the pressing rate is a constant, or to do an experiment where we
can get the same data. If the data check out with these deductions, a
very large pile of experimental data (and all the fancy analyses of
them) is bound for the garbage can.
---------------------------------
Bruce, you are a genius. When you thought of checking out the time per
reinforcement against the pressing rate, you pulled the foundations out
from under everything. Your initial deductions were right: the behavior
is not varying, and the curves we see are strictly an artifact.
This should not affect OPCOND5, because the collection delay is built
into it and I believe we can reproduce these effects. Maybe when we see
how the model parameters have to be set up to imitate the Staddon data,
we will get some insight into the kind of conditions needed to show a
true control phenomenon.
----------------------------------------------------------------------
Best,
Bill P.