# Ratio data

[From Bill Powers (950808.1640 MDT)]

Bruce Abbott (950807.1555 EST) --

It would appear that rats are able to control body weight to some
degree at least if the decline in earned opportunities to eat can
be compensated for by an increased amount eaten per opportunity.
Yet whether such compensation is possible or not, increasing the
response requirement ("cost" of a meal) drives down the number of
meals "purchased." This appears to suggest rather strongly that
rate of access to food is not a controlled variable in the rat.

The first thing we have to do here is to specify a test for control. The
method used in PCT is simple and quantitative. You start with some
baseline conditions and measure the behavior and the state of the
controlled variable under those conditions. Then you introduce a change
in the conditions, and postulating NO CHANGE in the behavior, you
compute the new state of the controlled variable. You then compare this
predicted new state of the variable with the state predicted on the
basis of no control -- that is, no change in behavior. If the predicted
new state of the variable matches the actual state under the new
conditions, there is no control.

Consider your method of analyzing the time per response and the
collection time. You measure the pressing rate (the true output) and the
rate of reinforcement at one ratio. Then, assuming this pressing rate
does NOT change, you calculate the new rate of reinforcement under a new
ratio. Comparing the predicted reinforcement rate at the new ratio with
an actually measured reinforcement rate, you find that there is no
difference. The new reinforcement rate is explained completely without
any change in the assumed (or measured) pressing rate. This proves that
there is no control of reinforcement rate when disturbed by changing the
ratio.

To demonstrate that there _is_ control, you must show that the actual
change in the proposed controlled variable is less than the change
predicted under the assumption of no change in the system's output. The
degree of control is indicated by the fraction of predicted
(uncontrolled) change that actually occurs because of a change in the
behavior. For example, if the actual change is 1/10 of the predicted
change or less, we can conclude that relatively good control exists. If
we can actually measure the behavior in question, we can verify that it
changes in just the right direction and amount to account for the
smallness of net change in the controlled variable.

In cases where there is a nonlinearity, as when we change ratios, the
evaluation of the existence of control may depend on the levels of the
variables. At FR-1 or FR-2, we may find that there is tight control, but
at higher ratios the difference between predicted and observed levels of
the controlled variable continually becomes less until we reach whatever
threshold we agree to use in distinguishing control from noncontrol.

In the present case, there are two possible contributions to the
controlled variable, the food intake. One is the amount eaten per meal,
and the other is the rate at which meals are produced. The intake rate
is the product of these two variables, which can be considered as
behavioral variables pending further analysis.

The controlled variable is the food intake; the behavioral variable is
the product of meal frequency and meal size. The basic test then amounts
to measuring some baseline values of these variables, then changing the
ratio and predicting what effect that would have on the food intake rate
given that the behavioral variable did not change. This predicted value
would then be compared with the actual rate of food intake under the
changed ratio. If there were no difference, we would have proven that
the rate of food intake is not controlled in the assumed way under
variations in the ratio.

If the food intake proves to be controlled, we have then defined a unit
of control that can be further analyzed. There are two behaviors
involved, and a change in either or both of them could explain the
control that is observed. We would then try to obtain data that would
show how much each of these behaviors varies as the disturbance is
applied. It is possible that both will vary or that only one will vary.
We are certain by this point that the _product_ varies in a way that
produces control, so at least the time per meal or the rate of producing
meals must vary.

Suppose we find that as the ratio increases, the rate of food intake
remains constant. Suppose we observe that the rate of eating goes up,
while the frequency of meals goes down. Obviously, for the food intake
to remain the same, the size of meals must increase as fast as the
frequency declines. However, it is still possible that there is some
control of meal frequency, even though the effect of the disturbance on
it is large. To determine this, we must look at meal frequency as a
controlled variable, and see whether its changes can be accounted for
completely by assuming a constant pressing rate between the meals. Once
again, we apply the same test, comparing the calculated change in meal
frequency as the ratio changes with the measured change in meal
frequency. If the measured change is less than the change calculated on
the basis of no change in pressing rate, there is some control, and we
can estimate how much quantitatively.

Actually, since we know the food intake rate, we need only know meal
frequency in order to deduce meal size.

Exactly the same procedure goes for testing the hypothesis that meal
size is a controlled variable, with eating rate being the behavior. Here
there should be little difficulty, since there is no disturbance and the
relationship is direct.

To model this whole situation, we would start with a control system for
controlling food intake rate. The error signal would be routed to two
other control systems, one controlling meal frequency and the other
controlling meal size. An auxiliary system would be required to gate
these two control systems on and off as appropriate, assuring that each
one corrects its error before the other is allowed to operate. We might
find that the reference signals should be given different weights
multiplying the error signal that drives both lowest-level systems.

With this model matched to the data, we could then propose a higher-
level control system controlling body weight or nutritional state, the
error signal providing the reference signal for the food intake control
system and that system's error signal providing reference signals for
both lower systems. So we end up with three levels of control connected
in the usual hierarchical manner. The bottom level contains two control
systems, the higher levels one each.

Instead, error in the nutrient-control (body weight?) system seems
to establish something like the "value" or "attractiveness" of the
food, which is diminished by the effort and/or time required to
obtain it.

Can't we get along without such made-up concepts? They aren't necessary
in a model, and add no meaning to what we already can find out. The gain
of the output function determines how much effect a given amount of
error will have. When we're finished with the model, we can work out
informal-language interpretations of what the various components of the
model mean, but that can be left until the very last.

If there is another explanation for why the rats are able to
control body-weight by increasing the amount consumed/meal but not
by increasing the meal rate, I'd like to hear it. We will
definitely have to take a look at what happens when we disturb meal
rate at a given ratio value in our experiments.

I think our explanations are in general agreement. We do not yet know
that the meal rate is invariant. We will not know that until we have
done the experiments ourselves and verified with the detailed press-by-
press data that this is true. Until then we will have no mutually-
agreeable opinion as to what is constant and what varies. If it turns
out that meal rate is constant, fine. If it doesn't, fine. But
notwithstanding the fine quality of the Collier et. al. experiments, the
only results I will trust will be ours. If our results agree with those
of Collier et. al., that will be a vindication of their results and will
multiply their value tenfold. If our result don't agree, let's make
damned sure that the correct data are ours.

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Best,

Bill P.