Collier et al. analysis

[From Bruce Abbott (950810.1040 EST)]

Here is a preliminary analysis of some of the Collier, Hirsch, and Hamlin
(1972) data from their Experiment 1.

Bill Powers (950808.1640 MDT) --

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.

Here are the food intake (grams), number of meals, and intake per meal
(grams) each day during free feeding in Exp. 1:

        Intake Meals Intake/Meal
Rat 1 22.9 10.6 2.16
Rat 2 24.9 8.9 2.80
Rat 3 21.9 14.4 1.52

As the ratio requirement increases, the number of meals earned per day was
observed to decline. The rat can compensate for the reduced number of meals
earned per day by increasing its intake per meal. If it did NOT do so, we
can estimate the daily intake at each ratio by multiplying the free-feeding
intake per meal by the number of meals earned at each ratio. We can then
compare these figures to the food intakes actually observed at those ratios.
In the following table, the "expected" intake (in grams) is the amount
expected under the assumption that intake amount is NOT controlled.

                   FOOD INTAKE/DAY (GRAMS)
        -----Rat 1 ----- -----Rat 2 ----- -----Rat 3 -----
Ratio Expected Actual Expected Actual Expected Actual
  FF 22.9 22.9 24.9 24.9 21.9 21.9
   1 19.2 20.0 25.5 23.2 17.3 17.7
   5 17.1 22.0 23.2 20.9 14.7 22.9
  10 14.7 19.9 19.6 20.9 14.1 16.7
  20 16.2 20.9 17.9 20.9 8.8 15.8
  40 11.2 18.8 15.1 20.9 9.1 17.6
  80 12.5 19.8 11.8 21.0 7.3 17.7
160 9.9 22.9 4.5 17.9 10.3 22.1
320 7.3 17.7 2.8 12.9 5.8 15.9
640 4.3 14.8 --- --- 3.3 13.8
1280 2.6 15.8 --- --- 3.3 14.8
2560 2.6 12.8 --- --- 2.1 16.9
5120 1.3 9.8 --- --- 1.7 8.8

The data support the view that food intake is controlled. If, for the
moment, we assume that the decrease in number of meals earned per day is not
under control but is simply suppressed by increased ratio requirement, then
we can forget about the ratios in the control analysis: from the point of
view of the food-intake control system, the changing number of meals per day
is simply a disturbance requiring a varying meal-size to offset. We would
see the same effect if the experimenter simply arranged for varying numbers
of meals per day, rather than its being a consequence of the varying ratio

One thing to note is that the above figures are rather sensitive to errors
in measurement, rounding errors, and errors in estimating from the figures.
Meal size, in particular, is very sensitive to these errors, especially as
the divisor (Meals) becomes small. One can probably improve on the
estimates by smoothing the observed data prior to conducting the remainder
of the analysis.

I conducted a further analysis to determine the relationship between meal
size (intake per meal in grams) and number of meals per day. If total
intake per day were under perfect control, one would expect the product of
meal size and number of meals per day to be constant:

     Intake = Mealsize * Meals = k, both Mealsize and Meals > 0

Solving for Mealsize, we get

     Mealsize = k/Meals

Taking the log of both sides of this equation yields

     Log(Mealsize) = Log k - b*Log(Meals), where b = 1.0

We can estimate k, the constant intake, from the intake value during free
feeding. If control is perfect, Log(Mealsize) will be a linear function of
Log(Meals) with an intercept of Log k and a slope of 1.0.

Linear regression produces the following results for the three animals in
this experiment (using common logs):

        inter slope r-sq k
Rat 1 1.10 -0.744 0.982 12.6
Rat 2 1.16 -0.782 0.985 14.5
Rat 3 1.07 -0.770 0.940 11.7

The value for k above is the antilog of the intercept and is in grams. At
the y intercept, Log(Meals) = 0, which corresponds to Meals = 1.0. Thus k
in this analysis is not equal to the free-feeding intake quantity but
rather, to the quantity consumed at a meal rate of one meal per day. It is
less than the free-feeding value because the slope of the line is shallower
than -1.0, the value expected under the assumption of perfect control. Of
course, we don't expect control systems to be perfect; there must be some
error between the reference signal and percepual signal or there will be no
action. On the other hand, the absence of control would imply no systematic
relationship between meal size and number of meals per day.

Bill P.: Perhaps you can work out how the slope of this line would relate
to the gain of the control system, if that seems worth doing.

It is noteworthy that the intercepts and slopes of the functions are very
similar across the three animals. I doubt that the slopes are significantly
different. This is all the more interesting when one recalls that Rat 2 did
not complete the series because the increasing ratio requirement had so
quickly suppressed the number of meals per day, compared to its effect on
the other two rats. Yet despite this difference in sensitivity of response
rates to the ratio requirement, all three rats produced essentially the same
function relating meal size to number of meals.

What we need now is a control-system model to check against these findings.