# Collier et al. analysis

**URL:** <http://discourse.iapct.org/t/collier-et-al-analysis/2662>\
**Category:** CSG1995\
**Created:** [August 10, 1995, 3:38pm UTC](http://discourse.iapct.org/t/collier-et-al-analysis/2662 "1995-08-10T15:38:05Z")\
**Posts on this page:** 1\
**Page:** 1

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**Author:** ![Bruce\_Abbott](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/b/ba8739/32.png) [@Bruce\_Abbott](http://discourse.iapct.org/u/Bruce_Abbott)\
**Post date:** [August 10, 1995, 3:38pm UTC](http://discourse.iapct.org/t/collier-et-al-analysis/2662/1 "1995-08-10T15:38:05Z")

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[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:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;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.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;FOOD INTAKE/DAY (GRAMS)  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;-----Rat 1 ----- -----Rat 2 ----- -----Rat 3 -----  
Ratio Expected Actual Expected Actual Expected Actual  
&nbsp;&nbsp;FF 22.9 22.9 24.9 24.9 21.9 21.9  
&nbsp;&nbsp;&nbsp;1 19.2 20.0 25.5 23.2 17.3 17.7  
&nbsp;&nbsp;&nbsp;5 17.1 22.0 23.2 20.9 14.7 22.9  
&nbsp;&nbsp;10 14.7 19.9 19.6 20.9 14.1 16.7  
&nbsp;&nbsp;20 16.2 20.9 17.9 20.9 8.8 15.8  
&nbsp;&nbsp;40 11.2 18.8 15.1 20.9 9.1 17.6  
&nbsp;&nbsp;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  
requirement.

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:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Intake = Mealsize \* Meals = k, both Mealsize and Meals \> 0

Solving for Mealsize, we get

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Mealsize = k/Meals

Taking the log of both sides of this equation yields

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;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):

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;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.

Regards,

Bruce