[From Bill Powers (960121.0900 MST)]

In exploring Bruce's various ways of handling the Killeen model of

operant behavior under ratio schedules I have been bothered by

intermediate variables and functions which seem to have been introduced

simply to fit a curve. These extra variables and functions involve

unobservable quantities; evaluating the parameters (in the same or other

experiments) requires accepting the existence of the auxiliary variables

_a priori_.

There is a more direct way of determining the system function from the

data without postulating invisible functions of unobservable variables.

In a plot of observed rates of reinforcement and behavior, we have one

set of straight lines that indicates the general relationship between

apparent behavior rate B and reinforcement rate R for each ratio N --

the apparatus equations. The lines begin at the origin and have slopes

equal to N, the ratio requirement. Along each line there is a single

point where the apparent behavior rate and reinforcement rate were

observed to lie for the corresponding ratio. These points, taken across

all ratios, form another curve, which is a direct representation of the

system function -- that is, the way in which behavior rate depends on

reinforcement rate. Given this direct graphical determination of the

system function, we can then try to write an equation that fits the data

points. The equation then approximates the way in which behavior depends

on reinforcement via the organism.

The apparent behavior rate has to be corrected for factors that we know

can make the apparent rate different from the actual rate (during

successive bar presses). The main factor, as Bruce Abbott showed last

year, is the collection time. The animals must cease pressing in order

to collect the food pellets, and during this time the actual pressing

rate is zero. The recorded pressing rate, however, is calculated as

total presses divided by total elapsed time, which gives a low estimate

of actual pressing rate because part of the time no pressing is

occurring. Finding the true system function requires plotting the data

points using the true behavior rate instead of the apparent rate.

If the collection time is C, the ratio requirement is N, and the

observed (apparent) behavior rate is B, we can calculate the true

pressing rate B' as follows.

The interval taken up by N presses at the true pressing rate B' is N/B'.

The total interval taken up in fulfilling the ratio requirement _and

collecting the food_ is N/B' + C. The apparent behavior rate B is the

total number of presses, N, divided by the total elapsed time, or

N

B = ----------

N/B' + C

Solving for the true behavior rate B' in terms of the apparent rate B we

have

1

B' = ------------

1/B + C/N

From the old Motheral data found in Staddon's book, I measured the data

points from the plot in the book (Units are per session of 1 hour).

After the observed behavior rate are four columns showing the corrected

behavior rate for collection times of 3, 4, 5, and 6 seconds. This table

just shows corrected behavior rates for each ratio and assumed

collection time. The reinforcement rates would then be used as the x

axis and the corrected behavior rates as the y axis to plot the organism

function, B' = f(R):

B'(calc)

N(est) B(obs) C=3.0 sec C=4.0 sec C=5.0 sec C=6.0 sec

1 210.0 254.5 273.9 296.5 323.1

2 390.0 465.7 497.9 534.9 577.8

3 1056.0 1237.5 1312.7 1397.6 1494.3

12 1872.0 2151.7 2264.5 2389.8 2529.7

21 2673.0 2990.2 3113.3 3247.0 3392.7

43 2989.0 3172.8 3239.2 3308.4 3380.7

90 1756.0 1785.0 1794.9 1804.9 1815.0

180 1320.0 1328.1 1330.8 1333.6 1336.3

In this case, the collection time doesn't make a large difference, but

in other cases it does.

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The ratios 1 through 21 produce points that can be fit pretty well with

a straight line. At higher ratios, the behavior rate falls further and

further below a straight line. A simple control model fits the low-ratio

data reasonably well; some other hypothesis (such as a cost-benefit

effect, or just as likely, the animal's spending less time at the lever)

will be needed to account for the rest of the curve.

If we _measure_ the collection time instead of guessing at it, we will

have a complete graphical form for the steady-state organism equation.

No other hypotheses are needed, nor can any theoretical curve be more

true to the organism.

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Best,

Bill P.