Graphical derivation of organism function

[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.

ยทยทยท

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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.