[From Bruce Abbott (960221.1710 EST)]

Bill Powers (960217.0100 MST) --

Bruce Abbott (960216.1625 EST)

The reason that the error between programmed and obtained

reinforcement rate is rather large at VI 15-s and decreases

progressively as the schedule interval size increases is that this

is the result of the limits imposed by the feedback function at the

rates McSweeney's rats tended to respond on these schedules.Beautiful, Bruce! Can you work this in reverse? That is, for each rat,

find the steady rate of responding that gives the best fit to the data

by using your formula? This should make the model fit the data even more

closely.

Below are the response rates (resp/min) implied by the observed

reinforcement rates using the linear function (VId = VIp + IRT)

Rat 15 30 60 120 240

161 12.0 8.5 9.0 2.0 0.6

162 10.0 8.0 9.1 2.0 0.2

163 10.3 12.2 15.8 3.6 0.4

164 6.4 7.1 6.1 5.7 0.4

## ···

----------------------------------

MEAN 9.7 9.0 10.0 3.3 0.4

These response rates give the EFFECTIVE rate of responding -- how fast the

rat would have to respond on average IF it responded with constant

probability in each successive time increment. Actual rates usually

differed from these values, which implies that the IRT distributions may

have varied within schedules (across components), across schedules, and from

rat to rat. However, as viewed from the perspective of the effective rate

of responding, the data do reveal some consistencies. First, none of the

rats responded at a high effective rate on the 240-s VI, and three out of

four showed a large decline in effective response rate on the 120-s VI as

compared to the rates sustained by the shorter intervals. There is no clear

pattern of rate-change from VI 15 to VI 60. For two rats the VI 15-s

schedule produced the highese effective rates, for one the VI 30-s schedule

did and for the remaining rat the VI 60-schedule did. From VI 15-s to VI

30-s increasing, level, and decreasing trends can be found; on average the

rates across these schedules was nearly constant at 9-10 resp/min. Rat 164,

which had the lowest effective response rates on these schedules, was the

only rat to maintain nearly the same effective rate on the VI 120-s

schedule; the other rats decreased their response rates considerably.

Killeen's formula, which assumes normally distributed IRTs, produces lower

effective response rates, but the general patterns are the same. For

example, Rat 162's effective response rates calculated on both the linear

and Killeen formulas were as follows:

15 30 60 120 240

linear 10.0 8.0 9.1 2.0 0.2

Killeen 5.6 4.3 4.6 0.9 0.1

If you can do this fitting, you can then do it for each 5-min segment of

the experiment, including the "satiation" regions, to get a picture of

how the actual rate of responding varies with time, for each rat. This

in turn might allow us to extrapolate to the total amount of obtained

food that would result in zero response rate -- the reference level for

each rat.

I think this can be done, but I would not be as simple as it should be owing

to the variations in _scheduled_ reinforcement rate across 5-min segments.

This becomes more of a problem for larger VIs. McSweeney's schedules

comprised 25 intervals each. On VI 15-s it required 25*15 = 375 s to

complete one cycle through the schedule if each reinforcer were collected as

soon as available; this is greater than the 300 s component length, so a

cycle cannot be completed in one component and the intervals programmed over

that time may not average to 15-s, depending on the sequencing of the

intervals. For VI 240-s it would require 6000 s of session time to complete

one cycle through the schedule, which permits much more variability in

programmed reinforcement rate across components. Still, here is an example

assuming VI 15-s actual average per component (for Rat 164):

Rft/5 min 7.0 14.4 14.6 13.6 11.8 10.2 6.8 6.4 4.8 2.2 2.8 2.6

Ex Rsp/ min 2.2 10.3 10.8 8.5 5.8 4.2 2.1 1.9 1.3 0.5 0.7 0.6

Ob Rsp/ min 17.1 17.9 12.6 8.6 7.9 4.2 3.1 1.5 1.3 1.4 1.5 2.0

The expected responses/min were computed using the linear formula. The

large initial difference between expected and observed rate probably

reflects a "warm-up" effect: the rat probably did not respond at all during

the first half of the component and then at a high rate thereafter. If a

long interval were being programmed then, the excess rate would not

translate into reinforcer deliveries. After the initial components, most of

the observed rates are not far from those prediced from the linear function,

even though the actual scheduled rates in each component were not taken into

account in the computations.

Re: Lab

I have obtained a video camera with which to photograph the activities

during each session and will have a few leftover rats from a colleague's

study available for my use in about two weeks. Between now and then I hope

to have the programs developed and tested that will schedule reinforcers and

record all responses and reinforcer-deliveries, along with the times at

which they occur.

Regards,

Bruce