[From Bruce Abbott (950725.0930 EST)]
Bill Powers (950724.1330 MDT) --
Sometimes I feel like Alice in Numberland, and I just can't get the numbers
to cooperate. Anyway, having given them a second look, I now understand how
you derived the numbers I said I was confused about in last night's post. I
assume you are probably setting me straight right about now and that our
posts will cross in cyberspace. Also, when I calculated the FR-2 rate for
Rat 1 the first time, I managed to pick off the wrong intercept, which is
why I was unable to get your result.
What you did was to get the intercept from the regression equation and
subtract it from the seconds per reinforcement number at each ratio value,
leaving the time required to complete the ratio. You then divided this into
the seconds per reinforcement to get the ratio of total time to
ratio-completion time, and used this as a multiplier of the original
response rate figures. You could have done the same by dividing the time
required to complete the ratio into 3600 and multiplying the result by the
ratio value.
The resulting function has a slight curvature in three of the four rats,
with a maximum value at FR-16. The other rat's data are less regular but
show a peak at FR-8. It is possible that the observed nonlinearity is real,
but to a first approximation the data are well fit by straight lines. (This
is especially apparent when you plot overall reinforcement rates rather than
response rates.)
A problem for interpretation of any curvature is that the values for the
lowest ratios are highly sensitive to the intercept of the fitted line (as
you noted). For example, if the intercept for Rat 1 were Rat 2's 5.77
instead of 5.36 (a difference of 0.47 s), the computed response rate
excluding collection time increases by over 1200. Even if the rates at
these values are in reality relatively low, this could simply reflect an
initial acceleration of response rate following return to the lever. At
higher ratios the rate would have time to reach a stable value and the
average over all responses would be higher. Even taking the apparent
nonlinearity into account, the variation in time to reinforcement it
accounts for is trivial compared to the variation attributable to the linear
effect of response requirement. Nearly all the variation in reinforcement
rate across the different ratios can be attributed to the constant
additional time required to complete each additional response in the ratio.
In fact, if you are willing to assume that the fitted line represents the
true relationship and that the deviations from this line are simply
experimental error, then _all_ the variation in reinforcement rate across
ratios can be attributed to this source.
The picture we get from these data is that the animals pressed the bar
at a very high rate at all ratios (while not collecting rewards), with
the fastest rate occurring at a ratio of 16. The collection time makes
less difference at the higher ratios because it is a smaller proportion
of the total time between reinforcements. The peak behavior rates change
less than 50% over the 32:1 range of ratios.
Actually the rates are not all that extreme: even the highest value you
calculated (9978.97) is only 2.77 responses/second or a bit more than a
third of a second per response. Rats can manage over 5 responses/second
with most standard levers. The lowest value (4013.36) gives 1.11
responses/sec or about 0.9 sec/response. Nearly a second per response is a
slow pace. Bear in mind that because of the collection time the rats are
not actually producing as many responses/hour as these numbers would imply.
All your calculations were based on the observed
behavior rate and the ratio; there is no independent direct data on
reinforcement rate. Since the only actual data refer to behavior rates,
we have to do the final correction in terms of behavior rates.
I'm not sure I follow. The data I presented were measured from graphs
showing response rate versus reinforcement rate. I used the response rate
because it provided a higher resolution and computed reinforcement rate from
this. But I could have used reinforcement rate directly. Since one is just
a multiple of the other, there can be no "independent" data on reinforcement
rate.
The actual position of the reference level is not revealed by the
present data, because the collection time makes the extrapolation to
zero problematic. Using the corrected "peak" behavior rates, we would
extrapolate to a very different zero point than we do with the
uncorrected data.
So how would the actual position of the reference level (if there is one) be
determined? Would this be the rate at which the rat consumed food pellets
if given free access to them?
In addition, varying the ratio directly varies the loop gain of the
control system, so the quality of control declines steeply as the ratio
grows from 2 to 64. If the rat is controlling well at FR-2, it may
hardly be controlling at all at FR-64. This is no problem for a control
model, because the model, too, will control poorly at FR-64, assuming
that its parameters remain the same and that it controls well at FR-2.
Excellent point. The implication would be that higher errors would be
necessary to sustain a given level of responding at high ratios than would
have been the case had the gain remained constant. If the constant-gain
relationship were linear, the decreasing-gain relationship would be
curvilinear, increasingly undershooting the straight line at higher ratios.
What does stay the same across all ratio requirements are (a) the
time required to collect the reinforcer and return to the lever
(about the same for all subjects) and (b) the average rate of
responding (which differs across subjects). At a given level of
deprivation, and for a given reward, the rat maintains a given rate
of responding.The rate of responding _between collections_ varies only slightly with
the schedule ratio, although the variations seem consistent with a peak
rate at FR-16. I find no simple proportional relationship between rate
of responding and reward rate. In fact, the rate of responding remains
essentially constant (plus-minus 25%) while the reward rate varies by a
factor of 5 to 6. Your generalizations above don't seem to fit the data
You seem to be misreading me. You just stated what I stated, and then
concluded that what I said was wrong. Let me restate:
For a given animal, the rate of responding between collections is
essentially constant regardless of the ratio requirement (I am assuming the
deviations are experimental error, to a first approximation). The time to
collect the reinforcer is essentially constant. Thus, the overall rate of
responding (including lever pressing and collection) is essentially
constant. Reward rate decreases linearly with the ratio requirement (thus
varying by a factor of 5 to 6). The decrease in reward rate is entirely
attributable to the increased time required to complete the increased
response requirement.
So this leads to the question: where is there any evidence for control of
reinforcement rate? It would appear instead that a given level of
deprivation, size of reward, etc. as provided in these experiments sustains
a particular rate of responding. I'm not really comfortable with that
conclusion, but it seems to be implied by the data.
These data don't include numbers concerning level of deprivation. I
would expect that the almost-constant rate of responding might show a
relationship to level of deprivation. Deprivation, of course, means a
_decrease_ in the mean sustained reinforcement rate. I would expect
behavior to increase with deprivation, up to a point, and thus to
decrease with increases in sustained reward rate.
Ettinger and Staddon do provide data comparing 80% and 95% free-feeding
weight conditions, but as the 95% functions were assessed only once they are
not nearly as stable as the averages I provided over four replications for
the 80% condition. The reliable effect apparent in the graphs (of rsp rate
vs rft rate) is a reduction in slope. The 80% and 95% lines tend to
converge as the ratio decreases toward FR-2. The implied "collection rate"
is either constant or, oddly enough, somewhat higher at 95% than at 80%.
(I'm just estimating visually from the plots.)
If you maintain the same rate of responding, the reward rate will
decrease as the ratio increases. That's just arithmetic. So it seems
that the reinforcement rate has no effect of its own on the behavior
rate; what does affect behavior rate is the error signal, the level of
deprivation.
Yep. But does this make sense?
Regards,
Bruce