[From Bruce Abbott (981018.1730 EST)]
Bill Powers (981018.0922 MDT) --
Bruce Abbott (981017.2155 EST)
What a strange thing to say, after I have proposed that availability of food
in the chamber is controlled, rather than its rate of occurrence.
Granted,
I've separated this from the rest of your sentence, so that it gets noticed.
But now comes a caveat:
although I have a hard time imagining how you measure
"availability." How did you prove that "availability", rather than, say,
amount of food in the cup, was controlled?
You're right--the experiment wasn't designed to test how _much_ food the rat
wanted to see in the cup before it stopped pressing and started eating. On
CRF (1 press per food-delivery), some rats did develop the habit of pressing
twice (and thereby getting two pellets into the cup) before moving to the
cup and consuming the food. But that falsify my statement that the rats
controlled for having food in the cup, it elaborates it by specifying a that
they wanted a specific quantity there. On schedules other than CRF, by the
way, the rat never allowed two pellets to accumulate in the cup before
moving to the cup and eating what was there.
It
is also obvious that ruling out one CV does not rule out the possibility
that the rat's actions may be controlling some other CV. It's not the sort
of logical error I'm likely to commit.
OK, I hoped not and now you say you don't make this sort of error. I can
only accept your claim until you disprove it (it doesn't take long: see
below).
I looked below: the promised disproof is not there. Instead, I find you
making the same error you made the first time. (See below.)
In that model, decreasing the reinforcement rate
(by lengthening the average interreinforcement interval of the VI schedule)
was supposed produce an increased rate of lever-pressing as the rat
attempted to offset the reduced reinforcement rate. In fact the opposite
happened. The model is incorrect.
No, the opposite did NOT happen, because the (apparent and illusory)
decrease in the rate of pressing was not caused by a decrease in the rate
of reinforcement, but by time away from the lever that affected both the
apparent rate of pressing and the apparent rate of reinforcement.
Bill, you're making a terrible mistake here; I only wish I could figure out
how to set your thinking strait about this. All I can do is try.
First, we are talking about performance on variable-interval schedules, not
ratio schedules. Both require the rat to press the lever in order to
receive food pellets, but -- and I'll separate this for emphasis -- the two
schedules
differ in their their environmental feedback functions.
On ratio schedules, each pellet is delivered only after a certain number of
lever-presses has been completed. The ratio of the ratio schedule specifies
the number of lever-presses per pellet delivery. Pellet delivery is thus
directly proportional to responses completed and consequently, rate of
pellet delivery depends directly on rate of responding.
On interval schedules, each pellet is delivered only after (a) a certain
interval of time has passed since the previous pellet delivery and (b) a
single lever-press has occurred _after_ the interval has elapsed. (Any
presses occurring prior to the end of the interval are counted but have no
effect on pellet delivery.) What is the nature of this environmental
feedback function?
To answer this question, I'll start with the case in which the interval is
of fixed length (a so-called fixed-interval or FI schedule). For
simplicity, assume that we replace the rat with a timer that closes the
switch on the lever every t seconds. The fixed interval is tau seconds.
From this information we can deduce the following three cases:
1. If t is equal to tau, then a pellet will be delivered for each
"response" (tick of the timer) at a rate of one pellet per tau
seconds. Every response will be immediately followed by food
delivery.
2. If t is greater than tau, then a pellet will be delivered for each
response at a rate of one pellet per t seconds. Every response will
be immediately followed by food delivery.
3. If t is less than tau, then a pellet will be delivered only when nt
is greater than or equal to tau, and n-1 responses will occur without
being followed immediately by food delivery.
Assume that tau = 30 seconds (an FI 30-s schedule) and t = 8 seconds. The
first three responses produce nothing (nt = 24 seconds, less than tau). The
fourth response occurs at nt = 32 seconds, 2 seconds greater than tau, and
is immediately followed by pellet delivery. The "programmed" interpellet
interval is tau (30 seconds), but the "observed" interpellet interval (for
this particular interval) is 32 seconds, or about 6.7% longer than it would
be if the timer were running at an infinite rate rather than at one tick
per 8 seconds.
But this analysis assumes that the rat can generate accurately timed
responses at a constant interval. In fact the intervals between responses
vary to some degree. In that case the time at which a given response occurs
will be unrelated to the time at which tau elapses. If the mean time
between responses is 8 seconds, then sometimes a response will occur just as
tau elapses, producing the reward, and sometimes it will occur just prior to
tau elapsing, producing nothing. In the latter case the next response will
occur on average 8 seconds later. It can be shown that on average the delay
between the end of tau and the next response will be t/2. Thus, for t = 8
seconds, the observed time between rewards will be tau + t/2 = 30 + 8/2 = 34
seconds.
_Doubling_ the response rate to 1 press per 4 seconds cuts the time between
rewards to 32 seconds. Doubling it again cuts it to 31 seconds. Doubling
it yet again cuts it to 30.5 seconds. So an increase in response rate from
7.5 responses per minute (1 press per 8 seconds) to 60 responses per minute
(1 press per second) reduces the average waiting time for food from 34
seconds to 30.5 seconds, or about a 10% improvement in waiting time at a
cost of an 8-fold increase in response rate. As tau increases, there is a
decline in the percentage improvement in waiting time produced by a given
increase in response rate. For example, on FI 300-s, the obtained reward
interval at 1 press per 8 seconds is 304 seconds (within 1.3% of the
theoretical optimum), and this falls to 300.5 seconds (about 1.2%
improvement) at 1 press per second.
In variable interval schedules, tau changes after each reward, but tau
averages to some specified value. The variation in tau makes it impossible
for the rat (or anyone else) to "time" responses to match tau, which makes
the analysis in terms of the rat's somewhat variable response rate have even
more force: the end of tau and occurrence of the next response can be
treated as randomly related with some specified mean difference.
So as you should be able to see, at relatively high rates of responding,
rate of reward is nearly independent of rate of responding (that is, reward
rate varies little with variation in response rate). Rats on VI schedules
typically maintain response rates sufficient to place them within this
region of the environmental feedback function. Until the increase in t
becomes significant, the reduced response rates associated with higher
values of tau are not accompanied by any appreciable change in reward rate.
The
independent variable is time away from the lever; the two apparent rates
both depend on time away from the lever and are not causally related to
each other (based on _that_ measurement).
The rat can reduce its response rate by (a) generating a longer average time
between presses wile standing at the lever or (b) maintaining a relatively
constant rate while at the lever but spending a greater time away from the
lever. I have indicated that both changes occur on VI schedules over the
range of values of tau investigated. What I don't see is why you think the
difference is so important, as if the rat's doing b rather than a somehow
invalidated my analysis. It doesn't.
The prediction from PCT is that a disturbance of a controlled variable and
the action that does the controlling will vary equally and oppositely. But
any application of that principle presupposes you have successfully
identified a controlled variable. PCT does not predict what variables will
prove to be controlled. PCT is not at fault for predicting that rate of
reinforcement is a controlled variable.
Do you imagine that I believe otherwise? Show me where I said that PCT was
at fault. Go ahead. It's time to back up your assertions -- or rather,
aspersions.
My assumption was at fault: that if
the rate of reinforcement varied, it was because the rate of pressing
varied as well as because of the change in ratio (this was all on FR
schedules). You then proved with data that the rate of pressing was
effectively a constant, and thus was actually independent of both rate of
reinforcement and schedule.
Yes -- for ratio schedules. Different ballgame altogether. Entirely
different environmental feedback function, you can't generalize from the
ratio-schedule result to the VI-schedule situation we are discussing.
I decided, and said a year and a half ago, that the particular model I came
up with couldn't be right because it assumed that the rate of pressing
changed with the (hypothetical) error. I acknowledged at the same time that
my fit of a control-system model to the Motheral data was spurious. Why are
you still harping on this? Do you want me to repeat once a week, or day,
that my first attempt at a specific model wasn't correct?
The VI result was reported to you -- privately -- last Spring. I haven't
mentioned it since -- until now. I brought it up because of this:
Bill Powers (981014.0522 MDT)
By the way, I recall that you were quite sure the story would be different
for Variable-Interval schedules. As I recall, your more recent experiments
with VI schedules seemed to be showing the same effect: no significant
variations in pressing rate on different schedules. Is that still holding up?
I mentioned your prediction because it was that prediction that led to my
interest in reevaluating the VI interval-length vs response rate data, and
because Rick had gone so far as to claim that this would be a crucial
experiment between PCT and reinforcement theory. If my memory serves, he
has made this claim again fairly recently, and I wanted to disabuse him of
the notion.
I'm sure that a perfectly good PCT model can be built that will handle these
data. However, it does give one pause for thought. The results confirm a
perfectly clear if only loosely qualitative prediction based on
reinforcement theory. That does not mean that reinforcement theory is
correct, but at least it made a definite prediction. Given our present
state of knowledge, PCT's predictions can only be made, as it were, after
the fact, that is, after you have learned what variables are being
controlled and how.
Or could it be -- this just occurred to me -- that while rate of food
delivery is not a controlled variable, you're assuming it might still be a
reinforcer? That, of course, is ruled out because the behavior rate is
constant and does not vary with rate of food delivery. The same data that
rule out rate of food delivery as a controlled variable rule it out as a
reinforcer. The behavior rate does not change with "reinforcement" rate.
Not so, reinforcement theory can handle these observations -- again because
of the difference between ratio and interval schedules. We went over this
once before, but you refused to listen to me then, and now you repeat the
assertion as if we'd never had the conversation. You leave me little hope
that a second attempt would make any greater impression on you, so I'll just
leave it at that.
Your VI example in which both behavior rate and reinforcement rate decline
means nothing because the actual cause of the decline is time away from the
lever; it is not the decline in reinforcement rate that causes the decline
in behavior rate.
You keep saying this but the logic of it -- if any there be -- escapes me.
See my above discussion of how VI schedules work.
There I go again, trying to pull the wool over everyone's eyes. How
fiendishly clever of me.
I'm beginning to think it's your own eyes you're pulling the wool over.
Bill, you're blind to your own blindness.
_Of course_ reinforcement rates decline with increases in the interval used
in the schedule -- changing the interval sizes is how we experimentally vary
the reinforcement rate.
That is not what I am talking about. If time away from the lever increases
with the interval or ratio, that alone will result in a decline in behavior
rate and reinforcement rate, when both are measured as total number during
a session divided by duration of the session. You do not see any
significant change in pressing rate (corrected for collection time) until
the animal starts spending significant time away from the lever in addition
to collection time. This occurs at the higher intervals or ratios.
Your statements apply correctly to ratio schedules. They are wrong when
applied to interval schedules. Why do you keep bringing up the ratio data
when we are discussing the interval study?
Perhaps so, but that is irrelevant. Time away from the lever increases with
increased intervals, doesn't it? That would automatically decrease both
apparent rates equally: pressing and reinforcement. If your mind-set tells
you that pressing rate depends on reinforcement rate, you will interpret
that simultaneous decline as causal, when it is not.
It is your mind-set, not mine, that is the problem. You keep thinking in
terms of ratio schedules, and drawing conclusions that do not apply to the
inteval schedules under discussion. Then, when the results do not agree
with my statements, you attribute the problem to _my_ thinking!
But on high VI schedules, most programmed interreinforcement
intervals are long enough that (a) reinforcement-events would not have
occurred during most pauses and (b) most pauses will end relatively soon
after a programmed interval has elapsed and the next reinforcement
opportunity set up. Reinforcement rate consequently is little affected.
That is mathematical gibberish, Bruce. What is little affected is the
_whole-session_ reinforcement (and behavior) rate. The short-term behavior
rate is drastically affected. You are rationalizing.
We're talking about slightly different things, Bill. When the rat is
pausing, it is not responding and of course "the short-term behavior rate is
drastically affected," if by short-term you mean the duration of the pause.
But what I'm talking about is a somewhat longer interval -- the time between
presses (some of which consist of pauses away from the lever) relative to
the durations of tau. So long as the pauses are much shorter than tau,
there will be little effect of these pauses on the obtained rate of reward.
And that, my friend, is not mathematical gibberish, and it is not rationalizing.
And don't forget that the data consist of the times at which each and every
response occurred, not session averages. However, it is convenient to
report the result in terms of session averages, because those averages
reflect the typical rates observed at any time within a session.
More gibberish. It is convenient because you get the answer you want, not
because there is either logical or mathematical justification for this
number-juggling. The data initially consist of each and every response.
When you use session averages, you throw out most of the relevant information.
If the session averages and averages computed for various times within a
session are similar, then one can argue strongly that one is as good as the
other for depicting the relationships observed. Tell me, Bill, if you were
assessing the performance of a control system at various levels of gain
while the system was under a relatively constant disturbance, how would you
proceed? Would you average over the noise in the CV being produced by the
small variations in the disturbance to find the level of the CV produced at
each value of gain for a given reference level? Is there anything terribly
wrong with this procedure, when the observations are taken after the system
has passed beyond the transient values of start-up? Because that's exactly
analogous to what my procedure involved.
But we can state with certainty that increasing X, the programmed interval
size (which is independent of the rat's behavior), reliably results in a
lowered response rate, Y.
That is because it reliably decreases the fraction of the total time the
rat spends pressing the bar. You are varying both numerators (total
presses, total reinforcements) while the demonominator stays the same
(session duration). There is no causal relation between X and Y.
There is no reason why Y (number of responses emitted) should fall just
because X (number of rewards programmed) has been reduced -- after all, the
denominator stays the same. Furthermore, as noted previously (and proven
above) X can change over a wide range while hardly affecting X' (number of
rewards obtained) at all. Thus the major reason X' changes is because X has
been manipulated by the experimenter.
If you
deliberately close your eyes to the obvious explanation of the decline in X
and Y, you can claim an overall effect, but in doing so you only reveal
that you want the conclusion to be true more than you want to arrive at it
by legitimate means.
Really, Bill, you're getting to be like Rick -- you think you can accurately
guess at other people's motives. Not only that, but it seems to me that you
have such confidence in your own thinking that it never occurs to you that
you might be mistaken; consequently you brashly accuse anyone whose
conclusions disagree with yours of muddy or even biased thinking. I've
never had that degree of confidence that I've correctly understood or
reasoned about something, and for that reason I've always been willing to
consider yours or anyone else's arguments on their merits, on the chance
that I've missed something significant. Sometimes I've discovered that my
thinking was wrong, but even then I can take comfort in the fact that I've
learned something. I've learned quite a bit from you, Bill. But I've also
learned that you don't know everything, and that you can on occasion be
quite wrong in your opinions. If you could bear that in mind, perhaps you
would be less inclined to treat me like the village idiot in your replies.
Regards,
Bruce