[From Bruce Abbott (950726.1230 EST)]
Bill Powers (950726.0545 MDT) --
Bruce Abbott (950725.2110 EST)
I still get an uncomfortable feeling that you do not quite follow
my line of thinking yet on these cyclic-ratio data.
I follow it, all right, but I believe you are making some conceptual
errors. Maybe I am -- but maybe, too, you are.
If we take the inverse of the reinforcement rates and convert, we
get seconds per reinforcement.
That is certainly true.
This is the average time required to complete one reinforcement
"cycle": complete each response, collect the pellet, and return to
the lever.Again, true and understood.
This number is NOT, repeat NOT b/m.
Right: it is m/b. The reinforcement rate is always exactly the behavior
rate divided by the ratio; the time per reinforcement is always exactly
the ratio divided by the behavior rate, which gives 1/r.
Sorry, I was thinking about the original conversion you insist on making.
Because, on the ratio schedule, b/m = reinforcement rate, you for some
reason that baffles me want to insist that we should substitute 1/ (b/m) for
1/r, which gives m/b. This implies that at FR-0 (m = 0, no responses
required), b should be zero, which is true if b is lever-pressing. But if r
= b/m, r should also be zero. In fact, under these conditions the rat is
happily collecting reinforcers at the rate of about one every 5.5 seconds.
As I managed to demonstrate above, I can get confused, and as you noted, so
can you; we're only human. About the logic of substituting m/b for r in my
graph, it's your turn to be confused. The problem with your substitution is
with the equation r = b/m. It doesn't apply here. I demonstrate this
mathematically below.
Let s = seconds per reinforcement,
m = responses required per reinforcement (the ratio),
L = seconds per lever-press, and
c = seconds to collect food.
r = reinforcement rate in rft/sec
b = response rate in rsp/sec
Now, the following is apparent:
(1) s = mL + c
That is, seconds required to collect a reinforcer equals the number of
required responses times the number of seconds per response plus the number
of seconds to collect the reinforcer. The inverse of this is the
reinforcement rate:
(2) r = 1/(mL + c)
But
(3) b = 1/L
thus
(4) r = 1/(m/b + c) = (b + c/m)/m
Yet you wish to represent
(5) r = b/m
This leads to a contradiction, since this implies that
(6) b = b + c/m
which is decidedly untrue.
The problem is that you have confused average behavior rate, which includes
collection time, with lever-pressing rate, which does not.
(Average) reinforcement rate equals (average) behavior rate divided by
the ratio. That is the physical fact. The reward rate is zero only if
the behavior rate is zero, which can happen. But for the reward rate to
equal the reference level, the behavior rate would have to be infinite
(by r = b/m), which is impossible. Between a ratio of 1 and 0 there
would have to be ratios of 0.5, 0.05, 0.0005, and so on to the limit,
with behavior approaching infinity as a limit. But the apparatus works
only with integer values.
An infinite behavior rate implies a zero delay, which is clearly not
impossible if there is no behavior required (FR = 0) to produce the pellet.
But this does not imply an infinite reinforcement rate, because it takes
time to collect the reinforcer once it becomes available. If we assume that
the rat is standing at the lever when we deliver the pellet, (as would be
the case at ratio completion for non-zero ratios), the time is about 5.5
sec, which implies a reinforcement rate of 654.5 rft/hour.
As to the x-axis, the ratio value is the number of responses
required to complete one reinforcement cycle. So we are plotting
the average time required to complete one cycle as a function of
the number of responses required to complete one cycle.
Yes, you CAN plot this if reinforcement rate is the independent
variable, but it is not. Of course you can plot it anyway, but then your
mathematics ceases to have any connection to the physical situation:
you're just pushing numbers around. When you divorce mathematical
manipulations from their physical meaning, you're doing numerology, not
science.
Absurd! The number of responses required to complete the ratio and collect
a reinforcer is the independent variable here, not the rate of
reinforcement. As to the connection of the mathematics to the physical
situation, it is clear and simple, as I described to Rick:
By plotting the data in the way I suggest, you get a relationship with
a simple and obvious interpretation. The time to complete a ratio and
collect the pellet equals the time required to leave the lever, pick up
the food, consume it, and return to the lever. This value is given by
the y-intercept of the line for a given animal. The slope of the line
indicates the time penality exacted per additional required response.
If this penalty is 1/2 second per additional response, then requiring,
say, 16 responses, will add 1/2 * 16 = 8 seconds to the time required to
complete the ratio, collect the pellet, and return to the lever. If the
collection-time (y-intercept) is 5.5 seconds, it will require 5.5 + 8 =
13.5 seconds on average to complete the ratio, collect the food, and
return to the lever, ready to begin a new ratio. Thus the total time
between reinforcements will be 13.5 seconds.
Once you push through this conceptual block, I think you will be amazed how
simple it all becomes. Perfectly sensible.
When Herrnstein "generalized" his matching law to
B1/(B1 + B2...Bn) = R1/(R1 + R2 ... Rn) and so forth,
he didn't, apparently, realize that the whole series is algebraically
identical to
B1/R1 = B2/R2 = ... Bn/Rn
which merely says that ALL the schedules are identical.
We discussed this issue a long time ago on the net. At that time I thought
we ultimately agreed that this was not the problem it appears to be. The
matching law applies to simultaneously-available alternative schedules, not
to schedules in isolation. More responses directed to one schedule mean
fewer responses directed to the others. On ratio schedules what emerges is
exclusive responding on the lower-ratio schedule, so obtained response rates
match obtained reinforcement rates, as expected under the law. On interval
schedules, responses are allocated across schedules in such a way that the
implied equivalence is obtained, at least to a first approximation (later
research revealed systematic deviations under certain conditions). There
are other problems with the matching law, but "numerology" is not one of
them. Nor is my analysis of the cyclic-ratio data another example of the
same, as I think I've shown pretty clearly above.
Rat intercept slope r r-sq
C1 5.35 0.679 1.000 1.000
C2 5.77 0.528 0.999 0.997
C3 5.47 0.458 0.999 0.999
C4 5.21 0.397 0.999 0.998You will note that the correlations are extremely high, indicating
an excellent linear fit to the data.This is not just an excellent linear fit to the data: it is (when
rounding errors are removed) a _perfect_ fit. There is no way you could
have estimated, or even measured, the data values with the implied
accuracy. What you have here is the result of computing an algebraic
identity. Just as Rick said, you have computed one function of a
variable, and then the inverse function of the result, ending up, within
computational limits, with the original values of the variable. If you
write out all the equations you used and solve them simultaneously, you
will find that you have proven that 0 = 0.
This is just the fit your program generated when you did it. So if I'm
guilty, we're both guilty. Here is your result:
Calculate seconds per reinforcement
Ratio Rat
1 2 3 4
2 7.15 6.86 6.62 6.40
4 8.15 7.89 7.63 7.12
8 10.62 9.52 9.23 8.18
16 15.74 13.89 12.20 10.99
32 27.07 23.90 19.79 17.76
64 48.91 39.10 35.04 30.80Calculate intercept of [sec/reinf vs ratio]
Rat 1 intercept = 5.36 slope = 0.68
Rat 2 intercept = 5.77 slope = 0.53
Rat 3 intercept = 5.47 slope = 0.46
Rat 4 intercept = 5.21 slope = 0.40
This is no trivial identity; it is the actual fit of four straight lines to
empirical data. Minitab confirms your analysis and adds the correlations
and r-squares. I'm hoping that at this point you are now in agreement, and
we can get back to discussing what these functions imply rather than whether
they mean anything at all.
To make you feel better, I initially did the same thing in my paper on
experimental measurement of purpose in Wayne Hershberger's book.
I will feel much better, thank you, when you finally see that the approach
I've taken with these data is correct. There's much more to do, but first
we have to get past this problem.
Regards,
Bruce