Cyclic-Ratio Data

[From Bruce Abbott (950721.1100 EST)]

Ettinger and Staddon (1983) reported the results of an investigation of
"operant regulation of feeding" using a new procedure they described as a
"cyclic-ratio" schedule, which I described in my last post (950718.2055
EST). Within each session, rats were exposed to an ascending and then
descending series of ratio values: 2, 4, 8, 16, 32, and 64
responses/reinforcement. Because each ratio is evaluated twice each session
in counterbalanced order, order effects such as satiation or fatigue are
minimized. Because all ratios are evaluated in a session, long-term changes
which might occur over sessions are also controlled.

Ettinger and Staddon reported data for the same four subjects (C1, C2, C3,
and C4) in three experiments. Experiment 1 simply compared the performances
of these subjects on the cyclic-ratio schedule to those of four other
subjects on the standard (single-ratio) schedule (in which ratio was
manipulated over sessions). Experiment 2 manipulated pellet taste (quinine
was added) and deprivation level (80% ad lib weight versus 95% ad lib
weight). Experiment 3 manipulated amphetamine dosage. Experiment 2
included a replication of the Experiment 1 cyclic-ratio conditions (80% ad
lib weight); experiment 3 included two such replications (one non-drug
control for each of two amphetamine levels); together this provides four
cyclic-ratio assessments of the same animals under the same conditions.

Examination of the four curves for each subject under this condition
revealed some variability from replication to replication but were generally
similar; on the other hand there were reliable differences between subjects
in the slopes of these curves. To provide more stable estimates of the
relationships I averaged the data for each subject across replications. The
resulting subject averages are given below.

               Responses/Hour
Ratio C1 C2 C3 C4
  2 1007 1050 1088 1125
  4 1767 1825 1887 2022
  8 2711 3026 3120 3522
16 3660 4146 4723 5243
32 4255 4820 5821 6486
64 4711 5893 6575 7481

Dividing each row by the ratio gives the reinforcers/hour:

              Reinforcers/Hour
Ratio C1 C2 C3 C4
  2 503.5 525.0 544.0 562.5
  4 441.8 456.3 471.8 505.5
  8 338.9 378.3 390.0 440.3
16 228.8 259.1 295.2 327.7
32 133.0 150.6 181.9 202.7
63 76.6 92.6 102.7 116.9

Plotting responses/hour as a function of reinforcements per hour as in the
Motheral data produces four curves, one for each subject, in which rate of
responding increases as rate for reinforcement decreases (i.e., typical
right limb of Motheral-type curve). The slope of the curve is lowest for
subject C1 and increases in order of subject number, so that the steepest is
for C4. Projecting the intercepts of these curves with the X axis gives
estimated reference levels for reinforcement rate in the 580-610 rft/hr
range, in order for subject number from lowest to highest. That is,
subjects whose curves have the lowest slopes also have curves with the
lowest intercepts.

These data appear consistent with a control model for reinforcement rate and
as I described in the previous post were interpreted essentially that way by
Ettinger and Staddon.

However. Converting reinforcements/hour to seconds/reinforcement (by
dividing the former numbers into 3600) gives the following data:

                 Seconds/Reinforcement
Ratio C1 C2 C3 C4
  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 15.73 13.89 12.20 10.99
32 27.07 23.90 19.79 17.76
64 48.91 39.09 35.05 30.80

If you plot Seconds/Rft as a function of the ratio requirement a strange and
wonderful thing happens. Within a small amount of error you can fit four
straight lines to these data, all having the same Y-intercept at 5.5 seconds
for 0 responses. I interpret this as the time required to move from the
lever, collect the reinforcer, and return to the lever. It would appear
that under 80% ad lib weight and using 45 mg noyes pellets as reinforcers,
all four animals required the same amount of time to perform this collection
of actions. The four lines have different slopes, which represent the
average time required to complete a lever press (including any pauses).
Parameters of the four lines are:

Subject Intercept Slope
  C1 5.5 s 0.68 s/rsp
  C2 5.5 s 0.52 s/rsp
  C3 5.5 s 0.46 s/rsp
  C4 5.5 s 0.39 s/rsp

Basically, these functions indicate that the time required to collect a
reinforcer is equal to the 5.5 seconds "collection" time plus the time
required to complete the ratio requirement. For example, at a ratio of 64,
subject C1 would be estimated to require 5.5 + 64*0.68 = 49.0 s on average
per reinforcement. The rate of reinforcement sustained would be estimated
to be 73.4 rft/hr. The actual rate was 73.6 rft/hr.

            Reinforcers/Hour: Predicted Vs. Observed
       ----C1---- ----C2---- ----C3---- ----C4----
Ratio Pred. Obs. Pred. Obs. Pred. Obs. Pred. Obs.
  2 525 504 550 525 561 544 573 563
  4 438 442 475 456 490 472 510 506
  8 329 339 373 378 392 390 418 440
16 220 229 260 259 280 295 307 328
32 132 133 163 151 178 182 200 203
64 73 74 93 92 103 103 118 117

You will note that the predictions at ratio 2 are consistently too high;
this is no doubt because the 5.5 s intercept estimate (by eyeball from the
chart) is just slightly too low. Even so, I think you'll agree that the
fits are impressive.

So what does it mean? The simplest interpretation is that reinforcement
rate is not being controlled. As the response requirement increases, the
time required to collect a reinforcer increases in direct proportion. Thus
there appears to be no opposition to the "disturbance" to reinforcer rate
produced by changing the ratio requirement.

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 decline in rate of
reinforcement with increasing ratio requirement is exactly that which would
be expected given the longer time required to complete a ratio while
maintaining the same rate of responding.

Back to the drawing board?

Regards,

Bruce

[From Bruce Abbott (950724.2135 EST)]

Bill Powers (950724.1340 MDT) --
    Bruce Abbott (950721.1100 EST)

A very successful 11th CSG meeting is over, the guests have departed,
and tranquillity descends once again on 73 Ridge Place. I will leave it
to others to review the highlights.

Have anyone in particular in mind? Those who have responded thus far have
spoken only in generalities (i.e., great time was had by all, etc.) Not too
informative, I must say. Is someone working on a more detailed account of
the week's events?

    Ettinger and Staddon reported data for the same four subjects (C1,
    C2, C3, and C4) in three experiments. Experiment 1 simply compared
    the performances of these subjects on the cyclic-ratio schedule to
    those of four other subjects on the standard (single-ratio)
    schedule (in which ratio was manipulated over sessions).
    Experiment 2 manipulated pellet taste (quinine was added) and
    deprivation level (80% ad lib weight versus 95% ad lib weight).
    Experiment 3 manipulated amphetamine dosage. Experiment 2 included
    a replication of the Experiment 1 cyclic-ratio conditions (80% ad
    lib weight); experiment 3 included two such replications (one non-
    drug control for each of two amphetamine levels); together this
    provides four cyclic-ratio assessments of the same animals under
    the same conditions.

This is my idea of shotgun research. When it's all over, you have a
giant collection of indigestible facts of no particular importance or
applicability.

Perhaps you have this view because you do not have the theoretical context
within which these manipulations make sense. As I was only interested at
this point in examining the baseline data from this study, I did not present
that context. Staddon's "regulatory" model predicted that certain
manipulations should affect the slopes of the functions without changing the
intercepts and others should change both. The manipulations were designed
to test these predictions.

    If you plot Seconds/Rft as a function of the ratio requirement a
    strange and wonderful thing happens. Within a small amount of
    error you can fit four straight lines to these data, all having the
    same Y-intercept at 5.5 seconds for 0 responses. I interpret this
    as the time required to move from the lever, collect the
    reinforcer, and return to the lever. It would appear that under
    80% ad lib weight and using 45 mg noyes pellets as reinforcers, all
    four animals required the same amount of time to perform this
    collection of actions.

The number of reinforcements per unit time r is calculated as the
behavior rate b divided by the ratio m: r = b/m. Therefore the time per
reinforcement is m/b (omitting constants for changing time units). When
you plot ratio against time per reinforcement, you are therefore
plotting the relationship of m/b with m. The meaning of such a plot is
not completely clear, nor is the meaning of the y-intercept for b/m = 0.
It is not usual to look at a relationship in which the same variable
appears on both sides of the equation without being separated first!

If you'll bear with me for a few moments perhaps we can find a clear meaning
in these data. Rather than plotting response rates as you have done in your
conversion, let's stick with the original data giving average time to
reinforcement as a function of the number of responses required to produce
reinforcement. The data for the four animals can be closely fit by four
straight lines having close to the same y intercept. The y intercept (at a
requirement of 0 responses/reinforcement indicates the time required to
leave the lever, collect the reinforcer, and return to the lever. It is
very close to the same value (about 5.5 seconds) for each animal. The slope
of each line gives the average rate of increase (per response) in the time
(in seconds) required to collect the reinforcer. For example, if the slope
is 0.50 sec/resp, then each additional response required by the ratio
schedule adds an additional 0.50 seconds to the time required to collect a
reinforcer. If the intercept is 5.5 seconds, then it would require 5.5 +
2*0.5 = 6.5 seconds on average to collect a reinforcer on the FR-2 schedule,
but 5.5 + 64*0.5 = 37.5 seconds to collect a reinforcer on the FR-64 schedule.

Because the functions are straight lines, this implies that each additional
response exacts a constant time penalty. In other words, the average _rate_
at which the ratios are completed is constant across ratios. Regardless of
the ratio size, the animal is completing the ratio in approximately 5.5 +
b*m seconds, where b is the slope and m is the ratio. If the rate of
responding is constant across all ratio requirements, then the rate of
reinforcement cannot be. In fact it will decrease as a linear function of
the ratio requirement. This explains the backward-sloping right limb of the
Motheral curve.

If the animal were attempting to keep the rate of reinforcement constant,
one would expect that the additional time penalty imposed by a higher ratio
requirement (disturbance) would be met by a compensatory increase in average
response rate. There seems to be little evidence that such is the case.

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 30.80

Calculate 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

Calculate peak behavior rate per hour with intercept time removed

                                 RATIO
          2 4 8 16 32 64

Rat 1 4013.36 5154.89 5467.55 5548.19 5304.33 5290.36
Rat 2 6599.83 6779.01 7677.32 7087.75 6352.63 6912.48
Rat 3 6286.10 6670.07 7662.62 8567.13 8045.73 7791.80
Rat 4 6070.50 7548.26 9719.11 9978.97 9181.19 9005.60

Your regression routine gives results not far from those I estimated by
eyeball from the graph. However, you need to check your routine for
converting to peak behavior rate. Here's a quick check at FR-2 for Rat 1:

7.15 s/rft - 5.77 s/rft = 1.38 sec/rft;

3600 sec rft
-------- * -------- = 2608.70 rft/hr * 2 rsp/rft = 5217.39 rsp/hr
   hr 1.38 sec

Alternatively, 0.68 sec/rsp = 1.471 rsp/sec * 3600 = 5294.118 rsp/hr;
the difference between this result and the previous one is due to rounding
error. Your figure is 4013.36.

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.

The "peak" rates do not include collection time (or so you indicated above)
so it can have no influence on the numbers at all. The actual "peak" rates
for a given animal do not change _at all_ over the 32:1 range of ratios; for
Rat 1 it is about 0.68 sec/response, or something in the 5200+ rsp/hr range
as calculated above, regardless of the ratio requirement.

Either that, or I'm completely misunderstanding what you're doing. Perhaps
you could describe the calculations.

Your measure of corrected reinforcements per hour is not appropriate,
because the reinforcements did not take place between collection times,
but during them.

I don't know what you're talking about. If by "measure of corrected
reinforcements per hour" you mean the rates implied by the linear fits to
the data, the reinforcements per hour given by these lines include the
collection times. For example, for Rat 1, you give an intercept of 5.36
sec/rft and a slope of 0.68 sec/rsp/rft. For FR-2 this gives an
interreinforcement time of 5.36 sec/rft + 2* 0.68 sec/rsp/rft = 5.26 sec/rft
+ 1.36 sec/rft = 6.62 sec/rft. 3600/6.62 = 543.81 rft/hr. The observed
value was 503.5 rft/hr.

By the way, not to open another can of worms, but you can estimate the
collection-time from the Motheral-type graph as the point where a straight
line fitted to the right portion of the curve meets the x-axis. This gives
the rate of reinforcement at FR-0. Invert and convert to seconds for the
collection-time. You can estimate the (constant) response rate by noting
where the line crosses the y-intercept. This gives the rate for
FR-infinity, when no reinforcement would ever be delivered (thus excluding
collection-time from the figure). Invert and convert to get the slope of
the line in the "sec/rft vs. ratio" plot.

Regards,

Bruce