[From Bruce Abbott (9500726.1040 EST)]
Rick Marken (950725.2310) --
All of these equations and your interpretations of them are fine.
The problem is in what you did with them, which was an exercise in curve
fitting that was presented as the prediction of a model. You said
(Bruce Abbott (950721.1100 EST)):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...
So what does it mean? The simplest interpretation is that
reinforcement rate is not being controlled.This makes no sense for several reasons. The first is that the data you
present do not suggest that reinforcement rate is _not_ being controlled
becuase there is no evidence of _lack_ of disturbance resistance.
Adding to the response requirement can be viewed as a disturbance, analogous
to, for example, changing the force required to depress the lever. If
reinforcement rate is being controlled, response rate should increase so as
to oppose the effect of this disturbance. It does not. The data show that
the rat continues to complete the ratio and collect the pellet at the same
_rate_. Because it takes longer to complete this cycle when more responses
are required, reinforcement rate declines. Because the number of responses
required to complete the ratio is higher and thus responding takes up a
greater proportion of the cycle, response rate increases. You get what
_looks_ like control-system behavior, but all the changes can be accounted
for entirely by the extra time required to complete a higher ratio at the
same within-ratio response rate. There is no resistance in evidence to the
disturbance to reinforcement rate produced by changing the ratio requirement.
If you are not following this, I give a detailed derivation later in this post.
But the real problem is that the predicted values of reinforcers/hr are not
model-based predictions. They are predictions in the sense that the linear
regression line gives the predicted seconds /reinforcement at each
ratio. They are a result of curve fitting. The "predicted" reinforcers/hr
at each ratio are based on the same data (the observed reinforcers/hr
at each ratio) as the predicted seconds/reinforcement.
Exactly true. As you know, in linear regression you can use the obtained
regression equation to generate the y-values which would be "predicted" from
the x-values if the true underlying relationship were in fact the one
described by the equation. If you subtract the observed y-values at each
corresponding x-value from the y-values predicted by the equation, you get
the differences between predicted and observed, which are called the
residuals. The better the "fit" of the line to the data, the smaller the
residuals (i.e., the closer the predicted values are to the observed
values). All I was doing with those "predictions" was to demonstrate how
well a straight line fits the data. If the fit is excellent, then a
straight line provides a good "summary" description of the relationship.
Linear regression demonstrates that a straight line is an excellent summary
of the relationship between seconds/reinforcement and
responses/reinforcement. Not only that, but all four lines converge to
almost the same intercept.
I'm not offering this fit as a model of the organism. I'm offering it as a
model of the behavior, which the model of the organism must reproduce if is
correct. I'm trying to understand what is going on in the data in a purely
descriptive sense in preparation for the development of that model. It is
this behavior that the organism-model must explain.
The data provide a test of the simple control model which says that the rat
is controlling rate of reinforcement against the disturbance produced by
changing ratio requirements. I'm sure there is control going on here, but I
can't find any evidence of control of reinforcement rate in these data. The
changes in overall response and reinforcement rates are exactly those
expected when a rat, responding at a constant rate and taking a constant
amount of time to collect the pellet, is required to complete different
numbers of responses to obtain the pellet.
Perhaps you could see that the "predictions" of reinforcement/hr are just
the predictions of a regression equation if you use the log of the ratio as
the predictor variable in a regression on reinforcements/hour. The fit to the
observed reinforcements/hour should be the same as what you report in
the table above.
Actually it isn't. If you graph reinforcements/hr as a function of log
ratio you get a set of slightly s-shaped functions. The inverse of
reinforcements/hr against the ratio itself gives straight lines. But of
course, I already know that the predictions are just those of a regression
equation. That's all I ever intended them to be.
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.
From this information you can derive the expected rate of reinforcement and
rate of responsing. The rate of reinforcement in rft/hr will = 3600/13.5 =
266.7 rft/hr, and the rate of responding will be 266.7 * 16 = 4266.7 rsp/hr.
If you run these calculations for each ratio value and plot rsp/hr against
rft/hr, you get a negatively-sloped straight line that looks just like what
you would expect to see if the rat were controlling rate of reinforcement.
But the changes in both values are simply a consequence of the extra time
required to complete the larger ratio in the absence of any compensatory
changes in response rate within the reinforcement cycle.
Regards,
Bruce