Curve fitting vs modelling

[From Rick Marken (950725.2310)]

Bruce Abbott (950725.2110 EST) --

If we take the inverse of the reinforcement rates and convert, we get
seconds per reinforcement. This is the average time required to
complete one reinforcement "cycle": complete each response, collect

OK.

As to the x-axis, the ratio value is the number of responses required to
complete one reinforcement cycle.

OK.

At the y intercept, no responses are required to complete one cycle.
Thus the value of the y intercept is the number of seconds required to

OK.

The slope of the line for a given animal is the increase in time per

OK.

If you plot the seconds/cycle as a function of the number of responses
per cycle, you get four essentially straight lines

OK.

Minitab gives the following regression results for these four lines:

Rat intercept slope r r-sq

>C1 5.35 0.679 1.000 1.000
...

OK.

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.

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.

Seconds/reinforcement were derived from reinforcements/hr (by
dividing into 3600). Then you did the regression to find the linear fit to
the seconds/reinforcement. Then you get the predicted
reinforcements/hr by converting the predicted seconds/reinforcement
(based on the regression equation) back into reinforcements/hour (by
dividing into 3600 again).

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.

Best

Rick