# Assessing rate-control

**URL:** <http://discourse.iapct.org/t/assessing-rate-control/1335>\
**Category:** CSG1997\
**Created:** [May 5, 1997, 5:16pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335 "1997-05-05T17:16:20Z")\
**Posts on this page:** 11\
**Page:** 1

<div class="post-metadata">

**Author:** ![Bruce\_Abbott](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/b/ba8739/32.png) [@Bruce\_Abbott](http://discourse.iapct.org/u/Bruce_Abbott)\
**Post date:** [May 5, 1997, 5:16pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335/1 "1997-05-05T17:16:20Z")

</div>

[From Bruce Abbott (9705.1220 EST)]

In the prototypical tracking experiment, a participant seated at a computer  
is asked to keep a cursor aligned with a target by moving the computer's  
mouse. Meanwhile, the cursor's position is being affected by a disturbance,  
which in the absence of counteracting mouse movements causes the cursor to  
drift from the target's position. The disturbance, mouse position, and  
cursor position are recorded during the run and then the the correlations  
between each possible pair of these three variables is computed. The  
classic result is a near-zero correlation between variation in cursor and  
mouse positions and a nearly perfect negative correlation (generally in  
excess of -.995) between variations in disturbance and mouse position. A  
typical example from an actual run appears below:  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Cursor Mouse  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Mouse 0.093  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Dist. -0.011 -0.997

This matrix demonstrates excellent control of cursor position; a plot of  
mouse position and disturbance as functions of time would produce two lines  
that were nearly mirror images of each other, whereas cursor position would  
appear to vary little and with no apparent relationship to either mouse  
position or disturbance. This is because mouse movements are essentially  
cancelling the effect of the disturbance on the position of the cursor.

This analysis assumes that the participant was simply controlling cursor  
position. However, there is another strategy participants could use in this  
task to help keep the cursor on-target. Because the disturbance was made to  
vary relatively smoothly, a participant could adjust the rate of  
mouse-travel as required to reduce the rate of cursor-travel to zero. If  
the cursor were already on-target, this strategy would tend to keep it  
there. Of course, if the cursor were elsewhere, the rate of mouse-travel  
would have to be adjusted so as to permit a residual drift of the cursor  
back to target and then changed so as to zero out further drift. This  
strategy would amount to a combination of position \_and\_ rate control of the  
cursor.

Is there evidence that the participant who generated the correlation matrix  
above used rate control in addition to position control? One way to get at  
this question is to take the difference between successive disturbance,  
cursor, and mouse positions; these differences provide the rates of change  
in each of the three variables. Here are the results:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;dCursor dMouse  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;dMouse 0.786  
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;dDist. 0.382 -0.270

These correlations suggest that control over cursor \_rate\_ was poor: cursor  
rate correlates moderately high with the rate of mouse movement and  
moderately low with the rate of disturbance change, indicating that  
mouse-rate was not very effective in zeroing out the rate of cursor  
movement, and the correlation between the rates of disturbance change and  
mouse movement, although negative, was very low. Nevertheless, I would  
suggest that this matrix provides some small measure of evidence for some  
very poor degree of rate control.

The analysis is complicated by the fact that positional control-actions  
would tend to occur at the expense of rate-control. In addition, it seems  
likely that positional control would entail a negative correlation between  
disturbance and mouse rates. To the extent that positional mouse-action  
fell behind (or got ahead of) the disturbance, the participant would have to  
play "catch-up" (or "fall-back") and this would reduce the correlation  
between disturbance rate and mouse rate. What these data show is that very  
high correlations between disturbance and mouse-position can be achieved in  
the absence of good control over cursor rate.

Comments?

Regards,

Bruce

---

<div class="post-metadata">

**Author:** ![Bill\_Powers1](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/b/8e7dd6/32.png) [@Bill\_Powers1](http://discourse.iapct.org/u/Bill_Powers1)\
**Post date:** [May 5, 1997, 7:36pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335/2 "1997-05-05T19:36:32Z")

</div>

[From Bill Powers (970505.1238 MST)]

Bruce Abbott (9705.1220 EST)--

> Is there evidence that the participant who generated the correlation  
> matrix above used rate control in addition to position control? One way  
> to get at this question is to take the difference between successive  
> disturbance, cursor, and mouse positions; these differences provide the  
> rates of change in each of the three variables. Here are the results:
> 
> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;dCursor dMouse  
> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;dMouse 0.786  
> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;dDist. 0.382 -0.270

...

> The analysis is complicated by the fact that positional control-actions  
> would tend to occur at the expense of rate-control. In addition, it \>seems

likely that positional control would entail a negative correlation \>between  
disturbance and mouse rates. To the extent that positional \>mouse-action  
fell behind (or got ahead of) the disturbance, the \>participant would have  
to play "catch-up" (or "fall-back") and this \>would reduce the correlation  
between disturbance rate and mouse rate.

This is a very good example of applying the Test. When you hypothesize that  
velocity is the controlled variable, you define the disturbance and the  
output in terms of their effects on velocity, which is the proper procedure.  
Your statistical analysis then shows that the correlation of mouse velocity  
with cursor velocity is almost 0.8, while mouse versus disturbance velocity  
is only -0.270. Both of these correlations argue for rejecting the hypothesis.

When you change hypotheses so it is position that is proposed to be under  
control, the correlations change radically. Mouse vs cursor position is  
0.093, and mouse vs. disturbance position is -0.997. The Test is clearly  
passed even if we set the criterion level (mouse-disturbance) to -0.95.

If you wanted to be formal about this, you would verify that the controlled  
variable is actually affected by the mouse (not very difficult), and that  
the controlled variable must be sensed in order to be controlled (slightly  
more difficult, but still easy).

> What these data show is that very high correlations between disturbance  
> and mouse-position can be achieved in the absence of good control over

cursor rate.

I wouldn't put it quite this way. If position is perfectly controlled, then  
rate is also controlled perfectly. If you were to construct a position  
control model without any noise in it, this is what you would find. However,  
there is noise in the real system, which amounts to uncorrelated  
disturbances of both rate and position. The controller brings the error to  
the level where it is comparable to the noise level, but that is true only  
when you measure position. When you measure velocity, the velocity error is  
larger than the noise component because in fact it isn't being directly  
controlled.

Your description of "playing catchup", however, is probably quite right: the  
catchup is due to position errors, which are corrected without regard to the  
effect on velocity.

The next thing to try is an experiment with actual velocity control. Can you  
think of an experiment in which a particular velocity rather than a  
particular position has to be maintained? If you're controlling velocity,  
then velocity errors can lead to cumulative changes in position, so the  
challenge is to arrange a display in which changes in position don't  
eventually run you off the screen!

Once you have the experiment, you can apply the same analysis as above to  
discriminate position control from velocity control, and you should then  
find the opposite outcomes, with position control being rejected.

This is what I call basic PCT research!

Best,

Bill P.

---

<div class="post-metadata">

**Author:** ![Human\_Factors\_Lab](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/h/b3f665/32.png) [@Human\_Factors\_Lab](http://discourse.iapct.org/u/Human_Factors_Lab)\
**Post date:** [May 6, 1997, 5:36pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335/3 "1997-05-06T17:36:09Z")

</div>

[From Rick Marken (970505.2130 PDT)]

Bruce Abbott (9705.1220 EST) --

> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;dCursor dMouse  
> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;dMouse 0.786  
> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;dDist. 0.382 -0.270
> 
> These correlations suggest that control over cursor \_rate\_ was  
> poor: cursor rate correlates moderately high with the rate of  
> mouse movement and moderately low with the rate of disturbance  
> change, indicating that mouse-rate was not very effective in  
> zeroing out the rate of cursor movement, and the correlation  
> between the rates of disturbance change and mouse movement,  
> although negative, was very low.

Bill Powers (970505.1238 MST) --

> This is a very good example of applying the Test...This is what  
> I call basic PCT research!

Yes. It also might provide a nice way for Bruce to illustrate the  
use of some fairly complex statistical methods in PCT research.  
If you did a multiple regression on the position control data,  
using Dist and Mouse to predict Cursor, you would probably find  
a regression equation like Cursor' = 1.0\*Mouse-1.0\*Dist. The  
coefficients would probably be slightly different than 1.0 and  
-1.0 and the intercept would probably be slightly different from 0.0.  
But they would all be close to these values, and R^2 would probably  
be around .98. This is the result expected if Cursor position is  
under control.

But it might be that the actual controlled variable is rate of  
change in Cursor position (dCursor) rather than Cursor position.  
This could be tested by adding dMouse and dDist to the prediction  
equation and making dCursor the dependent variable. The complete  
regression equation becomes:

dCursor = a\*b1\*Cursor+b2\*Mouse+b3\*Dist+b4\*dMouse+b5\*dDist

We are only interested in the partial regression of dMouse and  
dDist on dCursor but the other variables have to be included in  
the solution of the multiple regression in order to factor out  
their contribution to the variance in dCursor, dMouse and dDist.

If dCursor is under control, then the R^2 of the partial  
regression of dMouse and dDist on dCursor should be high  
(greater than .98) and, more importantly, the coefficients  
of dMouse and dDist should be about equal (1.0) and of  
\_opposite\_ sign.

I think this could be a nice way to do The Test when you can't  
actively manipulate disturances. In this tracking task, Dist  
and dDist are probably confounded (the correlation between them  
is \>0.0) and partial regression (like analysis of covariance)  
is a way to eliminate this confound statistically.

Whaddaya think, Bruce?

Best

Rick

---

<div class="post-metadata">

**Author:** ![Martin\_Taylor3](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/m/7ea924/32.png) [@Martin\_Taylor3](http://discourse.iapct.org/u/Martin_Taylor3)\
**Post date:** [May 6, 1997, 2:43pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335/4 "1997-05-06T14:43:58Z")

</div>

[Martin Taylor 970516 10:30]

> Bill Powers (970505.1238 MST)] to Bruce Abbott (9705.1220 EST)--

> If position is perfectly controlled, then  
> rate is also controlled perfectly. If you were to construct a position  
> control model without any noise in it, this is what you would find. However,  
> there is noise in the real system, which amounts to uncorrelated  
> disturbances of both rate and position. The controller brings the error to  
> the level where it is comparable to the noise level, but that is true only  
> when you measure position. When you measure velocity, the velocity error is  
> larger than the noise component because in fact it isn't being directly  
> controlled.

Yes, all that is true, and there's another effect, too.

Velocity is the derivative of position, and to take a derivative is  
equivalent to putting a 6 dB per octave high frequency emphasis onto  
the spectrum. The higher the frequency of the disturbance, the poorer  
the control (in a control loop with finite transport lag). So even if  
the velocity were to be independently controlled (if that were possible),  
it would still show poorer stability factor than would positional control  
against the same disturbance. (Here I mean the same physical disturbance  
affecting the physical entity whose position or velocity might be the  
complex environmental variable corresponding to the controlled perception).  
You could test this assertion merely by telling the subjects in one run to  
keep the cursor still at all times, wherever it might be, and then in another  
run with the same disturbance ask them to keep the cursor aligned with  
the target at all times. You can ask them to do both at once, but they  
won't be able to do it!

Independent parallel control of position and velocity is impossible, but  
positional control could easily be effected by altering the reference signal  
to a velocity control system (or vice-versa). See the Arm demo. What that  
means is that even if velocity is precisely controlled, the data you get  
will not show it in the correlations, because the velocity reference value  
is changing all over the lot as a function of the position perceptual error.  
(But if position were being controlled in order to implement velocity  
control, you would see relatively good correlations in the velocity data  
but not in the position data, allowing for the spectral high-frequency  
emphasis mentioned above).

Martin

---

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**Author:** ![Bill\_Powers1](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/b/8e7dd6/32.png) [@Bill\_Powers1](http://discourse.iapct.org/u/Bill_Powers1)\
**Post date:** [May 6, 1997, 6:35pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335/5 "1997-05-06T18:35:17Z")

</div>

[From Bill Powers (970506.1018 MST)]

Martin Taylor 970516 10:30--

> Velocity is the derivative of position, and to take a derivative is  
> equivalent to putting a 6 dB per octave high frequency emphasis onto  
> the spectrum. The higher the frequency of the disturbance, the poorer  
> the control (in a control loop with finite transport lag). So even if  
> the velocity were to be independently controlled (if that were possible),  
> it would still show poorer stability factor than would positional control  
> against the same disturbance.

I'm not sure if this is the right direction in hyperspace to generalize  
about rate control. If you do have rate control, it can seem just as good as  
position control over short periods of time, given the same basic components  
from which to build a control system and the same external disturbance. The  
difference is that a pure rate control system will drift under a steady  
disturbance (or the steady component of a disturbance that does not have a  
zero average value). Transport lag, as you say, limits how much gain a rate  
control system can have at the highest frequency that can be transmitted  
through the system, but that limit is the same in a proportional system or  
an integrating system. The main difference is in how the \_low\_ frequency  
components are handled.

> Independent parallel control of position and velocity is impossible, but  
> positional control could easily be effected by altering the reference  
> signal to a velocity control system (or vice-versa). See the Arm demo.

Yes, you're quite right about that. In fact, this brings up a point that  
we've touched on very briefly now and then: is there one rate control level,  
or two? In the Arm demo, as modeled, the position control system in the  
spinal control systems does set a reference velocity, but this is really an  
imaginary hierarchy. In fact, the annulospiral length sensors have a rate  
plus proportional response; we can separate these responses out into a rate  
signal and a proportional signal for computational convenience, but both  
signals actually reach the same motor neuron, the same comparator, so there  
is no physically separate level where a pure rate signal is compared against  
a rate reference signal. Instead, the effective rate reference signal is  
always zero. The gamma reference signal is a position reference signal, and  
the alpha reference signal is a force (or torque) reference signal. There is  
no separate rate reference signal that I know of.

This means, I think, that we don't really have a distinct velocity control  
level below the position control level. What we have is rate plus  
proportional sensing, which puts damping into the position control system.  
The result is not to add a rate component to the resulting closed-loop  
perceptual signal, but exactly the opposite: the damping \_suppresses\_ the  
high frequencies in the whole loop. The result is that the perceptual signal  
follows rapid changes in the position reference signal more closely than it  
would do without the rate component. Confusing.

However, when the Arm model's parameters are adjusted for the fastest stable  
response to square-wave reference signals, what we find is that the static  
loop gain is only 2 to 4, while the dynamic loop gain is anywhere from 50 to  
100. As a result, when gravity is turned on, the arm sags! So it seems that  
these spinal control loops could be viewed (as a whole) as rate control  
systems, with only a little proportional gain. In order to eliminate the  
sag, it is necessary to add another control loop using joint-angle  
perceptual signals, with an integrating output function. I have seen  
evidence that in the brainstep loops there are such integrations in the  
output functions (the motor nuclei of the brain stem).

Obviously, all these conjectures call for a much more extensive  
investigation of these peripheral control systems. I hope someone will take  
it up, once the Arm paper appears in IJHMS.

> (But if position were being controlled in order to implement velocity  
> control, you would see relatively good correlations in the velocity data  
> but not in the position data, allowing for the spectral high-frequency  
> emphasis mentioned above).

The \_second\_ level of velocity control is what I have called the fourth  
level, the transition level. The strongest evidence I know of for this level  
is the phenomenonon of stroboscopic motion. Here we have a series of static  
configurations which, presented in rapid sequence, give rise to a perception  
of motion. It's just as though there were a higher level of perception which  
takes first derivatives (or first differences?) of signals from the  
configuration level and constructs a velocity signal from them (and perhaps  
acceleration signals too?). In this case we would have what you describe: a  
control system maintaining perceived rates of change by varying  
configuration reference signals. This level is actually \_slower\_ than the  
configuration or position control level: I believe that the impression of  
motion begins to disappear when alternating positions appear 300  
milliseconds apart or less. That's a long time; the reaction time for  
position control is much less than that. If we were talking about true rate  
feedback at a lower level, the rate signal should \_increase\_ as the  
frequency of changes gets higher, and we should see a rate response at much  
higher frequencies.

This obviously needs a lot of experimental sorting out. Rick Marken's  
experiments with hierarchical perceptions are one good way to go; Bruce  
Abbott's statistical analysis approach is another.

Incidentally, transport lag in the peripheral systems (or anywhere in the  
brain, for that matter) is not really a simple phenomenon. Every control  
loop is actually made up of many parallel control loops, and in each loop  
the delays will be a little different. Some loops might have very short  
transmission times; others abnormally long ones. In addition, there's the  
"recruitment" effect (I don't know where neurologists get these irrelevant  
images): some paths will not begin transmitting until the signal is very  
strong, while others have nearly zero thresholds. The other end of this  
effect is that some pathways will saturate long before others. The result is  
that the composite response in the whole control loop can be very different  
from that of any one microloop (I read somewhere that in the stretch reflex,  
the resulting overall response is far more linear than what is found in any  
individual pathway). The dynamic response to a step input would not be a  
delayed step output, but a smeared-out rise along some curve.

These are \_continuous\_ systems: the transport lags do not all start at the  
same instant, so in fact there is no clear single delay. Any one signal in  
the loop would appear to vary continuously, at frequencies all the way up to  
the saturation limit and all the way down to where the "grain" of a signal  
carried by a train of impulses begins to matter.

Best,

Bill P.

---

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**Author:** ![Bruce\_Abbott](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/b/ba8739/32.png) [@Bruce\_Abbott](http://discourse.iapct.org/u/Bruce_Abbott)\
**Post date:** [May 6, 1997, 9:31pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335/6 "1997-05-06T21:31:47Z")

</div>

[From Bruce Abbott (970506.1430 EST)]

> Rick Marken (970505.2130 PDT) --

> If you did a multiple regression on the position control data,  
> using Dist and Mouse to predict Cursor, you would probably find  
> a regression equation like Cursor' = 1.0\*Mouse-1.0\*Dist. The  
> coefficients would probably be slightly different than 1.0 and  
> -1.0 and the intercept would probably be slightly different from 0.0.  
> But they would all be close to these values, and R^2 would probably  
> be around .98. This is the result expected if Cursor position is  
> under control.

The equation is Cursor' = 1.00\*Mouse + 1.00\*Dist, and R^2 = 100.0.

This equation tells us that the cursor position was the sum of the the mouse  
and disturbance values (which were zero-centered).

> But it might be that the actual controlled variable is rate of  
> change in Cursor position (dCursor) rather than Cursor position.  
> This could be tested by adding dMouse and dDist to the prediction  
> equation and making dCursor the dependent variable. The complete  
> regression equation becomes:
> 
> dCursor = a\*b1\*Cursor+b2\*Mouse+b3\*Dist+b4\*dMouse+b5\*dDist
> 
> We are only interested in the partial regression of dMouse and  
> dDist on dCursor but the other variables have to be included in  
> the solution of the multiple regression in order to factor out  
> their contribution to the variance in dCursor, dMouse and dDist.
> 
> If dCursor is under control, then the R^2 of the partial  
> regression of dMouse and dDist on dCursor should be high  
> (greater than .98) and, more importantly, the coefficients  
> of dMouse and dDist should be about equal (1.0) and of  
> \_opposite\_ sign.

Actual results:

dCursor' = 0.0\*Cursor + 0.0\*Mouse + 0.0\*Dist + 1.00\*dMouse + 1.00\*Dist,  
and R^ = 1.000.

also, dCursor' = - 4.94 + 0.0155 Cursor3 +0.000276 Mouse, Dist having been  
removed from the equation because of its redundancy with the other variables  
entered first. This equation's R^2 was 0.008: positional values account for  
essentially none of the variance in cursor rate.

Basically, what linear regression is finding is that Cursor is the sum of  
Mouse and Dist, and that dCursor is the sum of dMouse and dDist, as in fact  
they are. So what we are seeing are the feedback functions (mouse effect on  
cursor) and disturbance functions (disturbance effect on cursor). These are  
captured whether there is excellent control (position) or not (rate).

By the way, I tried to improve the correlations for rate control by  
consciously attempting to null out the cursor's motion without attempting to  
correct positional error. After two such attempts, I was still getting an  
indication of excellent position control, without seeing much improvement in  
the rate-control numbers. I found it very difficult to keep the rate of  
cursor motion at zero given the disturbance waveform I was using, which  
included some relatively fast changes in rate and direction.

Regards,

Bruce

---

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**Author:** ![Bill\_Powers1](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/b/8e7dd6/32.png) [@Bill\_Powers1](http://discourse.iapct.org/u/Bill_Powers1)\
**Post date:** [May 6, 1997, 10:21pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335/7 "1997-05-06T22:21:31Z")

</div>

[From Bill Powers (970506.1615 MST)]

Bruce Abbott (970506.1430 EST)--

> Basically, what linear regression is finding is that Cursor is the sum of  
> Mouse and Dist, and that dCursor is the sum of dMouse and dDist, as in fact  
> they are. So what we are seeing are the feedback functions (mouse effect on  
> cursor) and disturbance functions (disturbance effect on cursor). These are  
> captured whether there is excellent control (position) or not (rate).

Nice to know it works!

> By the way, I tried to improve the correlations for rate control by  
> consciously attempting to null out the cursor's motion without attempting to  
> correct positional error. After two such attempts, I was still getting an  
> indication of excellent position control, without seeing much improvement in  
> the rate-control numbers. I found it very difficult to keep the rate of  
> cursor motion at zero given the disturbance waveform I was using, which  
> included some relatively fast changes in rate and direction.

This is another interesting subject. Try doing a run while you imagine that  
your hand is pushing the mouse through very thick syrup. You can change your  
tracking characteristics considerably in this way. This is another reason I  
think we should deal only with practiced subjects trying to do their best:  
there are many ways to fake suboptimum control, but I can't think of any  
ways to pretend to control better than you possibly can!

Best,

Bill P.

---

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**Author:** ![Human\_Factors\_Lab](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/h/b3f665/32.png) [@Human\_Factors\_Lab](http://discourse.iapct.org/u/Human_Factors_Lab)\
**Post date:** [May 7, 1997, 4:45pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335/8 "1997-05-07T16:45:09Z")

</div>

[From Rick Marken (970506.2050)]

Me:

> If you did a multiple regression on the position control data,  
> using Dist and Mouse to predict Cursor, you would probably find  
> a regression equation like Cursor' = 1.0\*Mouse-1.0\*Dist.

Bruce Abbott (970506.1430 EST) --

> The equation is Cursor' = 1.00\*Mouse + 1.00\*Dist, and R^2 = 100.0.

Boy, is my face red. That's what I get for trying to conceptualize  
control in terms of conventional statistical models. The regression  
model led me to think of the variance in the dependent variable (Cursor  
position) as being partly the result of random error. That's why I  
thought R^2 would be only about .98. After all, the correlation between  
the behavior of control model and subject is usually no greater than  
.99 and .99^2 = .98.

I also thought the regression model would reveal the nature of the  
control law in the same way that it supposedly reveals cause-effect laws  
in conventional research; that's why I thought the coefficients would be  
about equal and of opposite sign. What I forgot was the physical fact  
that Cursor = Mouse + Dist. So if Mouse and Disturbance are used to  
predict Cursor position then the coefficients should be \_exactly\_ 1.0  
(and the same sign). Also, since this weighted sum  
is the exact equation used to determine Cursor position, it should  
predict Cursor position perfectly: R^2 should = 1.0 (assuming no  
rounding errors in computing the values of the variable) and it does.

So the moral (for me) is: the best was to study control is still  
the old fashioned way; Testing for Controlled Variables and  
comparing observed behavior to the behavior of models that control those  
same variables.

Best

Rick

---

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**Author:** ![Bill\_Powers1](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/b/8e7dd6/32.png) [@Bill\_Powers1](http://discourse.iapct.org/u/Bill_Powers1)\
**Post date:** [May 7, 1997, 12:08pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335/9 "1997-05-07T12:08:53Z")

</div>

[From Bill Powers (970507.0500 MST)]

Rick Marken (970506.2050) (replying to Bruce Abbott)--

> Mouse and Disturbance are used to  
> predict Cursor position then the coefficients should be \_exactly\_ 1.0  
> (and the same sign). Also, since this weighted sum  
> is the exact equation used to determine Cursor position, it should  
> predict Cursor position perfectly: R^2 should = 1.0 (assuming no  
> rounding errors in computing the values of the variable) and it does.

This is interesting. Given output and disturbance, we can predict cursor  
perfectly; in fact, given any two, we can predict the other. So a multiple  
regression analysis in this case amounts to an analysis of linear physical  
relationships in the environment (no organism required).

Hold that thought.

Now, what is it that psychologists try to do when they vary a stimulus and  
try to predict a response? They are varying disturbance and trying to  
predict output. This can't be done, because there is a missing term: cursor,  
the controlled variable. You must know two out of the three variables to  
predict the remaining one. However, it is possible to assume a value of  
cursor, namely, zero. Under that assumption, one can do a multiple  
regression and solve for the response (responses) given the stimulus  
(stimuli). This will yield non-zero regression coefficients. The intercept,  
in fact, will yield an assumed constant value of cursor that will depend on  
the zeros and signs of the measurement scales used. Is this right so far?

To assume a constant value of cursor is to assume a constant value of the  
reference signal (and no uncontrolled disturbances). This can mean only the  
\_average\_ value of the reference signal, over some period of time. If the  
reference signal actually varies in an unpredictable way with time, or  
across subjects, or both, the multiple correlations will be less than 1.0.  
The regression coefficients will show the equal-and-opposite effects of  
disturbance and output on the cursor, but only on the average. The signs of  
the regression coefficients and correlations will depend on how disturbance  
and output are measured; since measurement scales are arbitrary, either sign  
could be found (for example, one could measure "proximity" or "distance").

Conclusion: this seems to me an elegant way to discuss the behavioral  
illusion in language familiar to psychologists.

Let's turn now to operant conditioning. Here we have measures of output  
(responses) and cursor (reinforcements), and we know the relationship  
between them (the schedule). From these we should be able to calculate the  
disturbance signal, although not the disturbance itself. There are even  
measures that are related to the disturbance signal (deprivation, the  
"establishing conditions" if that's the right term). So it should be  
possible to calculate (roughly) the disturbance signal and relate it to  
measures like the degree of deprivation. A multiple regression should then  
show the expected correlations: a positive correlation between deprivation  
and responses (since positive deprivation has an effect on reinforcements  
opposite to the effect of responses under a normal schedule). The same  
assumption of a constant or average reference signal holds.

The results of a correlation analysis would then give us a picture of the  
environmental relationships involved. The organism would still not be in the  
picture, except for the value of the average reference signal which shows up  
as the intercept(s).

For operant behavior, there are some additional problems. The most popular  
schedules are the variable ones, which put a large random component into the  
dependence of reinforcement on behavior, as well as making the average  
relationship nonlinear. This greatly reduces the correlation coefficients  
that are to be expected, and requires taking much more data to make any  
stable relationship visible. Also, there have been unwarranted assumptions  
about the actual measure of behavior that is appropriate. Bruce Abbott has  
found that the most common measure -- rate of bar-pressing -- is not  
actually a variable in the experiments he has investigated. Instead, the  
rate of pressing is almost constant, while time away from the lever, meal  
size, and interval between meals are much more likely to vary (when  
possible). These alternative variables are hidden by the practice of  
counting total bar presses per session and dividing by session duration to  
get "responses per unit time." This crude measure does give us a behavioral  
variable to measure that has a positive effect on reinforcement rate, so a  
correlation analysis would still be possible, although less informative than  
it might be.

Since we observe the total responses and the total reinforcements, it should  
be possible to find the mean relationship between them even for  
variable-ratio or variable-interval schedules. A plot of one against the  
other, with long periods of observation for each point, should exactly match  
the theoretical response of the apparatus to lever-pressing at various mean  
rates. In fact, this response can be measured without the organism present,  
by generating artificial lever presses at various rates and observing the  
resulting rates of reinforcement, all in simulation but using the same  
program that is used to operate the real apparatus.

Once we have the relationship between mean pressing rate and mean rate of  
reinforcement, we can calculate, in real experiments, the magnitude of the  
disturbing signal under the assumption of a constant reference signal, or  
the magnitude of the reference signal assuming a constant disturbance (I  
suspect that the two will be the same). What we can't do is separate the  
effects of disturbances from the effects of changes in the reference signal.  
Or maybe we can -- that would be a most interesting result!

Anyway, this too would be another useful way to apply standard statistical  
methods to show the relationship between PCT and an accepted way of  
analyzing behavior.

I propose that Rick and Bruce (and anyone else who wants to volunteer) put  
their heads together and develop a rigorous statistical analysis for both of  
these cases. This has been accomplished in part already; what we need is the  
rest of the story. It would make a valuable addition to the PCT literature  
even if it never gets published anywhere else.

Best,

Bill P.

---

<div class="post-metadata">

**Author:** ![Bruce\_Gregory9](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/b/43a26b/32.png) [@Bruce\_Gregory9](http://discourse.iapct.org/u/Bruce_Gregory9)\
**Post date:** [May 8, 1997, 6:26pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335/10 "1997-05-08T18:26:31Z")

</div>

[From

Rick Marken (970506.2050)

> So the moral (for me) is: the best was to study control is still  
> the old fashioned way; Testing for Controlled Variables and  
> comparing observed behavior to the behavior of models that control those  
> same variables.

Rick,

Thought you might resonate with this quote from an article  
entitled "Does the Cosmos Have a Direction" in the 26 April  
\_Science News\_:

"I really think this is much ado about nothing," says  
cosmologist Michael S. Turner ... "The number one rule in  
astronomy is that you can't reanalyze someone else's data to  
look for an effect that [the observations] were not designed to  
measure."

Regards,

Bruce

---

<div class="post-metadata">

**Author:** ![Richard\_Marken](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/r/9fc348/32.png) [@Richard\_Marken](http://discourse.iapct.org/u/Richard_Marken)\
**Post date:** [May 8, 1997, 7:14pm UTC](http://discourse.iapct.org/t/assessing-rate-control/1335/11 "1997-05-08T19:14:33Z")

</div>

[From Rick Marken (970508.1215 PDT)]

Bruce Grergory (970508) --

> Thought you might resonate with this quote from an article entitled  
> "Does the Cosmos Have a Direction" in the 26 April \_Science News\_:

> says cosmologist Michael S. Turner ... "The number one rule in astronomy  
> is that you can't reanalyze someone else's data to look for an effect  
> that [the observations] were not designed to measure."

I'll put it in the frontpiece of my book, if I am ever able to write one.  
This is what I've been trying to say for years (with respect to PCT, of  
course); maybe some people will believe it now that it comes from (what  
I presume is) a \_real\_ scientist.

Best

Rick