[From Rick Marken (970521.1350 PDT)]
Some time ago, Bill Powers (970507.0500 MST) said:
I propose that Rick and Bruce...put their heads together and
develop a rigorous statistical analysis for both of these cases
[predicting responses from disturbance when the controlled
variable is not known oir predicting disturbance from responses
when controlled variable is known -- RM].
Since Bruce hasn't been pestering me to work this problem with him,
I thought I would pester Bruce to work on a different (but similar)
problem with me. The problem is one I ran into long ago while developing
the "Mind Reading" demo -- what is now called "The Test
for the Controlled Variable" in the Java demos. I just ran into it again
recently while trying to develop an improved version of the "Hierarchy
of Behavior and Control" demo.
The problem occurs when one tries to Test more than two hypotheses about
a controlled variable at the same time. In the "Mind Reading"
demo I was testing five hypotheses (just three in the Java version)
at the same time. The hypotheses are that the position of each of the
objects on the screen is under control. I test all hypotheses
simultaneously by appying a _different_ time varying disturbance to the
position of each object. The "Mind Reading" program concludes that the
object being controlled is the one with the _lowest_ correlation
between position (the hypothetical controlled variable) and
This approach to testing several simultaneous hypotheses about the
controlled variable works pretty well. The problem is that sometimes
the correlation between position and disturbance will be lowest for
one of the _uncontrolled_ objects; it looks like an object's position is
controlled when it is _not_. These spuriously low correlations result
from the fact that the disturbances to each of the hypothetical
controlled objects are _correlated with each other_ to some extent!
So occasionally the mouse movements that are negatively correlated with
the disturbance to the controlled object are even _more_ negatively
correlated with the disturbance to another object.
This problem would not arise if the disturbances were _not_ correlated
with each other. It's easy to select two disturbance waveforms that are
not correlated with each other -- a sine and cosine wave, for example.
But I think (and this is where a statistician would help)
that it is impossible to select more than 2 disturbances such
that the correlation between every disturbance and every other
disturbance is 0.0. That is, given three disturbance waveforms,
d1(t), d2(t) and d3(t), I think it is impossible to have it be true that
the correlations between d1(t) and d2(t), d1(t) and d3(t) and d2(t) and
d3(t) are _all_ 0.0.
If this is true, then one way to improve the accuracy of Tests of
more than two simultaneous hypotheses about controlled variables is
by "statistically" removing the covariation between disturbances that
might influence the results of these Tests. This is where Bruce
Abbott comes in. What I would like is a way to determine the correlation
between d1 and cv1 (controlled variable 1) while
factoring out any covariation between d1 and d2 that could
contribute to the existence of a spuriously low correlation between
d2 and cv2 (giving the impression that cv2 rather than cv1 is under
control). To do this, I think it would also be necessary to factor
out any correlation between d2 and m (mouse movements), d2 and cv2
and cv2 and cv1.
I think what I want to compute is called a "partial correlation"
between, say, d1 and cv1 with d2 (and all other disturbances and
other possible contributors to the corrlation between disturbance
and controlled variables) "partialled out". I think I can get
"running values" of all the relevant correlations; what I need is
the formula that will give me the partial correlations (between
d1 and cv1, d2 and cv2, etc), presumably as a function of the
relevant correlations. If I had such a method available I think I could
extend the capability of my "Mind Reading" program so that it
could test not only for object position but also for variables like
"distance between objects". I had problems testing for these more
complex variables _while_ testing for position because the disturbances
to these more complex variables tend to be _highly correlated_ with the
disturbances to position. If I could compute partial correlation I might
be able to include these more complex variables in the set of variables
that the subject could control.
Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: firstname.lastname@example.org