Using regression to estimate references

[From Rick Marken (950124.0845)]

Ok. The horse is dead but I can't resist one more small beat.

This whole c=d+h thing started because Bruce was using regression analysis to
determine the relationship between handle, h, and disturbance, d, in a
tracking task. The regression equation is:

h = r + kd

Bruce found that the intercept of this regression equation (r) is almost
exactly equal (in screen units) to the reference position at which the
subject is trying to maintain the cursor. That is, if the subject is trying
to keep the cursor at screen position 319 (the middle) then the value of r
that results from the regression analysis is almost exactly 319. But (as
Bruce discovered) this is only true if the predictor variable (d) is properly
reconstructed. If it is, then d will have an average value of 0. Since h is
stored in screen units, then r will be the value of h when d is 0, which is
the reference value of the controlled variable in screen units.

My last beat on the dead horse is just a caveat regarding the use of
regression analysis to estimate the reference state, r, of a controlled
variable. The intercept of the regression equation gives an estimate of r
only under (at least) the following special circumstances:

1. The average of the predictor (d) must be 0.

2. The dependent variable, h, must be measured in units of the controlled
variable, c. In the THREECV1 experiments, h = c, where c is the displayed
position of the cursor.

3. The subject must keep the cursor in one position (fixed reference). In
this case, the average position of the handle corresponds to the fixed
position where c is maintained (assuming the average disturbance is 0).

If any one of these conditions is not met then the r of the regression
equation is not an estimate of the reference position of the controlled