Corresponding Regressions
Fri, 8 Mar 1996 14:15:35 -0500
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Forgive me for not directly addressing your
request for the regression equations. I felt =

it necessary that the core of the method be
understood before going on to regression
techniques that might, on their own, mystify =

less mathematically informed readers. So on
now to the regressions!
Partitioning the data into extremes and
midrange groups is inexact. And in applied
research we may have only measured one
of the IVs. Regression analysis can help
make things more efficient. =

I will quote the procedure from my article
in JMB, as I am too tired right now to
"wing it":
NOTE: In the following x and y are not
necessarily the same as X1, X2 or Y as =

described in the Core posting. Each variable
is, in turned, treated as the x and y
in order to discover which is the otherwise =

unknown X1, X2 and Y. =


The procedure "requires conducting two
regression analyses on the same variables,
letting each serve as the predictor of the =

other in turn. First, either an x or y is =

used to predict the other, with the =

understanding that in eventual applied
research their status as IVs and DVs will
not be known. As the regression models =

are developed, the prediction errors are =

derived by subtracting the predicted values
from the actual values of the predicted =

variable. These errors of prediction are then =

converted to absolute values in order to =

reflect the extremity of the errors. Next, the
absolute values of the deviations from the =

mean of the predictor variable are determined
in order to reflect the extremity of the
predictor values. The correlation between =

these absolute deviations and the absolute
errors is found. When the predictor =

variable is x this correlation is symbolised
as rde(x)- the correlation between absolute
deviations from the mean of x and the
absolute errors predicting y. When y is the
predictor variable this index is referred to =

as rde(y)- the correlation between absolute =

deviations from the mean of y and the =

absolute errors from predicting x. The =

assessment of asymmetrical relations comes
from a comparison of rde(x) and rde(y). The
rde correlation should be more negative when =

the real dependent variable serves as the

That is a mouth full. I will try to explain,
although- as I said- I am tired and may
jumble things up. Its also been a long
time since I thought about all this.
Recall that our model in The Core posting
was that nature has synthesized Y by =

adding X1 and X2; Y=3DX1+X2. X1 and X2 =

were found to be more similar at the extremes
of Y. In the purest extreme case, both X1 and =

X2 would simply be duplicates and the same
as Y/2. In terms of regression lingo,
our equation would be X1=3DY*.5, at
the extremes of Y, but not in the midrange
of Y. In this extreme and pure case, there
would be no error at the extremes of Y
when predicting X1 from Y. At the extreme
of Y, the part of Y that is not X1 would just be
a duplicate of X1, ie. (X2), and the need for X2 =

drops out of the equation. There is just not
much left over in X1 that does not correlate
with Y (at the extremes of Y). This is why
rde(Y), as described above, will tend to be
negative. The more extreme the Y value, the
lower the error in predicting X1 from Y. =

This is not the case when we predict Y
from the extremes of X1. Extreme values
of X1 could be paired with any value of X2.
Only those random pairings where X1 and
X2 were similar produced extreme values
of Y. An extreme value of X1 could be paired
with any kind of X2. Since Y=3DX1+X2, and
since X2 is uncorrelated with X1, whether at
the extreme or midrange of X1, there will be
plenty in Y that is not explained by X1, where
ever in the range of X1 that we look.This error
is caused by X2 values that differ from X1. =

Thus we would expect the errors to be =

uncorrelated with the extremity of X1, =

when predicting Y from X1.
I am going to sign off now. I am tired.
Let me know if this makes sense or
is just jumbled jargon.
Bill =