Re: Correlation of X with XY ?
From: Charles Knapp (nowhere_at_nomailspam.com)
Date: 09/27/04
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Date: Mon, 27 Sep 2004 10:34:59 -0400
"Ray Koopman" <koopman@sfu.ca> wrote in
message news:1096263808.693743.183230@k17g2000odb.googlegroups.com...
> Charles Knapp wrote:
> > "Ray Koopman" <koopman@sfu.ca> wrote in message
> > news:1096241534.783487.155680@h37g2000oda.googlegroups.com...
> >>
> >> SD[x] E[y]
> >> Corr[x,xy] = ----------------------------------------------------.
> >> Sqrt(Var[x] Var[y] + Var[x] E[y]^2 + E[x]^2Var[y])
> >
> > Boy..... we're sure a long way apart..... I get:
> >
> > E{X^2}*E{Y}-E{X}*E{X}*E{Y}
> > Corr(x,xy) = ----------------------------------------------
> > sqrt( Var{X}*( E{X^2}*E{Y^2}-E{X}^2*E{Y}^2 ) )
> >
> > ?????????????
>
> Google is certainly doing some wonderful reformatting here!!
>
> Your last expression is ok as far as it goes.
> What you're objecting to are my simplifications.
>
> Your numerator is E{X^2}*E{Y}-E{X}*E{X}*E{Y}.
> Factor out E{Y} to get (E{X^2}-E{X}*E{X})*E{Y},
> rewrite that as (E{X^2}-E{X}^2)*E{Y},
> and recognize that E{X^2}-E{X}^2 = Var{X},
> giving Var{X}*E{Y}.
>
> Your denominator is sqrt( Var{X}*( E{X^2}*E{Y^2}-E{X}^2*E{Y}^2 ) ).
> Take Var{X} outside the sqrt as SD{X}
> and divide it into the numerator,
> giving SD{X}*E{Y}.
>
> That leaves E{X^2}*E{Y^2}-E{X}^2*E{Y}^2 inside the sqrt.
> Substitute E{X^2} = Var{X}+E{X}^2, and similarly for Y,
> to get (Var{X}+E{X}^2)*(Var{Y}+E{Y}^2)-E{X}^2*E{Y}^2.
> Expanding and cancelling gives
> Var{X}*Var{Y} + Var{X}*E{Y}^2 + E{X}^2*Var{Y},
> which is what I had inside the sqrt.
[Charles Knapp]
Agreed! We got the right answer. Thanks a million.
Using your (simpler) expression:
SD[x] E[y]
Corr[x,xy] = ----------------------------------------------------.
Sqrt(Var[x] Var[y] + Var[x] E[y]^2 + E[x]^2Var[y])
we see that while the denominator can vary for different Means and
Variances.... it can never go to zero, so we can write:
SD[x] E[y]
Corr[x,xy] = ----------------------------------------
Denominator>0
This tells us that:
1. In general Corr[x,xy] can be "anything" depending on SD[x] and E[y]
2. For Mean(y)=0 the correlation will in fact be zero.
3. Whatever the means and variances of x and y are (as long as they
are the same), x must correlate with xy the same as y does
(symmetric correlation... this is obvious).
All of this comes up in Factor Analysis in connection with Thurstone's
famous "Box Problem". There he utilizes the correlation of x, y and z
with the folloing functions:
x^2
y^2
z^2
xy
xz
yz
sqrt(x^2+y^2)
sqrt(x^2+z^2)
sqrt(y^2+z^2)
2x+2y
2x+2z
2y+2z
log x
log y
log z
xyz
sqrt(x^2++y^2+z^2)
exp x
exp y
exp z
Where x, y and z are the "3 sides" of a collection of 20 boxes.
x ranges from 1 to 3
y ranges from 2 to 4
z ranges from 3 to 5
so none of the Means are zero and none of the variances
are zero. Hence he comes up with a "positive non zero"
correlation between x, y and z and any of the above functions.
Because of point (3) above that:
3. Whatever the means and variances of x and y are (as long as they
are the same), x must correlate with xy the same as y does.
(symmetric correlation..... this is obvious)
it turns out that x, y and z correlate "symmetrically" with the above
functions, and when the vectors are plotted in "Factor Space"
(eigenvector space) the "Box" is recreated in that space and the
vectors fall on the "13 symmetry axes of a cube (box)".
Now... a dispute arose because actual Psychology Tests
(Intelligence and Personality tests) appear to form a "cube"
in Psychometry space (Factor Analytic eigenvector space)
and I have suggested that "Psychology"
is actually therefore a "Thurstone's Box Problem", but someone
pointed out that if you use "Z-scored variables" (mean=0,
variance=1) as they do in Psychometry the Box Problem
would not work!
Turns out it does work anyway, because the human head
does not utilize "simple algebraic functions" such as above
in Thurstone's Box Problem.... but instead uses "dipole monents"
between the various "corners of the cube", (corners of the cubic
brain) and so the correlations are never zero despite using
Z-scored data.
In other words, Thurston's "Box Problem" is a very good
model of human Personality Structure. Theoretically
accurate in fact!!
Anyway.... thanks a lot for your help... this has clarified a
great deal of confusion.
George Hammond aka "Charles Knapp".
PS: I opened a Cape Cod phone book and the first name
I saw as Charles Knapp. I have so many "paparazzi" following
me around these days I have to use a pseudonym to carry on a
private conversation on the internet.
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