-------- Original Message --------
Subject: Re: multiple correlated variables
Date: Wed, 27 Feb 2008 07:19:53 -0800 (PST)
From: Joseph Kunkel <[EMAIL PROTECTED]>
To: [email protected]
CC: Joseph Kunkel <[EMAIL PROTECTED]>
References: <[EMAIL PROTECTED]>
Jim,
I was thinking of rephrasing the Y=XB problem to correct the problem:
Redefine Y to be {Y1|X1| ...|Xp} then redefine X to be {Xp+1| Xp+2|...|
Xq}
Then ask is there any additional information with respect to the new X
of Y corrected for X1..Xp. This answers the question of whether the
X1..Xp add anything significantly different to the analysis. I
thought of that as the more important question although the author
might just have wanted to know the best estimate of his Y. I am, I
guess, more interested in the hypothesis X of how Y is explained and
is it different from another hypothesis.
I think the matrix equations still work whether the Xi are formally
thought of as dependent or independent. No? Causality questions are
independent of the math testing the correlations?
It is like multivariate analysis of covariance. I agree that there is
a possibility of singularity and then the approach would fail.
I need to look into your suggested options.
Joe
On Feb 27, 2008, at 9:10 AM, morphmet wrote:
-------- Original Message --------
Subject: RE: multiple correlated variables
Date: Tue, 26 Feb 2008 18:27:22 -0800 (PST)
From: F. James Rohlf <[EMAIL PROTECTED]>
Reply-To: [EMAIL PROTECTED]
Organization: Stony Brook University
To: [email protected]
References: <[EMAIL PROTECTED]>
It may not be that easy. Jason said he has only one dependent
variable and many independent variables. The problem for a general
analysis is that Jason says that the independent variables are
highly correlated - which means that standard regression methods may
fail because one cannot reliably invert their covariance matrix.
There are alternative methods such as ridge regression, partial
least squares, and principal components regression.
If the question really is just "which independent variable is more
important in explaining variation in the dependent one" then just
regress the dependent variable on each of the independent variables
one at a time and see which gives the highest R^2 value. However,
usually the question is what set of independent variables gives the
best prediction and that requires simultaneous testing and
introduces the problems mentioned in my first paragraph.
------------------------
F. James Rohlf, Distinguished Professor
Ecology & Evolution, Stony Brook University
www: http://life.bio.sunysb.edu/ee/rohlf
-----Original Message-----
From: morphmet [mailto:[EMAIL PROTECTED]
Sent: Tuesday, February 26, 2008 3:30 PM
To: morphmet
Subject: Re: multiple correlated variables
-------- Original Message --------
Subject: Re: multiple correlated variables
Date: Mon, 25 Feb 2008 17:16:01 -0800 (PST)
From: Joseph Kunkel <[EMAIL PROTECTED]>
To: [email protected]
CC: Joseph Kunkel <[EMAIL PROTECTED]>
References: <[EMAIL PROTECTED]>
Jason,
The appropriate approach is Analysis of Dispersion which is the
multivariate extension of Analysis of Variance. It allows multiple
Yi
observations that may be correlated via a Cov(Yi,Yj) covariance
matrix. These are related to a traditional X design matrix via the
generalized matrix equation Y = XB where Y is n by p for p dependent
characters and sample size n. X is n by k for k design elements or
independent variables. That results in a k by p matrix of
parameters,
B, to be estimated.
One can ask the question for design feature q, a portion of the k
design elements:
Given characters say 1 to 10 of the p available, if characters 1-5
are
corrected for characters 6-10 is there any additional information in
1-5 that is significant on the design feature q. This type of
question answers your query of whether one variable or a set of
variables is more important or more predictive than another set and
answers it with an F-test.
I have a set of R-scripts that carry out this analysis if you are
willing to use the Gnu Public R calculation environment and are
somewhat familiar with setting up the X design matrices. I have not
seen another convenient implementation of this although I am sure it
can be found somewhere else.
Analysis of Dispersion is well described in C. Radhakrishna Rao
(1965)
Linear Statistical Inference and Its Applications, 1st Edition.
Joe Kunkel
[EMAIL PROTECTED]
On Feb 25, 2008, at 11:30 AM, morphmet wrote:
> -------- Original Message --------
> Subject: multiple correlated variables
> Date: Sun, 24 Feb 2008 04:48:15 -0800 (PST)
> From: Jason S <[EMAIL PROTECTED]>
> To: [email protected]
>
>
>
>
> Dear all,
>
> I have a quick question, and I'd love to hear any suggestions you
> might
> have.
>
> I got one dependent variable (body size) and several independent
> variables, and I'm trying to figure out which independent
variable is
> more important in explaining variation in the dependent one
> (experiments
> are not feasible). The problem is that many of the independent
> variables
> are themselves highly correlated. I thought about calculating
partial
> correlation coefficients, but I'm not very confident with that
choice.
> Any suggestions?
>
> Thanks a lot!
>
> Jason
>
>
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-·. .· ·. .><((((º>·. .· ·. .><((((º>·. .· ·. .><((((º> .··.·
>=- =º}}}}}><
Joseph G. Kunkel, Professor
Biology Department
University of Massachusetts Amherst
Amherst MA 01003
http://www.bio.umass.edu/biology/kunkel/
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For more information visit http://www.morphometrics.org
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-·. .· ·. .><((((º>·. .· ·. .><((((º>·. .· ·. .><((((º> .··.·
=- =º}}}}}><
Joseph G. Kunkel, Professor
Biology Department
University of Massachusetts Amherst
Amherst MA 01003
http://www.bio.umass.edu/biology/kunkel/
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