Anova unequal variance

Hi:

On Thu, Jan 21, 2010 at 1:06 PM, Peng Yu <pengyu.ut at gmail.com> wrote:

On Thu, Jan 21, 2010 at 2:41 PM, Dennis Murphy <djmuser at gmail.com> wrote:

Hi:

On Thu, Jan 21, 2010 at 12:29 PM, Peng Yu <pengyu.ut at gmail.com> wrote:

On Thu, Jan 21, 2010 at 2:16 PM, Dennis Murphy <djmuser at gmail.com>
wrote:

Hi:

This paper was a prelude to his first book 'Exact Statistical Methods
for
Data Analysis'.
He uses what is called a generalized p-value approach to inference,
and
for
the
book he wrote commercial software. AFAIK, no R package implements his
methodology. The 'conventional' approach to unequal variance in ANOVA
is
to use generalized least squares, whose implementation is found in
gls()
in
the nlme package.

There are quite a few references on ?gls. Which one is the most
introductory material that I should start with, if I want to
understand the method?

GLS is a standard technique in linear model theory. It is well
documented.
Any good book on linear statistical models should have a discussion on
it.
(Probably Wikipedia, too). If unequal
variance is the only issue (meaning independent observations), the
technique
is called weighted least squares (WLS). GLS is more general in that it
can
be applied to correlated observations. Assuming the variances are known
(a big if), it is easy to convert from WLS to ordinary least squares -
divide all
the responses by the group standard deviation to which it belongs. The
transformation in GLS (again, assuming variances known) involves a
matrix
transformation (Cholesky, when appropriate). When the variances are
unknown,
as they usually are, the estimation problem is a lot messier and one
needs
to resort to approximations.

Would you please recommend a good book to me?

Here are a couple: if you haven't been exposed to the matrix approach to
regression,
these will be over your head, but it's necessary to develop GLS:

(1) Ravishanker? & Dey: A First Course in Linear Model Theory. GLS starts on
p. 122
(2) Myers: Classical and Modern Regression with Applications. See Chapter 7.

There are a number of other good books that discuss GLS, but these are
pretty
good. Myers is on a lower mathematical level than R & D.

Do you have any simple explanation that may help me understand what is
the difference between the method in 'Exact Statistical Methods for
Data Analysis' and the method in gls()?

No. They're quite different approaches. Weerahandi's is conditional;
GLS is unconditional.

Would you please elaborate what you mean by "conditional" and
"unconditional"?

Conditional means given the observed data; unconditional means over all
potential
sets of data (of the same size, from the same population) that could be
observed.
These are two different forms of inference.

Take the simple linear regression of Y on X. Regression analysis aims to
estimate
the conditional mean E(Y|x); i.e., we treat the observed x's as fixed and Y
as random.
If we didn't make this assumption, then X would also be a random variable
and
we would have what is called 'errors in variables' regression, where the
objective
is to estimate E(Y), among other things. This topic arises more often in
econometrics.

HTH,
Dennis

Dennis

HTH,
Dennis

On Thu, Jan 21, 2010 at 12:03 PM, Peng Yu <pengyu.ut at gmail.com>
wrote:

I found this paper on ANOVA on unequal error variance. Has this be
incorporated to any R package? Is there any textbook that discuss
the
problem of ANOVA on unequal error variance in general?

http://www.jstor.org/stable/2532947?cookieSet=1