[R] Anova unequal variance

[Peng Yu]

On Fri, Jan 22, 2010 at 3:33 PM, Dennis Murphy <djmuser at gmail.com> wrote:
> 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.

Is gls() with only two factor levels the same as t.test() with var.equal=F?

>> >> 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
>> >> >>
>> >> >> ______________________________________________
>> >> >> R-help at r-project.org mailing list
>> >> >> https://stat.ethz.ch/mailman/listinfo/r-help
>> >> >> PLEASE do read the posting guide
>> >> >> http://www.R-project.org/posting-guide.html
>> >> >> and provide commented, minimal, self-contained, reproducible code.
>> >> >
>> >> >
>> >>
>> >> ______________________________________________
>> >> R-help at r-project.org mailing list
>> >> https://stat.ethz.ch/mailman/listinfo/r-help
>> >> PLEASE do read the posting guide
>> >> http://www.R-project.org/posting-guide.html
>> >> and provide commented, minimal, self-contained, reproducible code.
>> >
>> >
>
>