Gregor Gorjanc <

[hidden email]> writes:

> > WPhantom <

[hidden email]> writes:

> >

> >>> Thanks Brian for the reference.

> >>> I just discover that it is available in our

> >>> library so I going to take it & read it soon.

> >>> Actually, I don't even know the difference

> >>> between a multistratum vs a single-stratum AOV. A

> >>> quick search on google returned me the R materials so that I imagine

> >>> that these concepts are quite specific to R.

> >

> > You have to be careful not to confuse Google's view of the world with

> > Reality...

> >

> > The concept of error strata is much older than R, and existed for

> > instance in Genstat, anno 1977 or so. However, Genstat seems to have

> > left little impression on the Internet.

> >

> >>> I will read the book first before asking for more informations.

> >

> > The executive summary is that the concept of error strata relies

> > substantially on having a balanced design (at least for the random

> > effects), so that the analysis can be decomposed into analyses of

> > means, contrasts, and contrasts of means. For unbalanced designs, you

> > usually get meaningless analyses.

> >

>

> Can you (prof. Dalgaard) please point us to relevant book with these

> topics. I am very interested in it since my data are often unbalanced.

Hmm, the Danish tradition is highly based on lecture notes, so I don't

have a specific book for you. One possible starting point is

Tue Tjur (1984): Analysis of variance designs in orthogonal designs.

Int.Statist.Review 52, 33-81.

The thing to notice in relation to that paper is that the

decomposition (p.55) of the covariance matrix as sum(lambda_B Q_B^0)

is highly dependent on having an orthogonal design. Without the

orthogonality, it still defines a model, but typically one without a

sensible interpretation.

Look at a simple 1-way anova with three groups of equal size. The Q

matrices will be the projections P_X and I-P_X, where X is the design

matrix for the grouping factor, e.g.

> X <- model.matrix(~factor(rep(1:3,each=2)))

> X

(Intercept) factor(rep(1:3, each = 2))2 factor(rep(1:3, each = 2))3

1 1 0 0

2 1 0 0

3 1 1 0

4 1 1 0

5 1 0 1

6 1 0 1

...

P_X can be found in the following semi-secret way:

> P <- stats:::proj.matrix(X)

> P

1 2 3 4 5 6

1 0.5 0.5 0.0 0.0 0.0 0.0

2 0.5 0.5 0.0 0.0 0.0 0.0

3 0.0 0.0 0.5 0.5 0.0 0.0

4 0.0 0.0 0.5 0.5 0.0 0.0

5 0.0 0.0 0.0 0.0 0.5 0.5

6 0.0 0.0 0.0 0.0 0.5 0.5

Suppose we put a random component of 10 on P_X and 1 on (I-P_X).

We then get

> diag(6) - P + 10*P

1 2 3 4 5 6

1 5.5 4.5 0.0 0.0 0.0 0.0

2 4.5 5.5 0.0 0.0 0.0 0.0

3 0.0 0.0 5.5 4.5 0.0 0.0

4 0.0 0.0 4.5 5.5 0.0 0.0

5 0.0 0.0 0.0 0.0 5.5 4.5

6 0.0 0.0 0.0 0.0 4.5 5.5

which is a perfectly sensible covariance for within-group correlated

data.

Now try the same stunt with unbalanced data:

> X <- model.matrix(~factor(rep(1:3,1:3))-1)

> P <- stats:::proj.matrix(X)

> diag(6) - P + 10*P

1 2 3 4 5 6

1 10 0.0 0.0 0 0 0

2 0 5.5 4.5 0 0 0

3 0 4.5 5.5 0 0 0

4 0 0.0 0.0 4 3 3

5 0 0.0 0.0 3 4 3

6 0 0.0 0.0 3 3 4

I.e. we are de facto assuming that observations in the smaller group

have a larger variance than observations in the larger groups.

> >>> Thanks

> >>>

> >>> Sylvain Cl?ment

> >>>

> >>> At 12:38 14/02/2006, you wrote:

> >>

> >>>> >More to the point, you are confusing

> >>>> >multistratum AOV with single-stratuam AOV. For

> >>>> >a good tutorial, see MASS4 (bibliographic

> >>>> >information in the R FAQ). For unbalanced data

> >>>> >we suggest you use lme() instead.

>

> I do not have the whole book in my head as prof. Ripley probably does,

> but I can not recall to read about this in MASS4. I am sure I am wrong

> and would you (prof. Ripley) be please so kind and point us to relevant

> chapters/pages.

>

> Many thanks.

>

> --

> Lep pozdrav / With regards,

> Gregor Gorjanc

>

> ----------------------------------------------------------------------

> University of Ljubljana PhD student

> Biotechnical Faculty

> Zootechnical Department URI:

http://www.bfro.uni-lj.si/MR/ggorjan> Groblje 3 mail: gregor.gorjanc <at> bfro.uni-lj.si

>

> SI-1230 Domzale tel: +386 (0)1 72 17 861

> Slovenia, Europe fax: +386 (0)1 72 17 888

>

> ----------------------------------------------------------------------

> "One must learn by doing the thing; for though you think you know it,

> you have no certainty until you try." Sophocles ~ 450 B.C.

> ----------------------------------------------------------------------

>

--

O__ ---- Peter Dalgaard Øster Farimagsgade 5, Entr.B

c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K

(*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918

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