From: Peter Dalgaard <p.dalgaard_at_biostat.ku.dk>

Date: Thu 16 Feb 2006 - 21:55:47 EST

Date: Thu 16 Feb 2006 - 21:55:47 EST

Gregor Gorjanc <gregor.gorjanc@gmail.com> writes:

> > WPhantom <wp1@tiscali.fr> 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.
*

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 ุster Farimagsgade 5, Entr.B c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 ~~~~~~~~~~ - (p.dalgaard@biostat.ku.dk) FAX: (+45) 35327907 ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.htmlReceived on Thu Feb 16 22:18:27 2006

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