[R] A concrete type I/III Sum of square problem

[Peter Dalgaard]

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

> > WPhantom <wp1 at 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.

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
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