Good points, well worth our attention. I haven't
seen your references, but Bill Kruskal's coordiate
free approach treats the same ideas. He was
teaching it in the late 50's and early 60's at
Chicago and wrote it up in the fourth Berkeley.
The first part of Scheff� treats projections into
the various subspaces for those who might be
interested.
"George W. Cobb" wrote:
>
> I, too, think of ANOVA and regression as variations on a common
> theme. Here's an additional way in which they differ: For balanced
> ANOVA, the decomposition of the data into sums of squares and degrees of
> freedom is determined by a group of symmetries. For example, consider a
> one-way randomized complete block design with R rows as blocks and
> C columns as treatments. The analysis is invariant under all row
> permutations, and all column permutations, i.e., interchanging any two
> rows of the data, or any two columns of the data, won't change the
> analysis. If you now think of the data as a vector in RxC-dimensional
> space, the symmetries (row permutations, column permutations) determine
> invariant subspaces; these are precisely the subspaces you project the
> data vector onto to get the SSs and dfs. In regression, the subspaces
> you project onto are determined directly by a spanning set of carrier
> variables; in balanced ANOVA, the subspaces are uniquely determined by the
> symmetries, and the spanning sets are somewhat arbitrary. (I claim
> no credit for this lovely way of looking at things; I learned it from
> Peter Fortini and Persi Diaconis. It's written up in Fortini's
> dissertation from the 1970s, and Diaconis's IMS lecture notes on group
> theory and statistics.)
>
> Of course you only have such clean sets of symmetries for balanced
> designs, and the approach via symmetries doesn't address such things as
> the difference between fixed and random effects, which Bob
> Wheeler raises. Nevertheless, to the extent that I think of ANOVA
> as distinct from regression, I find the role of symmetries worth
> keeping in mind.
>
> George
>
> George W. Cobb
> Mount Holyoke College
> South Hadley, MA 01075
> 413-538-2401
>
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--
Bob Wheeler --- (Reply to: [EMAIL PROTECTED])
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