The space created by four columns is a tetrahedron in 3-dimensional space
and all row variables are projected onto the column-space,
meaning that all row variables are contained inside the tetrahedron. 
Similarly, if the data matrix has three columns, the space is a triangle
in 2-dimensional
space and all row vriables fall inside the triangle.  To stretch the
context to that of  multiple correspndence analysis of n column variables
and 
if all column variables have three categories each, then n triangles of n
column variables span 2n-dimensional space with generally different
oriientations,
where Cramer's coefficient can be regarded as a projection of one triangle
onto the space of the other triangle. I hope this might be of some use.
Shizuhiko Nishisato  

"Classification, clustering, and phylogeny estimation"          
   <[log in to unmask]> writes:


>Dear All, 
>
>I am doing a simple correspondence analysis on a contingency table having
>more than 5000 lines (that are representing genes) and 4 columns. I am
>then doing a clustering analysis on the coordinates of all genes on the
>factorial axes. The results of the correspondence analysis are quite
>surprising to me. The coordinates of all rows on the 3 axes are strictly
>within a simplex that is a perfect tetrahedron. Of course, the
>extremities of the tetrahedron are corresponding to the coordinates of
>the column in the factorial space. How could such a structure be
>obtained? 
>  
>I first thought that this was due to a specific structure in my original
>dataset, so I’ve decided to do the analysis again on a randomly drawn
>contingency table of the same size (each cell was drawn from a uniform
>distribution between 0 and 100). I collected again such a triangle
>structure. 
>  
>It thus seems that a simple correspondence analysis is always producing
>such a triangular structure on a space with a small number of dimensions,
>but I never heard about this before. I suspect that people are usually
>not getting this because either they have more dimensions, or less rows
>in the original table, leading to points that look to be spread on
>factorial plans in a more homogeneous way.
>
>Any explanation on this will be welcomed! 
>
>Eric 
>-- 
>    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~   
>    Eric Wajnberg
>    Chair of the ESF Scientific programme on
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>    (Universite du Quebec a Montreal)
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>]http://www.sophia.inra.fr/perso/wajnberg/
>
>    Editor-in-Chief of BioControl, Published by Springer.
>
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_______________________________________________________________
Shizuhiko Nishisato, Professor Emeritus, OISE/University of Toronto
   Email:  [log in to unmask]
  




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