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F. James Rohlf, Distinguished Professor

Dept. Ecology and Evolution, Stony Brook University, NY 11794-5245

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.