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I have a table of the following kind:

     Cat1 cat2  cat3 ...
Var1 6.3  5.3   8.3
Var2 5.2  5.7   6.1
Var3 2.2  3.9   7.6
.
.
.

It is a table of means for different continuous variables across a categorical variable. My question is: I want a visualization of how the categories relate to the variables. For this purpose I have been using Correspondence analysis. I get a result that makes sense to me, but I am wondering if it is OK to use correspondence analysis on a table of non-categorical data?

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I take it you mean "is it OK to use correspondence analysis for a table of other 'contingency' data, not just frequences?" Yes, it is OK. Your entries may me means or other values somehow reflecting affinity between rows and columns. Entries should be positive values (if negative values are there, add a constant to make them all positive).

Standard CA of frequency table usually uses chi-square distance to map it as euclidean one on the biplot. Since your data are not frequences you are more warranted to use euclidean distance instead of chi-square distance. Also, try playing with options such as centering columns and/or rows or equalizing marginal totals until you get the "best" (both visually and reflecting the table's structure) result.

Also you might consider using multidimensional unfolding as an alternative rows vs columns mapping technique.

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