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I'm using the FactoMineR package in R to do a Multiple Correspondence Analysis on a large set of data. Specifically, I'm looking for correlations among a set of five categorical variables.

I used the plot.MCA command to create asymmetric biplots of the variables (NOT the variable categories) in the 15 possible pairs of dimensions from 1-6 (which covers 90% of the total variance). Trying to interpret graphs like this (example below) has led to two questions.

enter image description here

Question 1: What values is the system using for coordinates to draw the plots? By inspection, it appears to be using the eta^2 values. That would be surprising, because as I understand it, the eta^2 value for a variable-dimension pair indicates the percentage of total variance explained in that dimension by that variable. In that light, using eta^2 to indicate correlation only makes sense if one assumes that position (0,0) on the graph means that neither variable is correlated with either dimension and that (1,1) means both variables are perfectly correlated with each of the dimensions. That's counter to what the FactoMineR videos by Francois Husson seem to say. As far as I can tell, Husson says that the closer the plotted point is to an axis, the greater the correlation between that variable and that dimension.

  • In either case, what conclusions would one draw from the sample graph? According to what Husson's video says, the graph indicates that:

  • greedcat is moderately correlated with dimension 1 and weakly correlated with dimension 2,

  • stratcat is weakly correlated with dimension 1 but strongly correlated with dimension 2,
  • trustcat is negligibly correlated with both dimensions, and
  • fearcat and sympcat are negligibly correlated with dimension 1 and weakly correlated with dimension 2.

Obviously the interpretation would be very different if FactoMineR is plotting eta^2 values. In that case:

  • trustcat would be sternly correlate with both dimensions,
  • fearcat and sympact would both be at least moderately correlated with both dimensions,
  • greedcat would be moderately correlated with dimension 1 and weakly correlated with dimension 2, and
  • stratcat would be moderately correlated with dimension 1 but not correlated with dimension 2.

So which interpretation is correct?

Question 2: Can one use the distance between points to infer the strength and/or direction of any correlation between the variables and each other? If yes, is it true that the smaller the distance, the greater the correlation? I know that when evaluating variable categories on a biplot, it's the angular distance that indicates the strength of correlation, but am not sure that works for the variables.

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Well, I've managed to answer my own questions by finding and reading Husson, Lê and Pagès (2011). Exploratory Multivariate Analysis By Example Using R.

FactoMineR uses the square correlation ratios (which in curvilinear relationships are equal to the eta^2 values) to plot the variables.

When interpreting the biplot, the greater the perpendicular distance from the axis to the point, the stronger the correlation between the axis and the point.

Also the cosine of the angular distance between any two variables indicates the magnitude of correlation between them. You have to do a similar analysis on categories to gain insight into the direction of effect.

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