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I have a basic question, which hopefully you all can resolve for me. What is the best way to determine correlations between 3 or more variables? I have a dataset in which 5 continuous variables each correlate (Pearson, R ~= 0.7) with another continuous variable (which I'll call Z).

I know I can use partial correlation between groups of 2 variables and Z, but I don't think this is exactly what I'm looking for, if I understand it correctly.

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You may be looking for the coefficient of determination of a regression of $z$ on $x_1, \dots, x_5$: But beware, interpreting it can be misleading. – Stephan Kolassa Jan 17 '13 at 19:30
If I read that correctly, the coefficient of determination gives an idea about how several measurements correlate together with a single variable (Z in this case). What if I want to get a feel for the relationship between these other variables, without reference to Z? – learner Jan 17 '13 at 21:01
@learner, if you want summarize the strengths of intercorrelations among p variables in a single number you could compute, for example, (1) average abs. correlation or (2) geometric mean abs. correlation, or (3) determinant of abs(corr. matrix). These three are very different ways to conceptualize p-variate association by a single value. – ttnphns Jan 18 '13 at 9:45
@ttnphns, I have heard that the difference between the arithmetic mean and the geometric mean is that the geometric mean reduces the impact of unusually high and low values in the quantities examined (correlations, here). What insight is the determinant of the correlation matrix providing? – learner Jan 18 '13 at 16:32
Geometric mean has also another obvious sense: the product is really big when its factors are big jointly. Determinant is the "volume" of the correlation matrix, - the product of its eigenvalues. When correlations are small, det is high; when either they are big or there is collinearities, det is close to 0. – ttnphns Jan 18 '13 at 18:06
up vote 3 down vote accepted

Since you seem to be saying that a matrix of all 10 correlations (those that don't involve z) is insufficient, it sounds as if you want to distill all these correlations into some underlying themes. The way to do that might be with factor analysis or principal component analysis, depending largely on whether your variables are measured with or without much error. Multidimensional scaling could be used as well. All of these techniques typically require a fair amount of expertise, so you may want to ask the help of a mentor or consultant.

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Am I right in thinking that I would want to use factor analysis in instances where I expect a high degree of error? – learner Jan 18 '13 at 16:26
Right, relatively speaking. See… – rolando2 Jan 18 '13 at 17:31

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