# Metrics of correlation for 3+ variables

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.

• You may be looking for the coefficient of determination of a regression of $z$ on $x_1, \dots, x_5$: en.wikipedia.org/wiki/Coefficient_of_determination But beware, interpreting it can be misleading. Jan 17, 2013 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? Jan 17, 2013 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. Jan 18, 2013 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? Jan 18, 2013 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. Jan 18, 2013 at 18:06