# Canonical correlation analysis with rank correlation

Canonical correlation analysis (CCA) aims to maximize the usual Pearson product-moment correlation (i.e. linear correlation coefficient) of the linear combinations of the two data sets.

Now, consider the fact that this correlation coefficient only measures linear associations - this is the very reason why we also use, for example, Spearman-$\rho$ or Kendall-$\tau$ (rank)correlation coefficients which measure arbitrary monotone (not necessarily linear) connection between variables.

Hence, I was thinking of the following: one limitation of CCA is that it only tries to capture linear association between the formed linear combinations due to its objective function. Wouldn't it be possible to extend CCA in some sense by maximizing, say, Spearman-$\rho$ instead of Pearson-$r$?

Would such procedure lead to anything statistically interpretable and meaningful? (Does it make sense - for example - to perform CCA on ranks...?) I am wondering if it would help when we are dealing with non-normal data...

• Will OVERALS - linear canonical analysis which optimally scales (monotonically transforms) variables to maximize canonical correlations - be to your liking? – ttnphns Apr 2 '13 at 16:17
• @ttnphns : Thanks for the idea, I haven't heard of it before, and looks really interesting! However, I don't think it addresses the point: as far as I understand, it is essentially a combination of optimal scaling and CCA - but optimal scaling makes really sense only for categorical variables. It doesn't seem to change much for continuous variables measured on ratio scale (which I have in my mind!). But correct me, if I'm wrong. – Tamas Ferenci Apr 2 '13 at 19:21
• @ttnphns : Well, the very same way you sometimes use Spearman correlation on continuous variables! (Of course it handles the data as being ordinal... but we neverthless use it on definitely continuous variables to characterize general monotone (and not only linear) association between the variables.) That's why I thought this would make sense within CCA as well... – Tamas Ferenci Apr 3 '13 at 5:35
• @Glen_b, You are right. Of course the rank correlations are for any monotonicity - be it ordinal or continuous data. I'm so much surprised at my own comment above that I'm deleting it. – ttnphns Oct 13 '13 at 1:43
• You could try using Kernel CCA which specifically when used with radial basis functions enables us to project the data into a infinite dimensional subspace. – roni Feb 25 '16 at 13:32

## 2 Answers

I used restricted cubic spline expansions when computing canonical variates. You are adding nonlinear basis functions to the analysis exactly as you would be adding new features. This results in nonlinear principal component analysis. See the R Hmisc package's transcan function for an example. The R homals package takes this much further.

• Thank you! The approach described in homals was novel to me, but definitely interesting. – Tamas Ferenci Aug 6 '14 at 7:47

The standard method of CCA works with product moment correlation coefficient matrix. For largest mgnitude CC it constructs two composite variables z1(n) and z2(n) by linear combination of two matixes (with n rows and m1 and m2 variables) such that abs(correlation(z1,z2)) is maximized. This objective function may be maximized directly even if correlation(z1,z2) is not product moment but defined differently.

Mishra, SK (2009) "A Note on the Ordinal Canonical Correlation Analysis of Two Sets of Ranking Scores"

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1328319