Phylogenetic canonical correlation analysis

I just posted a new function to do phylogenetic canonical correlation analysis. Canonical correlation is a procedure whereby, given two sets of variables (say, a set of Xs and a set of Y), we identify the orthogonal linear combinations of each that maximize the correlations between the sets. This type of analysis is most naturally used in an evolutionary study to analyze, say, a set of morphological variables and a set of environmental or ecological variables.

The phylogenetic version of this analysis takes the phylogeny and a (explicit or implicit) model of evolution into account to find the linear combination of Xs and Ys that maximizes the evolutionary correlations (that is, the inferred correlation of evolutionary changes) between the two sets (Revell & Harrison 2008).

The program is very simple. Direct link to the code is here. To use the function, first load the source:

> source("http://faculty.umb.edu/liam.revell/phytools/phyl.cca/v0.1/phyl.cca.R")
> result<-phyl.cca(tree,X,Y)

Here, tree is a phylogenetic tree; and X & Y are two data matrices containing values for one or multiple characters in columns and species in rows. Rows should be named by species.

The results are returned as a list with the following elements:

> result
$cor
[1] 0.3764753 0.1852836 0.1054606
$xcoef
CA1 CA2 CA3
[1,] 0.04497549 -0.09956576 -0.45926364
[2,] -0.18997199 0.46065246 -0.07810429
[3,] -0.42425815 -0.16063677 -0.18791902
[4,] 0.25374826 0.29822455 -0.06176255
$ycoef
CA1 CA2 CA3
[1,] -0.2704762 -0.3841450 0.1029158
[2,] -0.1048448 0.2502089 0.5860655
[3,] 0.3736474 -0.2580132 0.2743137
$xscores
CA1 CA2 CA3
1 0.27821077 -0.33344726 0.94985154
2 -0.23088044 0.78905936 0.26050453
3 -1.44525534 -0.22803129 -0.64071476
...
$yscores
CA1 CA2 CA3
1 0.55710619 -0.850905958 0.300282830
2 1.41482268 1.237829442 -0.446763906
3 -1.40453596 0.227361557 -0.964307876
...
$chisq
[1] 8.9531203 2.0752824 0.5032912
$p
[1] 0.7069293 0.9126462 0.7775202

Here, $cor is the set of canonical correlation; $xcoef & $ycoef are the canonical coefficients; $xscores & $yscores are the canonical scores, in terms of the original species; and $chisq & $p are Χ² with corresponding p-values. The p-values are properly interpreted as the probability that the ith and all subsequent correlations are zero.

A few years ago I released a C program that does more or less the same thing; however there are a few differences.

1) My C program globally optimizes the λ parameter. I will add this to the present function promptly.

2) My C program first transforms the data into a phylogeny free space, and then computes the canonical correlations. This means that, although the correlations are the same in both methods, the scores are no longer in terms of species and will be different than in this function.

It is also possible to conduct a non-phylogenetic canonical correlation analysis with this function by giving phyl.cca() a star phylogeny with unit length. To do this with our previous example, we can just do the following:

> result<-phyl.cca(starTree(species=tree$tip.label, branch.lengths=rep(1,length(tree$tip))),X,Y)

This should produce the same result as cc() in the "CCA" package. (Note, though, that for cc() the rows of X & Y must be in the same order - they do not for phyl.cca(), but must contain row names.)

This function will be added to the next update of "phytools".