Correlation coefficient for three variables in r

Question

For three n-dimensional non-zero-variance variables a, b, and c, n > 2, if r(ab), r(bc), and r(ac) are Pearson's correlation coefficients between a and b, between b and c, and between a and c, respectively, then correlation coefficient r(abc) among a, b, and c is defined as:

r^2(abc) = ( r^2(ab) + r^2(bc) + r^2(ac) ) - ( 2 xr(ab) xr(bc) xr(ac) )

I was able to get the code for a manual way of doing it:

a <- c(4, 6, 2, 7)
b <- c(8, 1, 3, 5)
c <- c(6, 3, 1, 9)

al <- data.frame(a, b, c)
al


ab_cor <- cor(al$a, al$b, method = c("pearson"))
bc_cor <- cor(al$b, al$c, method = c("pearson"))
ac_cor <- cor(al$a, al$c, method = c("pearson"))

abc_cor <- sqrt( ( (ab_cor)^2 + (bc_cor)^2 + (ac_cor)^2 ) - ( 2 * ab_cor * bc_cor * ac_cor) )
abc_cor

But I was wondering if this could be done with less lines of code, for example with a for loop. Addittionaly, how would I write it so that I could do it with more than 3 variables as well, for example, r(abcd) ie r(ab), r(ac), r(ad), r(bc), r(bd), and r(cd).

Answer 1

The cor function already creates a matrix of the correlations. You just need to pick out the relevant ones and then use some vector operations.

cs <- cor(al, method = "pearson")

cs <- cs[upper.tri(cs)]

#sqrt(sum(cs^2)) - 2*prod(cs)
# apparently it's
sqrt(sum(cs^2) - 2*prod(cs))

This generalizes to your larger case as well assuming that you have all the variables you want in your al data.frame.