# Re: [R] Testing the equality of correlations

From: Paul Hewson <paul.hewson_at_plymouth.ac.uk>
Date: Wed 27 Sep 2006 - 15:40:09 GMT

Marc,

Off the top my head (i.e. this could all be horribly wrong), I think Anderson gave an asymptotic version for such a test, whereby under the null hypothesis, the difference between Fisher's z for each sample, z1 - z2, is normal with zero mean. If I recall correctly, the 1984 edition gave a test statistic something like:

$\frac{\vert z_{1} - z_{2} \rvert}{ \sqrt{\frac{1}{N_{1} - 3} + \frac{1}{N_{2} - 3}}}$,

as the test. I ***think*** N = n+1. Assuming that part is correct, it can be coded up quite simply (and crudely and hopefully correctly) as:

## estimate Fisher's z for the sample

fishz <- function(x1,x2){
r <- cor(x1,x2)
z <- 0.5 * log( (1+r)/(1-r) )
}

## apply the correlation test

cortest <- function(x1,x2,x3,x4){

numer <- abs(fishz(x1,x2) - fishz(x3,x4) )    denom <- sqrt( 1/( length(x1) - 2) + 1/( length(x2) - 2) )    test.stat <- numer / denom
return(test.stat)
}

A quick demo with some simulated data:

require(MASS)
X1 <- mvrnorm(1000, c(0,0), matrix(c(1,0.7,0.7,1),2,2)) X2 <- mvrnorm(1000, c(0,0), matrix(c(1,0.9,0.9,1),2,2))

cortest(X1[,1], X1[,2], X2[,1], X2[,2])

Is above 1.96 indicating they are different

X2 <- mvrnorm(1000, c(0,0), matrix(c(1,0.7,0.7,1),2,2)) cortest(X1[,1], X1[,2], X2[,1], X2[,2])

Is below 1.96 indicating they are not different. All that needs to be done for these pair is to get the pooled estimate of the populations Fisher's z $\frac{ (N_{1} - 3)z_{1} + (N_{2} - 3)z_{2}}{N_{1} + N_{2} - 6}$ and solve fisher's z to get an estimate of rho.

Just dabbling around with this suggests either I've missed something, or that we need quite a large sample size before the asymptotics are any use. If this is of any use I'll double check on N.

Paul

-=-=-=-=-=-=-=-=-=-=-=-=
Paul Hewson
Lecturer in Statistics
School of Mathematics and Statistics
University of Plymouth
Drake Circus
Plymouth PL4 8AA

tel (01752) 232778 (Campus)
tel (01752) 764437 (Tamar Science Park)
fax (01752) 232780



email: paul.hewson@plymouth.ac.uk
web: http://www.plymouth.ac.uk/staff/phewson -=-=-=-=-=-=-=-=-=-=-=-=

Dear All,

I wonder if there is any implemented statistical test in R to test the equality between many correlations. As an example, let X1, X2, X3 X4 be four random variables. let
Phi(X1,X2) , Phi(X1,X3) and Phi(X1,X4) be the corresponding
correlations.
How to test Phi(X1,X2) = Phi(X1,X3) = P(X1,X4)?

Many thanks in advance,

Bernard

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