# Are two Pearson correlation coefficients different?

I am aware of this question here, which mine was listed as a duplicate of, but it does not fully answer my question. It did however help me progress a little further so thanks, I couldn't find it before. The online calculator on the above answer also disagrees (though marginally) with the vassarstats.net calculator so I think that further backs up my reasoning for not using a black box.

So I'll re-explain my problem and hope that this gets opened to answers:

I have two Pearson correlation coefficients which I would like to compare. Each comes from 2 sets of 40 genetically identical lines and are correlations between male and female trait values.

I have managed to do $z$-transformations on my correlation coefficients in R using the function atanh() and replicated that with a home-made function RtoZ <- function (r) 0.5*log((1+r)/(1-r)).

The problem is what to do next: how do I actually test, in R, whether the two correlations are different?

• @AndyW this question has been rewritten, is it possible to open it now? Jul 15, 2013 at 8:45
• @gung see above Jul 15, 2013 at 8:45
• Why use 0.5*log((1+r)/(1-r) when you can use atanh(r)? Jul 16, 2013 at 8:30
• @NickCox I used both, the use of 0.5*log((1+r)/(1-r)) was to help me (and others reading the question) understand the fisher's z transformation Jul 16, 2013 at 9:11

Just in case that someone (else) has to perform a comparison of correlation coefficients on multiple pairs of variables, here's a function based on rg255's helpful reply to copy:

cor.diff.test = function(x1, x2, y1, y2, method="pearson") {
cor1 = cor.test(x1, x2, method=method)
cor2 = cor.test(y1, y2, method=method)

r1 = cor1$estimate r2 = cor2$estimate
n1 = sum(complete.cases(x1, x2))
n2 = sum(complete.cases(y1, y2))
fisher = ((0.5*log((1+r1)/(1-r1)))-(0.5*log((1+r2)/(1-r2))))/((1/(n1-3))+(1/(n2-3)))^0.5

p.value = (2*(1-pnorm(abs(fisher))))

result= list(
"cor1" = list(
"estimate" = as.numeric(cor1$estimate), "p.value" = cor1$p.value,
"n" = n1
),
"cor2" = list(
"estimate" = as.numeric(cor2$estimate), "p.value" = cor2$p.value,
"n" = n2
),
"p.value.twosided" = as.numeric(p.value),
"p.value.onesided" = as.numeric(p.value) / 2
)
cat(paste(sep="",
"cor1: r=", format(result$cor1$estimate, digits=3), ", p=", format(result$cor1$p.value, digits=3), ", n=", result$cor1$n, "\n",
"cor2: r=", format(result$cor2$estimate, digits=3), ", p=", format(result$cor2$p.value, digits=3), ", n=", result$cor2$n, "\n",
"diffence: p(one-sided)=", format(result$p.value.onesided, digits=3), ", p(two-sided)=", format(result$p.value.twosided, digits=3), "\n"
))
return(result);
}

• Can I apply your function for more than 2 comparisons? thanks! Nov 14, 2016 at 0:37
• I only rewrote the function of rg255 to make it easier to use. For statistical questions, better ask @rg255 :) Nov 14, 2016 at 11:04
• my understanding is that this is only for independent samples. for dependent samples, would have to consider the correlation between the shared data between groups as well. Jun 4, 2020 at 16:01

Once the Fisher's z transformations are done it is just a case of obtaining p-values

# Correlations
cor.test (df1$a, df1$b, method = "p")
cor.test (df2$a, df2$b, method = "p")

# function to do fisher transformations
fisher.z<- function (r1,r2,n1,n2) ((0.5*log((1+r1)/(1-r1)))-(0.5*log((1+r2)/(1-r2))))/((1/(n1-3))+(1/(n2-3)))^0.5

# or this (either version will suffice)
fisher.z<- function (r1,r2,n1,n2) (atanh(r1) - atanh(r2)) / ((1/(n1-3))+(1/(n2-3)))^0.5

#input n and r from correlations manually (two tailed test)
2*(1-pnorm(abs(fisher.z(r1= ,r2= ,n1= ,n2= ))))


See the final four slides of this presentation and the pnorm() function in r.

• Mind the assumptions, though. I'd try to look at the bootstrap distribution as well.
– Erik
Jul 16, 2013 at 9:33
• @erik thanks, could you possibly direct me towards a guide on bootstrapping to test if two correlations are different - 3 hours of googling is proving unsuccessful for me Jul 16, 2013 at 12:58
• Unfortunately, I personally do not know any good online guide. I use "An Introduction to the Bootstrap" by Efron & Tibshirani as my main manual for the bootstrap, but if you can't get it from the local library it has a rather steep price.
– Erik
Jul 16, 2013 at 14:04
• @erik I've ordered it from my department library :) Jul 16, 2013 at 14:20

In case people are still looking for an easy way to compare two $r$ Pearson correlations. There is a function called paired.r in the package psych for R exactly for that.

Usage:

paired.r(xy, xz, yz=NULL, n, n2=NULL,twotailed=TRUE)


Or as a simple example: For r1 and r2 and a sample size of n just do:

paired.r(r1,r2,n=n)