# [R] How to test the difference between paired correlations?

Ebert,Timothy Aaron tebert @end|ng |rom u||@edu
Thu Mar 23 01:23:17 CET 2023

```If you are open to other options:
The null hypothesis is that there is no difference.
If I have two equations y=x and y=z and there is no difference then it would not matter if an observation was from x or z.
Randomize the x and z observations. For each randomization calculate a correlation for y=x and for y=z.
At each iteration calculate the absolute value of the difference in the correlations.
Generate a frequency distribution from 100,000+ randomizations.
Find the observed difference in the frequency from random distributions.
What proportion of observations are as large or larger than the observed. This is your p-value.

Tim

-----Original Message-----
From: R-help <r-help-bounces using r-project.org> On Behalf Of Luigi Marongiu
Sent: Wednesday, March 22, 2023 5:12 PM
To: r-help <r-help using r-project.org>
Subject: [R] How to test the difference between paired correlations?

[External Email]

Hello,
I have three numerical variables and I would like to test if their correlation is significantly different.
I have seen that there is a package that "Test the difference between two (paired or unpaired) correlations".
However, there is the need to convert the correlations to "z scores using the Fisher r-z transform". I have seen that there is another package that does that [https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Fsearch.r-project.org%2FCRAN%2Frefmans%2FDescTools%2Fhtml%2FFisherZ.html&data=05%7C01%7Ctebert%40ufl.edu%7C35f2e7d6d9e844553c6408db2b1a337f%7C0d4da0f84a314d76ace60a62331e1b84%7C0%7C0%7C638151163767327230%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=gI3vIHV5UnFbPSmeMyuCVvg9hpFCdF33qNgAXmOQOXU%3D&reserved=0].
Yet, I do not understand how to process the data. Shall I pass the raw data or the correlations directly?

I have made the following working example:
```
# define data
v1 <- c(62.480,  59.492,  74.060,  88.519,  91.417,  53.907,  64.202,  62.426,
54.406,  88.117)
v2 <- c(56.814, 42.005, 56.074, 65.990, 81.572, 53.855, 50.335, 63.537, 41.713,
78.265)
v3 <- c(54.170,  64.224,  57.569,  85.089, 104.056,  48.713,  61.239,  60.290,
67.308,  71.179)
# visual exploration
par(mfrow=c(2, 1))
plot(v2~v1, ylim=c(min(c(v1,v2,v3)), max(c(v1,v2,v3))),
xlim=c(min(c(v1,v2,v3)), max(c(v1,v2,v3))),
main="V1 vs V2")
abline(lm(v2~v1))
plot(v3~v1, ylim=c(min(c(v1,v2,v3)), max(c(v1,v2,v3))),
xlim=c(min(c(v1,v2,v3)), max(c(v1,v2,v3))),
main="V1 vs V3")
abline(lm(v3~v1))
## test differences in correlation
# convert raw data into z-scores
library(psych)
library(DescTools)
FisherZ(v1) # I cannot convert the raw data into z scores (same for the other variables):
>  NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN Warning message:
> In log((1 + rho)/(1 - rho)) : NaNs produced
# convert correlations into z scores
# (the correlation score of 0.79 has been converted into 1.08; is this correct?)
FisherZ(lm(v2~v1)\$coefficients)
>      v1
> 1.081667
lm(v2~v1)\$coefficients
>       v1
> 0.7938164
# apply test
v1_v2 = FisherZ(lm(v2~v1)\$coefficients)
v1_v3 = FisherZ(lm(v3~v1)\$coefficients)
paired.r(v1_v2, v1_v3, yz=NULL, length(v1), n2=NULL, twotailed=TRUE)
> Call: paired.r(xy = v1_v2, xz = v1_v3, yz = NULL, n = length(v1), n2 = NULL,
>    twotailed = TRUE)
>  "test of difference between two independent correlations"
> z = NaN  With probability =  NaNWarning messages:
> 1: In log((1 + xy)/(1 - xy)) : NaNs produced
> 2: In log((1 + xz)/(1 - xz)) : NaNs produced
```

What is the right way to run this test?
Shall I apply also yz?
Thank you

--
Best regards,
Luigi

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