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

Luigi Marongiu m@rong|u@|u|g| @end|ng |rom gm@||@com
Thu Mar 23 10:30:04 CET 2023

```Thank you, but this now sounds more difficult: what would be the point
in having these ready-made functions if I have to do it manually?
Anyway, How would I implement the last part?

On Thu, Mar 23, 2023 at 1:23 AM Ebert,Timothy Aaron <tebert using ufl.edu> wrote:
>
> 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):
> > [1] 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[2])
> >      v1
> > 1.081667
> lm(v2~v1)\$coefficients[2]
> >       v1
> > 0.7938164
> # apply test
> v1_v2 = FisherZ(lm(v2~v1)\$coefficients[2])
> v1_v3 = FisherZ(lm(v3~v1)\$coefficients[2])
> 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)
> > [1] "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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