A new correlation coefficient (Chatterjee)

[vincentarelbundock]

Have not read yet, but this looks fun: https://arxiv.org/pdf/1909.10140.pdf

[bwiernik]

Neat! Looks straightforward!

[mattansb]

From here

ksaai <- function(X, Y, ties = TRUE){
  n <- length(X)
  r <- rank(Y[order(X)], ties.method = "random")
  set.seed(42)
  if(ties){
    l <- rank(Y[order(X)], ties.method = "max")
    return( 1 - n*sum( abs(r[-1] - r[-n]) ) / (2*sum(l*(n - l))) )
  } else {
    return( 1 - 3 * sum( abs(r[-1] - r[-n]) ) / (n^2 - 1) )    
  }
}

I don't like that it's not symmetrical - shouldn't correlation coefficients be symmetrical?

x <- rnorm(100, sd = 4)
y <- sin(x) + rnorm(100, sd = 0.2)

plot(x, y)

ksaai(x, y)
#> [1] 0.6306631
ksaai(y, x)
#> [1] -0.1710171

Also the maximal value isn't 1 and seems to depend on the sample size?

z10 <- runif(10)
z100 <- runif(100)
z1000 <- runif(1000)

ksaai(z10, z10)
#> [1] 0.7272727
ksaai(z100, z100)
#> [1] 0.970297
ksaai(z1000, z1000)
#> [1] 0.997003

^{Created on 2024-04-14 with reprex v2.1.0}

[vincentarelbundock]

Your note about sample size is presumably what he means by "converges to a limit" in point 4 of the screenshot in my original post. Since there's theory to provide confidence intervals, maybe that's not a big deal? Maybe even good?

And on symmetry:

(1) Unlike most coefficients, ξn is not symmetric in X and Y . But that is intentional. We would like to keep it that way because we may want to understand if Y is a function X, and not just if one of the variables is a function of the other. If we want to understand whether X is a function of Y , we should use ξn(Y, X) instead of ξn(X, Y ). A symmetric measure of dependence, if required, can be easily obtained by taking the maximum of ξn(X, Y ) and ξn(Y, X).

[mattansb]

Cool (👍

1. I don't see any mention of a confidence interval - should we just use Fisher's Z?
2. In theory, xi is non-negative, but it sometimes is - should we return 0 in such cases?

[vincentarelbundock]

I don’t see any mention of a confidence interval

Sorry, I misread about the CI. The XICOR package does provide a SD, but it feels wrong to just compute a symmetric interval using that.

should we just use Fisher’s Z?

I’ve only really skimmed the paper, and don’t truly understand it. Until I grok this better (realistically: never), I would be reticent to report a quantity not explicitly endorsed by the author.

In theory, xi is non-negative, but it sometimes is - should we return 0 in such cases?

“In the limit” != “In theory”. I’d say report the actual output of the equation, rather than an ad hoc hack.

I ran into some errors with your ksaai() function with large N. However, the paper authors have published a XICOR package on CRAN. It seems fast and is published under Apache License which, I believe, is compatible with GPL3.

library(XICOR)
N <- 100
x <- rnorm(N, sd = 4)
y <- sin(x) + rnorm(N, sd = 0.2)
xicor(y, x, pvalue = TRUE)

    $xi
    [1] 0.03840384

    $sd
    [1] 0.06325978

    $pval
    [1] 0.2718984

[mattansb]

In theory == I mean the estimand is non-positive.

I'll run some simulations to see if the Fisher Z CIs work well enough.

[bwiernik]

The author did a small simulation in section 4.2 and concluded that sqrt(n) * xi is asymptomatically normal (when n = 1000). That's not unexpected, but also not very helpful for more realistic sample sizes.

The author's XICOR package defaults to using the specified mean and SD values with a normal distribution. They also offer a permutation test.

I'd be okay with reporting normal-theory intervals and p values to start given that's what the author does, but we should ideally do some simulations to confirm good performance of the intervals at smaller n (or use a z transform if that works nicely).

I don't compare the code above from the blog post and the XICOR package to be sure they aligned, but we should follow XICOR https://github.com/cran/XICOR/blob/master/R/xicor.R

[TarandeepKang]

Hi All,

Just to mention that this preprint has now been published:

Chatterjee, S. (2021). A New Coefficient of Correlation. Journal of the American Statistical Association, 116(536), 2009–2022. https://doi.org/10.1080/01621459.2020.1758115