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I have a weird question I couldn't find the answer on the Internet.

Lets consider a data set for which the points describe a function with some noise around it. Assuming that the function in question is monotonous.

In that case, is the Pearson coefficient necessarily inferior of the Spearman coefficient (in absolute value). If it's not, would you have a counter example ?

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  • $\begingroup$ What do you mean by "inferior"? Do you mean (a) that it is less sensitive (less likely to detect a significant correlation) or that (b) r smaller than ρ? $\endgroup$
    – January
    Nov 22, 2018 at 10:55
  • $\begingroup$ Yes, I mean smaller. In all of the examples I could come across, the Pearson coefficient was smaller in absolute value than the Spearman coefficient unless the function that seems to be correlated isn't monotonous. Do you have a counter example ? $\endgroup$
    – Quego
    Nov 22, 2018 at 15:04
  • $\begingroup$ Monotonic, not monotonous. $\endgroup$
    – Nick Cox
    Nov 22, 2018 at 15:47

1 Answer 1

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As an example, consider the following situation:

  • the true relationship is linear (since the Pearson will only 'see' a linear approximation of any relationship, make it linear so the Pearson isn't penalized by that property)

  • the largest and smallest few observations in both variables are relatively far from the bulk of the points (there's a concentration of points near the middle and a "long tail" either side).

e.g. in R:

  x <- rgamma(100,.5)-rgamma(100,.5)
  y <- x+rnorm(100,0,.2)
  plot(x,y) 
  cor(x,y)
 [1] 0.9845002
  cor(x,y,method="spearman")
 [1] 0.9509631

Plot of generated data, a strongly linear relationship with concentration of points in the middle and long tails

The reason why this works is that the "tail" increases the variation in the direction of the linear relationship relative to the variation around it; taking ranks "pulls in" that tail (ranks don't have a 'tail'), making the 'noise'/variation around the relationship loom relatively larger.

[It's not necessary that the relationship be quite this strong, though.]

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