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I am looking for some clarification on whether the formulas listed in the title are equivalent or not, and if so why do the formulas look so different?

For example, in a guide I found on Laerd (my rep is too low to post the link) for Spearman correlation, it states the formula for calculating the Spearman correlation with tied ranks is:

enter image description here

Which is the same formula I use when calculating the Pearson product-moment correlation.

Yet a book I am using as a reference, Introduction to Modern Nonparametric Statistics by James J. Higgins, lists the formula for calculating the Spearman correlation with tied ranks as:

enter image description here

Overall, are these formulas interchangeable? Is there any benefit in using one over the other? Or am I totally missing the point, and they are calculating different values?

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  • $\begingroup$ No, they are not equivalent. The Pearson formula makes no allowance for ties. The best link between Spearman and Pearson -- surprisingly often not explained or understood -- is that the (usual) formula for Spearman is just a short-cut formula and that the Spearman correlation is just the Pearson correlation applies to the ranks, so long as ranks are calculated to assign tied ranks their average so that the sum of ranks is preserved. So they are really the same method, one applied to data and the other their ranks. $\endgroup$
    – Nick Cox
    Oct 16, 2016 at 22:05

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I have discovered that the two formulas are equivalent when the Pearson correlation is applied to the ranks adjusted for ties. The Pearson correlation formula in its current state is not equivalent. However, when you simply switch all the X and Y variables out with their rank equivalent, i.e. in the numerator to:

SUMMATION**(Rx - Rx(mean))(Ry - Ry(mean))**

Where Rx are the ranks applied to X and Ry are the Ranks applied to Y the two formulas become equivalent.

Discovered in p. 155, Introduction to Modern Nonparametric Statistics by James J. Higgins

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