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Why is it correct not to measure the correlation between two distance matrices by calculating the Pearson's correlation coefficient between all distance pairs but to use the Mantel test instead?

Edit: Wikipedia says that because "distances are not independent of each other – since changing the "position" of one object would change n-1 of these distances (the distance from that object to each of the others) – we can not assess the relationship between the two matrices by simply evaluating the correlation coefficient between the two sets of distances and testing its statistical significance. The Mantel test deals with this problem."

This argument is not clear to me. Why is dependence between the distances a problem?

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    $\begingroup$ Can you expound this interesting idea in your question? Or at least leave references? $\endgroup$
    – ttnphns
    Nov 20, 2012 at 14:10

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If you have $n$ objects you will have $\frac{n(n-1)}2$ distances. So, unless $n<4$, you always have more distances than objects. This artificially inflates the number of independent samples when using Pearson's correlation.

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