5.2 Rank correlation

Where the association is non-linear, replacing the observations by their ranks often turns it into a linear one. A relationship that is merely monotonic — one variable rising whenever the other does, at whatever rate — becomes exactly linear in the ranks, so a correlation coefficient computed on ranks measures something the coefficient computed on the raw values would miss.

Pearson’s significance test cannot be applied to ranks, because ranks are not a sample from a bivariate normal distribution: they are a permutation of \(1,2,\dots ,n\), with fixed mean and fixed variance. The null distribution has to be worked out for ranks specifically, which is what the tables of \(r_s\) and \(\tau \) provide.

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