6.2 Spearman’s Rho

Spearman’s ranked correlation (Spearman’s rho) is a measure of the strength and direction of a monotonic relationship between two variables that are at least ordinal. Spearman’s correlation is a non-parametric alternative to Pearson when one or more of its conditions are violated. Unlike Pearson, the relationship need not be linear (it only needs to be monotonic), and has no outliers or bivariate normality conditions.

Spearman’s correlation is Pearson’s correlation applied to the ranks of variables (for ordinal variables, their value already is a rank). However, there is also a second definition that gives the same result, at least when there are no ties in the ranks:

\[\rho = 1 - \frac{6 \sum_i d^2_i}{n(n^2 - 1)}\]

where \(d_i\) is the difference in ranks of observation \(i\).