Spearman's Rank Correlation Coefficient

A rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them.

We have two random variables $X$ and $Y$: $X=\{x_i, x_2, x_3, ..., x_n\}$ $Y=\{y_i, y_2, y_3, ..., y_n\}$

if $Rank_X$ and $Rank_Y$ denote the respective ranks of each data point, then the Spearman's rank correlation coefficient, $r_s$, is the Pearson correlation coefficient of $Rank_X$ and $Rank_Y$.

Example

$X=\{0.2, 1.3, 0.2, 1.1, 1.4, 1.5\}$
$Y=\{1.9, 2.2, 3.1, 1.2, 2.2, 2.2\}$

$$ Rank_X \quad \begin{bmatrix} X: & 0.2 & 1.3 & 0.2 & 1.1 & 1.4 & 1.5 \\ Rank: & 1 & 3 & 1 & 2 & 4 & 5 \end{bmatrix} \quad $$

so, $Rank_X = \{1, 3, 1, 2, 4, 5\}$

similarly, $Rank_Y=\{2,3,4,1,3,3\}$

$r_s$ equals the Pearson correlation coefficient of $Rank_X$ and $Rank_Y$, meaning that $r=0.158114$

Special case : $X$ and $Y$ don't contain duplicates

$$r_s=1-\frac{6\sum d_i^2}{n(n^2-1)}$$

Where, $d_i$ is the difference between the respective values of $Rank_X$ and $Rank_Y$.

Example

Special case : \(X\) and \(Y\) don't contain duplicates