Spearman correlation coefficient¶

The Spearman rank correlation coefficient measures how strongly two random variables with finite variance are correlated. Spearman’s correlation assesses monotonic relationships between both variables.

Let $(X,Y)$ be two random variables which CDF are denoted by $F_X$ and $F_Y$ . Spearman’s rank correlation coefficient $\rho_S(X,Y)$ is defined by:

$\rho_S(X,Y) = \dfrac{\Cov{F_X(X),F_Y(Y)}}{\sqrt{\Var{F_X(X)}\Var{F_Y(Y)}}}$

where $\Cov{.}$ is the covariance operator and $F_X$ and $F_Y$ are the respective CDF of $X$ and $Y$ .

The Spearman correlation between two variables is equal to the Pearson correlation coefficient between the rank values of the variables:

$\rho_S(X,Y) = \rho_P(F_X(X), F_Y(Y))$

If $C$ is the CDF of the copula of the random vector $(X,Y)$ , then we get:

$\rho_S(X,Y) = \rho_P(F_X(X),F_Y(Y)) = 12 \iint_{[0,1]^2} C(u,v)\,du\,dv - 3$

which shows that the Spearman correlation is linked to the copula only.

Let $((x_1, y_1), \dots, (x_\sampleSize, y_\sampleSize))$ be a sample generated by the bivariate random vector $(X,Y)$ . We denote by $(r_1, s_1), \dots, (r_\sampleSize, s_\sampleSize)$ the rank sample, which means that $r_k$ is the rank of the value $x_k$ within the sample $(x_1, \dots, x_\sampleSize)$ and $s_k$ is the rank of the value $y_k$ within the sample $(y_1, \dots, y_\sampleSize)$ . The estimator $\hat{\rho}_S(X,Y)$ is equal to the estimator $\hat{\rho}_P(X,Y)$ computed on the rank sample $(r_1, s_1), \dots, (r_\sampleSize, s_\sampleSize)$ . It is estimated as follows:

(1)¶ $\hat{\rho}_S(X,Y) = \dfrac{\sum_{k=1}^\sampleSize (r_k- \bar{r})(s_k- \bar{s})} {\sqrt{\sum_{k=1}^\sampleSize(r_k- \bar{r})^2\sum_{k=1}^\sampleSize(s_k- \bar{s})^2}}$

where $\bar{r} = \dfrac{1}{\sampleSize} \sum_{k=1}^\sampleSize r_k$ and $\bar{s} = \dfrac{1}{\sampleSize} \sum_{k=1}^\sampleSize s_k$ are the empirical mean rank of each sample.

We sum up some interesting features of the coefficient:

The Spearman correlation coefficient takes values between -1 and 1.
If $|\rho_S(X,Y)|=1$ then there exists a monotonic function $\varphi$ such that $Y=\varphi(X)$ .
The closer $|\rho_S(X,Y)|$ is to 1, the stronger the indication is that a monotonic relationship exists between $X$ and $Y$ . The sign of the Spearman coefficient indicates if the two variables increase or decrease in the same direction (positive coefficient) or in opposite directions (negative coefficient).
If $X$ and $Y$ are independent, then $\rho_S(X,Y)=0$ .
If $\rho_S(X,Y)=0$ , it does not imply the independence of the variables $X$ and $Y$ . It may only means that the relation between both variables is not monotonic.