Spearman's Rho

Definition

Let $(X_1,X_1),(X_2,X_2),\dots ,(X_n,X_n)$ be a Random Sample from a bivariate distribution with cdf $F(x,y)$

Instead of a sample Pearson Correlation Coefficient, we define another statistic using the ranks of samples $r_S=\frac{{\sum }_{i=1}^n(R(X_i)-\frac{n+1}{2})(R(Y_i)-\frac{n+1}{2})}{n(n^2-1)/12}$ which is called Spearman’s $\rho$

$r_s$ is bounded by $\pm 1$. If $X,Y$ are independent, then $r_S=0$

Facts

Let $(X_1,X_1),(X_2,X_2),\dots ,(X_n,X_n)$ be a Random Sample from a bivariate distribution with cdf $F(x,y)$, and $r_S=\frac{{\sum }_{i=1}^n(R(X_i)-\frac{n+1}{2})(R(Y_i)-\frac{n+1}{2})}{n(n^2-1)/12}$, then under $H_0:X,Y$ are independent

• $r_S$ is distribution free with a symmetric pdf
• $E_{H_0}(r_S)=0$
• ${\mathrm{Var}}_{H_0}(r_S)=\frac{1}{n-1}$
• $\frac{r_S}{\sqrt{{\mathrm{Var}}_{H_0}(r_S)}}\stackrel{D}{\to }N(0,1)$