The CORR Procedure

Spearman Rank-Order Correlation

Spearman rank-order correlation is a nonparametric measure of association based on the ranks of the data values. The formula is

\[  \theta =\frac{\sum _ i (\, (R_ i-\bar{R})(S_ i-\bar{S})\, )}{\sqrt {\sum _ i (R_ i-\bar{R})^2 \,  \sum (S_ i-\bar{S})^2}}  \]

where $R_ i$ is the rank of $x_ i$, $S_ i$ is the rank of $y_ i$, $\bar{R}$ is the mean of the $R_ i$ values, and $\bar{S}$ is the mean of the $S_ i$ values.

PROC CORR computes the Spearman correlation by ranking the data and using the ranks in the Pearson product-moment correlation formula. In case of ties, the averaged ranks are used.

Probability Values

Probability values for the Spearman correlation are computed by treating

\[  t \,  = \,  {(n-2)}^{1/2} \,  {\left(\frac{r^{2}}{1-r^{2}}\right)}^{1/2}  \]

as coming from a t distribution with $(n-2)$ degrees of freedom, where $r$ is the sample Spearman correlation.