16.7 Spearman Rank Correlation Coefficient

Gordon E. Sarty

16. Non-parametric Tests

16.7 Spearman Rank Correlation Coefficient

This is a rank alternative to the Pearson correlation coefficient that may be used when the assumption of normality is not met for hypothesis testing. It is defined by

$r_{s} = 1 - \frac{6 \sum_{i=1}^{n} d_{i}^{2}}{n(n^{2}-1)}$

where

$\begin{eqnarray*} n & = & \mbox{sample size}\\ \\ d_{i} & = & \mbox{difference in ranks for data point } i \\ & = & r_{x_{i}} - r_{y_{i}} \end{eqnarray*}$

where $r_{x_{i}}$ = $x$ rank of point $i$ and $r_{y_{i}}$ = $y$ rank of point $ii$ .

To test $H_{0}: r_{s} = 0$ versus $H_{1}: r_{s} \neq 0$ use $r_{s}$ itself as the test statistic and $r_{s,\mbox{crit}}$ from the Rank Correlation Coefficient Critical Values Table as the critical statistic. (Note that the Rank Correlation Coefficient Critical Values Table requires $n < 30$ .) Reject $H_{0}$ if $r_{s} > r_{s,\mbox{crit}}$ .

Example 16.9 : Determine if the Spearman correlation between two textbook ratings, data given below, is significant.

Book	rating 1 ( $x$ )	rating 2 ( $y$ )	rank $x$	rank $y$	$d$	$d^{2}$
A	4	4	2	1	1	1
B	10	6	5	2	3	9
C	18	20	7	8	-1	1
D	20	14	8	6	2	4
E	12	16	6	7	-1	1
F	2	8	1	4	-3	9
G	5	11	3	5	-2	4
H	9	7	4	3	1	1
$n=8$						$\sum d^{2} = 30$