Rank correlation

In statistics, a rank correlation is the relationship between different rankings of the same set of items. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess its significance.

Correlation coefficients

Some of the more popular rank correlation statistics include

1. Spearman's ρ
2. Kendall's τ
3. Goodman and Kruskal's γ

An increasing rank correlation coefficient implies increasing agreement between rankings. The coefficient is inside the interval [−1, 1] and assumes the value:

• −1 if the disagreement between the two rankings is perfect; one ranking is the reverse of the other.
• 0 if the rankings are completely independent.
• 1 if the agreement between the two rankings is perfect; the two rankings are the same.

Following Diaconis (1988), a ranking can be seen as a permutation of a set of objects. Thus we can look at observed rankings as data obtained when the sample space is (identified with) a symmetric group. We can then introduce a metric, making the symmetric group into a metric space. Different metrics will correspond to different rank correlations.

General Correlation Coefficient

Kendall Kendall (1944) showed that can be defined a general correlation coefficient, of which his tau and the Spearman rho are particular cases.

Suppose we have a set of $n$ objects, which are being considered in relation to two properties, represented by $x$ and $y$, forming the sets of values $\{x_i\}_{i\le n}$ and $\{y_i\}_{i\le n}$. To any pair of individuals, say the $i$-th and the $j$-th we assign a $x$-score, denoted by $a_{ij}$, and a $y$-score, denoted by $b_{ij}$. The only requirement made to this functions is anti-symmetry, so $a_{ij}=-a_{ji}$ and $b_{ij}=-b_{ji}$. Then the generalised correlation coefficient $\Gamma$ is defined by

$\Gamma = \frac{\sum_{i,j = 1}^n a_{ij}b_{ij}}{\sqrt{\sum_{i,j = 1}^n a_{ij}^2 \sum_{i,j = 1}^n b_{ij}^2}}$

Kendall's $\tau$ as a particular case

If $r_i$ is the rank of the $i$-member according to the $x$-quality, we can define

$a_{ij} = \sgn(r_j-r_i)$

and similarly for $b$. The sum $\sum a_{ij}b_{ij}$ is twice the amount of concordant pairs minus the discordant pairs (see Kendall tau rank correlation coefficient). The sum $\sum a_{ij}^2$ is just the number of terms $a_{ij}$, equal to $n(n-1)$, and so for $\sum b_{ij}^2$. It follows that $\Gamma$ is equal to the Kendall's $\tau$ coefficient.

Spearman's $\rho$ as a particular case

If $r_i$, $s_i$ are the ranks of the $i$-member according to the $x$ and the $y$-quality respectively, we can simply define

$a_{ij} = r_j-r_i$

$b_{ij} = s_j-s_i$

The sums $\sum a_{ij}^2$ and $\sum b_{ij}^2$ are equal, since both $r_i$ and $s_i$ range from $1$ to $n$. Then we have:

$\Gamma = \frac{\sum (r_j-r_i)(s_j-s_i)}{\sum(r_j-r_i)^2}$

now

$\sum_{i,j = 1}^n (r_j-r_i)(s_j-s_i)= \sum_{i=1}^n \sum_{j=1}^n r_is_i + \sum_{i=1}^n \sum_{j=1}^n r_js_j - \sum_{i=1}^n \sum_{j=1}^n (r_is_j+r_js_i)$

$=2n\sum_{i=1}^n r_is_j - 2 \sum_{i=1}^n r_i \sum_{j=1}^n s_j$

$=2n\sum_{i=1}^n r_is_j - \frac12 n^2(n+1)^2$

since $\sum r_i$ and $\sum s_j$ are both equal to the sum of the first $n$ natural numbers, namely $\frac12n(n+1)$.

We also have

$S = \sum_{i=1}^n (r_i-s_i)^2 = 2 \sum r_i^2 - 2\sum r_is_i$

and hence

$\sum(r_j-r_i)(s_j-s_i) = 2n\sum r_i^2 - \frac12n^2(n+1)^2 - nS$

Being $\sum r_i^2$ the sum of squares of the first $n$ naturals, the last equation reduces to

$\sum(r_j-r_i)(s_j-s_i) = \frac16n^2(n^2-1) - nS$

Further

$\sum(r_j-r_i)^2 = 2n\sum r_i^2-2\sum r_ir_j$

$= 2n\sum r_i^2-2(\sum r_i)^2 = \frac16n^2(n^2-1)$

and thus, substituting into the original formula these results we get

$\Gamma = 1-\frac{6 S}{n^3-n}$

which is exactly the Spearman's rank correlation coefficient $\rho$.

References

• Everitt, B. S. (2002), The Cambridge Dictionary of Statistics, Cambridge: Cambridge University Press, ISBN 0-521-81099-X 
• Diaconis, P. (1988), Group Representations in Probability and Statistics, Lecture Notes-Monograph Series, Hayward, CA: Institute of Mathematical Statistics, ISBN 0-940600-14-5 
• Kendall, M. G. (1970), Rank Correlation Methods, London: Griffin, ISBN 0-85264-199-0