Definition:Rank Correlation Coefficient

Definition

A rank correlation coefficient is a statistical coefficient which measures the degree of agreement of a set of subjective rankings.

Spearman's Rank Correlation Coefficient

Let $X$ and $Y$ be two rankings assigned to the same set of entities.

The Spearman's rank correlation coefficient is the Pearson correlation coefficient between $X$ and $Y$.

Kendall's Rank Correlation Coefficient

Kendall's rank correlation coefficient is a test for consistency of $2$ sets of rankings $\sequence a_n$ and $\sequence b_n$ on a set $S$ of $n$ objects.

The set $R$ of ordered pairs $\tuple {a_i, b_i}$ is assembled:

$R = \set {\tuple {a_i, b_i}: i \in \set {1, 2, \ldots, n} }$

and ordered according to $\sequence a$.

The number $Q$ of elements of $S$ out of ranking order from $\sequence b$ is counted.

Kendall's rank correlation coefficient is then formed:

$K = 1 - \dfrac {4 Q} {n \paren {n + 1} }$

which takes values between $-1$ (complete disagreement) and $+1$ (complete agreement).

Complete disagreement happens when $\sequence a_n$ is in reverse order to $\sequence b_n$.

Kendall's Coefficient of Concordance

Kendall's coefficient of concordance is a test for consistency of more than $2$ sets of rankings.

Let $m$ judges independently award ranks $1$ to $n$ to a set of $n$ competitors.

Let $s_i$ be the sum of the rankings awarded to competitor $i$.

The mean $M$ of the values of $s_i$ is $\dfrac 1 2 m \paren {n + 1}$.

The sum of the squares of the deviations from $M$ is given by:

$S = \ds \sum_{i \mathop = 1}^n \paren {s_i - M}^2$

and Kendall's coefficient of concordance is given by:

$W = \dfrac {12 S} {m^2 n \paren {n^2 - 1} }$

Also see

• Results about rank correlation coefficients can be found here.

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): rank correlation coefficient