Rank correlation
In statistics, a rank correlation is any of several statistics that measure the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the labels "first", "second", "third", etc. to different observations of a particular variable. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them.
Contents
Context
If, for example, one variable is the identity of a college basketball program and another variable is the identity of a college football program, one could test for a relationship between the poll rankings of the two types of program: do colleges with a higherranked basketball program tend to have a higherranked football program? A rank correlation coefficient can measure that relationship, and the measure of significance of the rank correlation coefficient can show whether the measured relationship is small enough to likely be a coincidence.
If there is only one variable, the identity of a college football program, but it is subject to two different poll rankings (say, one by coaches and one by sportswriters), then the similarity of the two different polls' rankings can be measured with a rank correlation coefficient.
Correlation coefficients
Some of the more popular rank correlation statistics include
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 agreement between the two rankings is perfect; the two rankings are the same.
 0 if the rankings are completely independent.
 −1 if the disagreement between the two rankings is perfect; one ranking is the reverse of the other.
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 (1944) showed that his tau and Spearman's rho are particular cases of a general correlation coefficient.
Suppose we have a set of objects, which are being considered in relation to two properties, represented by and , forming the sets of values and . To any pair of individuals, say the th and the th we assign a score, denoted by , and a score, denoted by . The only requirement made to this functions is antisymmetry, so and . Then the generalised correlation coefficient is defined by
Kendall's as a particular case
If is the rank of the member according to the quality, we can define
and similarly for . The sum is twice the amount of concordant pairs minus the discordant pairs (see Kendall tau rank correlation coefficient). The sum is just the number of terms , equal to , and so for . It follows that is equal to the Kendall's coefficient.
Spearman's as a particular case
If , are the ranks of the member according to the and the quality respectively, we can simply define
The sums and are equal, since both and range from to . Then we have:
now
since and are both equal to the sum of the first natural numbers, namely .
We also have
and hence
being the sum of squares of the first naturals equals . Thus, the last equation reduces to
Further
and thus, substituting into the original formula these results we get
which is exactly the Spearman's rank correlation coefficient .
References
 Everitt, B. S. (2002), The Cambridge Dictionary of Statistics, Cambridge: Cambridge University Press, ISBN 052181099X
 Diaconis, P. (1988), Group Representations in Probability and Statistics, Lecture NotesMonograph Series, Hayward, CA: Institute of Mathematical Statistics, ISBN 0940600145
 Kendall, M. G. (1970), Rank Correlation Methods, London: Griffin, ISBN 0852641990

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