Kendall Rank Correlation

Notations and Definition

Let S be a set of N objects,

When the order of the elements of the set is taken into account, we obtain an ordered set that can also be represented by the rank order given to the objects of the set. For example, with the following set of N = 4 objects,

the ordered set O1 = [a, c, b, d] gives the ranks R1 = [1, 3, 2, 4]. An ordered set on N objects can be decomposed into ½N(N–1) ordered pairs. For example, O1 is composed of the following six ordered pairs:

In order to compare two ordered sets (on the same set of objects), the approach of Kendall is to count the number of different pairs between two ordered sets. This number gives a distance between these sets called the symmetric difference distance (the symmetric difference is a set operation that associates to two sets the set of elements that belongs to only one set).

The symmetric difference distance between two sets of ordered pairs P1 and P2 is denoted dΔ(P1, P2).

The Kendall coefficient of correlation is obtained by normalizing the symmetric difference such that it will take values between −1 and +1, with −1 corresponding to the largest possible distance (obtained when one order is the exact reverse of the other order) and +1 corresponding to the smallest possible distance (equal to 0, obtained when both orders are identical). Taking into account that the maximum number of pairs that can differ between two sets with 1/2N(N–1) elements is equal to N(N–1), this gives the following formula for the Kendall rank correlation coefficient:

How should the Kendall coefficient be interpreted? Because τ is based upon counting the number of different pairs between two ordered sets, its interpretation can be framed in a probabilistic context. Specifically, for a pair of objects taken at random, τ can be interpreted as the difference between the probability of these objects being in the same order [denoted P(same)] and the probability of these objects being in a different order [denoted P(different)]. Formally, we have

An Example

Suppose that two experts order four wines called {a, b, c, d}. The first expert gives the following order: O1 = [a, c, b, d], which corresponds to the following ranks: R1= [1, 3, 2, 4]; and the second expert orders the wines as O2 = [a, c, d, b], which corresponds to the following ranks: R2 = [1, 4, 2, 3]. The order given by the first expert is composed of the following six ordered

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