Kendall’s Tau Basic Concepts

Basic Concepts

Definition 1: Let x₁, …, x_n be a sample for random variable x and let y₁, …, y_n be a sample for random variable y of the same size n. There are C(n, 2) possible ways of selecting distinct pairs (x_i, y_i) and (x_j, y_j). For any such assignment of pairs, define each pair as concordant, discordant, or neither as follows:

• concordant if (x_i > x_j and y_i > y_j) or (x_i < x_j and y_i < y_j)
• discordant if (x_i > x_j and y_i < y_j) or (x_i < x_j and y_i > y_j)
• neither if x_i = x_j or y_i = y_j (i.e. ties are not counted).

Now let C = the number of concordant pairs and D = the number of discordant pairs. Then define tau as

Observations

If there are no ties, then C(n, 2) = C + D. Thus

Alternatively

To facilitate the calculation of C – D it is best to first put all the x data elements in ascending order. If x and y are perfectly positively correlated, then all the values of y would be in ascending order too, and so if there are no ties then C = C(n, 2) and τ  = 1.

Otherwise, there will be some inversions. For each i, count the number of j > i for which x_j < x_i. This sum is D. If x and y are perfectly negatively correlated, then all the values of y would be in descending order, and so if there are no ties then D = C(n, 2) and τ  = -1.

Properties

Property 1:

Proof: This is a result of the fact that there are C(n, 2) pairings with C(n, 2) = C + D + T where T = the number of tied pairs. Thus

τ is maximum when D = T = 0 and so τ = 1. τ is minimum when C = T = 0 and so τ = -1.

Worksheet Functions

Real Statistics Functions: The following functions are provided in the Real Statistics Resource Pack where the samples for z, x, and y are contained in R, R1, and R2 respectively.

KCORREL(R1, R2) = Kendall’s tau τ_x,y

PART_KCORREL(R, R1, R2) = partial correlation τ_zx,y of variables z and x holding y constant

SEMIPART_KCORREL(R, R1, R2) = semi-partial correlation τ_z(x,y)

Here, τ_zx,y and τ_z(x,y) are defined exactly as for r_zx,y and r_z(x,y) except that Kendall’s tau is used instead of Pearson’s correlation see Multiple Correlation).

References

eGyanKosh (2017) Unit 2: other types of correlation
http://egyankosh.ac.in/bitstream/123456789/20956/1/Unit-2.pdf

Wikipedia (2015) Kendall rank correlation coefficient
https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient