Kendall’s Tau Basic Concepts

Basic Concepts

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

  • concordant if (xi > xj and yi > yj) or (xi < xj and yi < yj)
  • discordant if (xi > xj and yi < yj) or (xi < xj and yi > yj)
  • neither if xi = xj or yi = yj (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

Kendall's tau

Observations

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

image7219Alternatively

image7220

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 xj < xi. 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:
image3664

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

image7221

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 τx,y

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

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

Here, τzx,y and τz(x,y) are defined exactly as for rzx,y and rz(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

6 thoughts on “Kendall’s Tau Basic Concepts”

  1. dear charles ,

    thanks for your site may i ask you send me a formula for ties in ranking
    other than arithmetic mean , if possible
    best regards

    Reply
  2. Hi Charles,
    I think the p-value calculation shown in your first example (Fig. 1, with no ties) should be =NORMSDIST(I11) not F11?
    Thanks, KEH

    Reply
    • Hi KEH,
      Thanks for catching this typo. I have now changed the formula on the webpage as you have suggested.
      Charles

      Reply

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