Kendall tau rank correlation coefficient

The Kendall tau rank correlation coefficient (or simply the Kendall tau coefficient, Kendall's τ or tau test(s)) is a non-parametric statistic used to measure the degree of correspondence between two rankings and assessing the significance of this correspondence. In other words, it measures the strength of association of the cross tabulations.

It was developed by Maurice Kendall in 1938.

Definition

The Kendall tau coefficient (τ) has the following properties:

* If the agreement between the two rankings is perfect (i.e., the two rankings are the same) the coefficient has value 1.
* If the disagreement between the two rankings is perfect (i.e., one ranking is the reverse of the other) the coefficient has value −1.
* For all other arrangements the value lies between −1 and 1, and increasing values imply increasing agreement between the rankings. If the rankings are completely independent, the coefficient has value 0 on average.

Kendall tau coefficient is defined

: $au\; =\; frac\{n\_c-n\_d\}\{frac\{1\}\{2\}\{n(n-1)$

where "$n\_c$" is the number of concordant pairs, and "$n\_d$" is the number of discordant pairs in the data set.

The denominator in the definition of $au$ can be interpreted as the total number of pairs of items. So, a high value in the numerator means that most pairs are concordant, indicating that the two rankings are consistent. Note that a tied pair is not regarded as concordant or discordant. If there is a large number of ties, the total number of pairs (in the denominator of the expression of $au$) should be adjusted accordingly.

Tau a, b and c

*"Tau a" — This tests the strength of association of the cross tabulations when both variables are measured at the ordinal level but makes no adjustment for ties.
*"Tau b" — This tests the strength of association of the cross tabulations when both variables are measured at the ordinal level. It makes adjustments for ties and is most suitable for square tables. Values range from −1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association.
*"Tau c" — This tests the strength of association of the cross tabulations when both variables are measured at the ordinal level. It makes adjustments for ties and is most suitable for rectangular tables. Values range from −1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association.

See also

* Correlation
* Kendall tau distance
* Kendall's W
* Spearman's rank correlation coefficient

References

*cite paper | author = Abdi, H. | title = [http://www.utdallas.edu/~herve/Abdi-KendallCorrelation2007-pretty.pdf] (2007) Kendall rank correlation. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.| year = 2007 |
* Kruskal, W.H. (1958) "Ordinal Measures of Association", Journal of the American Statistical Association, 53(284), 814-861.
* Kendall, M. (1948) "Rank Correlation Methods", Charles Griffin & Company Limited
* Kendall, M. (1938) "A New Measure of Rank Correlation", Biometrika, 30, 81-89.

External links

* [http://rsscse.org.uk/ts/bts/noether/text.html Why Kendall tau?]
* [http://www.wessa.net/rwasp_kendall.wasp Online software: computes Kendall's tau rank correlation]