**KENDALL'S TAU**
Kendall's Tau (Kendall's Rank Correlation Coefficient) is a measure of nonlinear dependence between two random variables. If random variables
and
have joint distribution
_{ }
and random vectors
_{ }
and
_{ }
are independent realizations from that distribution, then Kendall's tau of
and
equals

If
and
have continuous marginal distributions then
_{}
has the same units as Pearson's correlation. Just like Pearson's correlation it covers the whole range of [-1,1], but now -1 corresponds to a perfect negative relationship ( is *any* decreasing deterministic function of ) and 1 corresponds to a perfect positive relationship ( is *any* increasing deterministic function of ). When
or
has a discrete mass, interval [-1,1] is not covered fully. For example, if variable
takes a given value with positive probability *p*, then with probability of at least *p*^{2} there is a tie:
_{}
And so
_{}
falls into interval [-1 + *p*^{2}, 1 - *p*^{2}]
no matter what the bivariate relationship is. There are several proposals on how to adjust for ties, the most obvious one being to divide formula (1) by

Still, no single generalization has been widely accepted.

Note that definition (1) depends on ranks only. We only care if
_{}
is bigger than
_{}
the actual values being irrelevant. So Kendall's tau is invariant to any monotonically increasing nonlinear transformations of
and
If we raise
to the third power Kendall's tau will stay the same. This is very important. Kendall's tau is naturally built to capture the strength of highly nonlinear relationships, where traditional linear association measures fail. The following graph illustrates the fact.

Kendall's tau has direct relation to the copula function
_{}
generated by random variables
and

The copula function does not depend on marginal distributions and captures what happens to
and
if they are transformed into random variables uniformly distributed on [0,1]. The formula above signals once again that Kendall's tau does not depend on marginal distributions of
and
and is invariant to any monotonically increasing transformations of
and

Several sample estimators have been developed.
Let
_{ }
denote observations from the joint distribution of
and
Pairs
_{}
and
_{}
are called

*concordant* if their ranks agree:
_{}
or
_{}

*discordant* if their ranks disagree:
_{}
or
_{}

*tied* if
_{}
or
_{}

Let

_{} number of concordant pairs,

_{} number of discordant pairs,

_{} number of unique values in
_{ }

_{} number of unique values in
_{ }

_{} number of tied values in the *i*-th group of ties in
_{ }

_{} number of tied values in the *j*-th group of ties in
_{ }

The estimators of
_{}
are defined as

Estimators
_{}
and
_{}
make adjustments for ties and are suitable for all distributions.
Estimator
_{}
does not adjust for ties and is suitable only for continuous distributions measured with high precision.
Each of the estimators
_{}
is *nonparametric* in the sense that it makes little or no assumptions about the joint distribution of
and
In particular, no functional form is postulated for the conditional expectation of
given
and the conditional expectation of
given
For each of the estimators
_{}
tests have been developed, telling us if
_{}
equals 0. A typical test is based on a transformation of the estimator which is *asymptotically* normal (its distribution converges to a normal distribution when the sample size grows big).

**KENDALL'S TAU REFERENCES**

Nelsen, R. B. (2006). An Introduction to Copulas (2nd ed). New York: Springer.

Salvadori, G., De Michele, C., Kottegoda, N. T., & Rosso, R. (2007). Extremes in Nature: An Approach Using Copulas. Springer.

Gibbons, J. D., & Chakraborti, S. (2003). Nonparametric Statistical Inference (4th ed). New York: Marcel Dekker.

Nešlehová, J. (2007). On Rank Correlation Measures for Non-continuous Random Variables. Journal of Multivariate Analysis, Vol. 98, Issue 3, pp. 544-567.

Kendall, M. (1938). A New Measure of Rank Correlation. Biometrika, Vol. 30 (1–2), pp. 81–89.

**BACK TO THE ****STATISTICAL ANALYSES DIRECTORY**

**IMPORTANT LINKS ON THIS SITE**