# 18.3 - Kendall Tau-b Correlation Coefficient

18.3 - Kendall Tau-b Correlation Coefficient

The Kendall tau-b correlation coefficient, $$\tau_b$$, is a nonparametric measure of association based on the number of concordances and discordances in paired observations.

Suppose two observations $$\left(X_i , Y_i \right)$$ and $$\left(X_j , Y_j \right)$$ are concordant if they are in the same order with respect to each variable. That is, if

1. $$X_i < X_j$$ and $$Y_i < Y_j$$ , or if
2. $$X_i > X_j$$ and $$Y_i > Y_j$$

They are discordant if they are in the reverse ordering for X and Y, or the values are arranged in opposite directions. That is, if

1. $$X_i < X_j$$ and $$Y_i > Y_j$$ , or if
2. $$X_i > X_j$$ and $$Y_i < Y_j$$

The two observations are tied if $$X_i = X_j$$ and/or $$Y_i = Y_j$$ .

The total number of pairs that can be constructed for a sample size of n is

$$N=\binom{n}{2}=\dfrac{1}{2}n(n-1)$$

N can be decomposed into these five quantities:

$$N = P + Q + X_0 + Y_0 + (XY)_0$$

where P is the number of concordant pairs, Q is the number of discordant pairs, $$X_0$$ is the number of pairs tied only on the X variable, $$Y_0$$ is the number of pairs tied only on the Y variable, and $$\left(XY\right)_0$$ is the number of pairs tied on both X and Y.

The Kendall tau-b for measuring order association between variables X and Y is given by the following formula:

$$t_b=\dfrac{P-Q}{\sqrt{(P+Q+X_0)(P+Q+Y_0)}}$$

This value becomes scaled and ranges between -1 and +1. Unlike Spearman it does estimate a population variance as:

$$t_b \text{ is the sample estimate of } t_b = Pr[\text{concordance}] - Pr[\text{discordance}]$$

The Kendall tau-b has properties similar to the properties of the Spearman $$r_s$$. Because the sample estimate, $$t_b$$ , does estimate a population parameter, $$t_b$$ , many statisticians prefer the Kendall tau-b to the Spearman rank correlation coefficient.

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