Rank correlation

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In statistics, rank correlation is the study of relationships between different rankings on the same set of items. It deals with the measurement of correspondence between two rankings, and the calculation of the significance of the correspondence.

Correlation coefficients

Suppose we rank a group of eight people by height and by weight:

Person A B C D E F G H
Rank by Height 1 2 3 4 5 6 7 8
Rank by Weight 3 4 1 2 5 7 8 6

We can see that there is some correlation between the two rankings but that the correlation is far from perfect, and we would like some way of objectively measuring the degree of correspondence. In the 1940s Maurice Kendall developed a coefficient, τ, for this purpose that has the following properties:

• If the agreement between the two rankings is perfect, ie. the two rankings are the same, the value of the coefficient is equal to 1.
• If the disagreement beween the two rankings is perfect, ie. one ranking is the reverse of the other, the coefficient is equal to -1.
• For all other arrangements the value lies between -1 and 1, and increasing values imply increasing agreement between the rankings.
• A value of 0 implies that the two rankings are independent.

It is defined by

$\tau = \frac{2P}{\frac{1}{2}n(n-1)} - 1$

where n is the number of items, and P is a quantity derived from the rankings as follows:

In the Weight ranking above, the first entry, 3, has five higher ranks to the right of it; the contribution to P of this entry is 5. Moving to the second entry, 4, we see that there are four higher ranks to the right of it and the contribution to P is 4. Continuing this way, we find that

P = 5 + 4 + 5 + 4 + 3 + 1 + 0 + 0 = 22.

Thus $\tau= \frac{44}{28}-1 = 0.57$. This result indicates that there is strong agreement between the rankings, as expected.

References

• Kendall, M. (1948) Rank Correlation Methods, Charles Griffin & Company Limited

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