Measuring Order: The Mathematics of Concordance and Discordance
In short: How do you mathematically prove that two different rankings are actually telling the same story? Instead of looking at raw values, statisticians use Kendall's tau to see if items maintain their relative positions. It is a non-parametric way to detect hidden patterns without assuming your data follows a perfect bell curve.
Developed by Maurice Kendall in 1938—though similar concepts date back to Gustav Fechner in 1897—the Kendall rank correlation coefficient (tau) measures the ordinal association between two quantities. Unlike methods that focus on the distance between numbers, tau focuses on their relative order. If two variables are 'concordant,' it means they move in the same direction; if they are 'discordant,' one increases while the other decreases.
The coefficient is calculated by comparing all possible pairs of observations within a dataset. In a set of unique values with no ties, tau is the difference between concordant and discordant pairs divided by the total number of possible pairs. This results in a score ranging from -1 to 1. A perfect agreement yields a 1, while a perfectly reversed ranking yields a -1. If two variables are completely independent and not constant, the expected value for tau is zero.
The nuance lies in how we handle 'ties'—instances where values are identical. Kendall originally proposed Tau-a, which does not adjust for ties; under Tau-a, even a perfect equality of variables cannot reach a score of 1. To solve this, he later introduced Tau-b (originally called Tau-w) in 1945. Tau-b adjusts the denominator to account for tied pairs, allowing it to reach a full +1 or -1. Because it is non-parametric, tau is prized for its ability to establish statistical dependence without making rigid assumptions about the underlying distribution of the data.
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