Kendall rank correlation coefficient

Kendall rank correlation coefficient, also known as Kendall's tau coefficient (after the Greek letter τ), is a statistic used to measure the ordinal association between two measured quantities. A tau test is a non-parametric hypothesis test for statistical dependence based on the Kendall rank correlation coefficient.

Definition

The Kendall tau coefficient is defined as the difference between the probability that two randomly selected data points are in the same order for both variables and the probability that they are in a different order, divided by the total number of possible pairings. Mathematically, if we have a set of observations of the form \((x_i, y_i)\) where \(i = 1, 2, ..., n\), and there are no tied ranks, the Kendall tau coefficient (\(\tau\)) can be calculated using the formula:

\[\tau = \frac{2}{n(n-1)} \sum_{i<j} \text{sgn}(x_i - x_j) \cdot \text{sgn}(y_i - y_j)\]

where \(sgn\) is the sign function, which is +1 if the argument is positive, −1 if it is negative, and 0 if it is zero.

Interpretation

The value of \(\tau\) ranges from -1 to 1. A \(\tau\) value of 1 indicates a perfect agreement, -1 indicates a perfect disagreement, and 0 indicates the absence of association between the variables.

Comparison with Spearman's rho

Kendall's tau and Spearman's rank correlation coefficient are both measures of rank correlation: they assess how well the relationship between two variables can be described using a monotonic function. However, they are calculated differently and may give slightly different results for the same data. Kendall's tau is often considered more interpretable since it directly reflects the proportion of concordant and discordant pairs.

Applications

Kendall's tau is widely used in the fields of statistics, epidemiology, and social sciences to test the association between two measured quantities when the assumption of a normal distribution cannot be met. It is particularly useful for small sample sizes or data with many tied ranks.

Advantages and Limitations

One of the main advantages of Kendall's tau is its non-parametric nature, meaning it does not assume a specific distribution for the data. However, it can be less powerful than parametric methods when those assumptions are met. Additionally, calculating Kendall's tau can be computationally intensive for large datasets.

See Also

• Pearson product-moment correlation coefficient
• Spearman's rank correlation coefficient
• Goodman and Kruskal's gamma
• Somers' D