Kendall’s tau correlation is a non-parametric measure of the strength and direction of the association between two variables. It measures the degree of correspondence between two sets of rankings or orders. Kendall’s tau is denoted by the symbol ‘τ’ and ranges between -1 and +1.
A Kendall’s tau correlation coefficient of +1 indicates a perfect positive correlation, which means that both variables move in the same direction with a similar magnitude. A Kendall’s tau correlation coefficient of -1 indicates a perfect negative correlation, which means that both variables move in opposite directions with a similar magnitude. A Kendall’s tau correlation coefficient of 0 indicates no correlation between the variables.
The formula to calculate Kendall’s tau correlation is:
τ = (2 / (n(n-1))) * Σ(i=1 to n-1) Σ(j=i+1 to n) sign[(x_i – x_j) * (y_i – y_j)]
Where:
- n is the number of observations
- x_i and y_i are the ranks of the ith observation for the two variables being compared
- Σ is the summation symbol, which means “add up all the values”
- sign is the sign function, which returns +1 if the argument is positive, -1 if the argument is negative, and 0 if the argument is 0.
Kendall’s tau correlation is often used in situations where the data does not meet the assumptions of parametric correlation measures such as Pearson’s correlation. It is widely used in fields such as ecology, economics, and psychology to analyze the relationship between two variables.
Kendall’s tau assume the following:
- Independence: The data points being analyzed should be independent of each other.
- Random sampling: The data should be collected using a random sampling technique.
- Ordinal measurement: The variables being correlated should be measured on an ordinal scale.
- Ties: Ties may exist in the data, which occurs when two or more data points have the same value. Kendall’s tau accounts for ties in the data and adjusts the correlation coefficient accordingly.