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Python | Kendall Rank Correlation Coefficient

  • Last Updated : 20 Jul, 2021

What is correlation test?
The strength of the association between two variables is known as the correlation test. For instance, if we are interested to know whether there is a relationship between the heights of fathers and sons, a correlation coefficient can be calculated to answer this question.

For know more about correlation please refer this.

Methods for correlation analysis:
There are mainly two types of correlation:

  • Parametric Correlation – Pearson correlation(r) : It measures a linear dependence between two variables (x and y) is known as a parametric correlation test because it depends on the distribution of the data.
  • Non-Parametric Correlation – Kendall(tau) and Spearman(rho): They are rank-based correlation coefficients, are known as non-parametric correlation.

Kendall Rank Correlation Coefficient formula:

\tau=\frac{\text { Number of concordant pairs-Number of discordant pairs }}{n(n-1) / 2}

where,



  • Concordant Pair: A pair of observations (x1, y1) and (x2, y2) that follows the property
    • x1 > x2 and y1 > y2 or
    • x1 < x2 and y1 < y2
  • Discordant Pair: A pair of observations (x1, y1) and (x2, y2) that follows the property
    • x1 > x2 and y1 < y2 or
    • x1 < x2 and y1 > y2
  • n: Total number of samples

Note: The pair for which x1 = x2 and y1 = y2 are not classified as concordant or discordant and are ignored.

Example: Let’s consider two experts ranking on food items in the below table.

ItemsExpert 1Expert 2
111
223
336
442
557
664
775

The table says that for item-1, expert-1 gives rank-1 whereas expert-2 gives also rank-1. Similarly for item-2, expert-1 gives rank-2 whereas expert-2 gives rank-3 and so on.

Step1:
At first, according to the formula, we have to find the number of concordant pairs and the number of discordant pairs. So take a look at item-1 and item-2 rows. Let for expert-1, x1 = 1 and x2 = 2. Similarly for expert-2, y1 = 1 and y2 = 3. So the condition x1 < x2 and y1 < y2 satisfies and we can say item-1 and item-2 rows are concordant pairs.

Similarly take a look at item-2 and item-4 rows. Let for expert-1, x1 = 2 and x2 = 4. Similarly for expert-2, y1 = 3 and y2 = 2. So the condition x1 < x2 and y1 > y2 satisfies and we can say item-2 and item-4 rows are discordant pairs.

Like that, by comparing each row you can calculate the number of concordant and discordant pairs. The complete solution is given in the below table.

1
2C
3CC
4CDD
5CCCC
6CCCDD
7CCCCDD
1234567

Step 2:
So from the above table, we found that,
The number of concordant pairs is: 15
The number of discordant pairs is: 6
The total number of samples/items is: 7

Hence by applying the Kendall Rank Correlation Coefficient formula

tau = (15 – 6) / 21 = 0.42857



This result says that if it’s basically high then there is a broad agreement between the two experts. Otherwise, if the expert-1 completely disagrees with expert-2 you might get even negative values.

kendalltau() : Python functions to compute Kendall Rank Correlation Coefficient in Python

Syntax:
kendalltau(x, y)

  • x, y: Numeric lists with the same length

Code: Python program to illustrate Kendall Rank correlation

Python




# Import required libraries
from scipy.stats import kendalltau
  
# Taking values from the above example in Lists
X = [1, 2, 3, 4, 5, 6, 7]
Y = [1, 3, 6, 2, 7, 4, 5]
  
# Calculating Kendall Rank correlation
corr, _ = kendalltau(X, Y)
print('Kendall Rank correlation: %.5f' % corr)
  
# This code is contributed by Amiya Rout

Output:

Kendall Rank correlation: 0.42857

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