Python

What is correlation test?
The strength of the association between two variables is known as 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:

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.

Items	Expert 1	Expert 2
1	1	1
2	2	3
3	3	6
4	4	2
5	5	7
6	6	4
7	7	5

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
2	C
3	C	C
4	C	D	D
5	C	C	C	C
6	C	C	C	D	D
7	C	C	C	C	D	D
	1	2	3	4	5	6	7

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

# 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

My Personal Notes

Python – Kendall Rank Correlation Coefficient

Python

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