# Kendall’s Tau and Spearman’s Rank Correlation Coefficient

There are two accepted measures of non-parametric rank correlations:** Kendall’s tau **and **Spearman’s (rho) rank correlation** coefficient.

Correlation analyses measure the strength of the relationship between two variables.

**Kendall’s Tau **and Spearman’s rank correlation coefficient assess statistical associations based on the ranks of the data. Ranking data is carried out on the variables that are separately put in order and are numbered.

Correlation coefficients take the values between minus one and plus one. The positive correlation signifies that the ranks of both the variables are increasing. On the other hand, the negative correlation signifies that as the rank of one variable is increased, the rank of the other variable is decreased.

Correlation analyses can be used to test for associations in hypothesis testing. The null hypothesis is that there is no association between the variables under study. Thus, the purpose is to investigate the possible association in the underlying variables. It would be incorrect to write the null hypothesis as having no rank correlation between the variables.

Kendall’s Tau: usually smaller values than Spearman’s rho correlation. Calculations based on concordant and discordant pairs. Insensitive to error. P values are more accurate with smaller sample sizes.

Spearman’s rho: usually have larger values than Kendall’s Tau. Calculations based on deviations. Much more sensitive to error and discrepancies in data.

The main advantages of using Kendall’s tau are as follows:

- The distribution of Kendall’s tau has better statistical properties.
- The interpretation of Kendall’s tau in terms of the probabilities of observing the agreeable (concordant) and non-agreeable (discordant) pairs is very direct.
- In most of the situations, the interpretations of Kendall’s tau and Spearman’s rank correlation coefficient are very similar and thus invariably lead to the same inferences.

**Spearman’s rank correlation** coefficient is the more widely used rank correlation coefficient.

Symbolically, Spearman’s rank correlation coefficient is denoted by r_{s} . It is given by the following formula:

r_{s} = 1- (6∑d_{i}^{2} )/ (n (n^{2}-1))

**Here *d_{i}* represents the difference in the ranks given to the values of the variable for each item of the particular data*

This formula is applied in cases when there are no tied ranks. However, in the case of fewer numbers of tied ranks, this approximation of Spearman’s rank correlation coefficient provides sufficiently good approximations.

**Key terms:**

**Non-parametric test: **it does not depend upon the assumptions of various underlying distributions; this means that it is distribution free.

**Concordant pairs:** if both members of one observation are larger than their respective members of the other observations

**Discordant pairs: **if the two numbers in one observation differ in opposite directions

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