There are two accepted measures of rank correlations, namely Kendall’s tau and Spearman’s rank correlation coefficient.
Kendall’s tau is a measure of correlation. Kendall’s tau measures the strength of the relationship between the two variables. Like Spearman’s rank correlation coefficient, Kendall’s tau is carried out on the ranks of the data. In other words, Kendall’s tau is carried out on the variables that are separately put in order and are numbered.
Like other measures of correlation, Kendall’s tau takes 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.
It is feasible to calculate the confidence intervals and carry out the hypothesis tests with the help of Kendall’s tau.
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.
In Spearman’s rank correlation coefficient, the measure of rank correlation is the more widely used rank correlation coefficient.
Symbolically, Spearman’s rank correlation coefficient is denoted by rs . It is given by the following formula:
rs = 1- (6∑di2 )/ (n (n2-1))
*Here di 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.
The Spearman’s rank correlation coefficient table, called the Spearman table, is constructed by taking all kinds of possible happenings into consideration.
The Spearman’s rank correlation coefficient, as the name suggests, is used to test for association. However, there are some specific relationships that are picked up by Spearman’s rank correlation coefficient test.
For this test to work effectively, the underlying relationship must be monotonic. In other words, for the efficient result generation, the variables should either increase in values together, or when one gets increased, then the other should get decreased.
Spearman’s rank correlation coefficient is a non parametric test. In other words, it does not depend upon the assumptions of various underlying distributions, this means that it is distribution free.
Spearman’s rank correlation coefficient can be used to test for association in the testing of hypothesis. 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.
One difficulty is with the data having very large or very small samples. In the case of very large samples, it is very time consuming to perform since it requires the ranking of the data of both of the variables.
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