Spearman’s Rank Correlation
The concept of rank is used when the actual measurement or accurate
assessment is not possible or the available information is qualitative in
nature. In such a situation, the correlation failed, and an improvised
statistical measure was developed to measure the strength and direction of
association between two qualitative variables.
For example, two persons are assigned to rank the 6 fruit juices (1
denotes first, the best taste).
The spearman rank correlation is used when the following
assumptions are not met.
i. The data is not an interval or ratio.
ii. The data are not linearly related.
iii. The data is not bivariately distributed.
The spearman rank correlation between two sets of ranks is given by:
Practice Question
The two professors assign marks to five students on the basis
of past experience in the two subjects, which are given below:
Find the coefficient of rank correlation.
Solution: The marks are not actual, so it
is better to find spearman rank correlation instead of spearman correlation.
Step 1: Assign ranks
Step 2: Replace each observation by their respective rank.
Practice Question
Compute the rank correlation
coefficient for the following data of the marks obtained by 8 students in statistics and mathematics.
Solution: Arrange the of both subjects and
assigned ranks.
In statistics (X), 23
is repeated 2 times, so m = 2
Kendall’s Tau Coefficient
Kendall’s Tau correlation coefficient is a statistical measure
that assesses the strength of association between ranked variables. The Kendall tau rank correlation is a
non-parametric alternative to the coefficient of Pearson correlation.
The Kendall coefficient can be calculated as:
Procedure to Perform Kendall’s test
Step 1: Assign ranks to both variables (X & Y).
Step 2: Arrange the ranks of X (column - X) in
ascending order (from least to greatest) and rank of Y as it is.
Step 3: Count the number of concordant pairs (i.e., C)
and discordant pairs (i.e., D). The sum of concordant pairs (C) & discordant pairs (D).
Step 4: Use the
following formula to calculate
The large Tau value suggests a strong association.
The test of significance is given by:
Practice Question
The two experts ranked the quality of watermelon’s as
follows:
Test the association by Kendall’s coefficient.
Solution:
Step 1: Assign ranks
Step 2: The rankings for expert 1 should be in ascending order.
Step 3: Find the concordant & disconcordant.
Kendall’s W Coefficient of
Concordance
The Kendall's test the degree of
association between two sets of ranks. Now if the data is ranked by more than
two judges, then the coefficient of concordance is used. The Kendall’s W statistic
is given by:
The value of the W statistic lies between 0 & 1.
If W = 0, that means the judge ranked the list differently (or
randomly).
If the W = 1, then everyone ranked the list in exactly the
same order.
Where:
Practice Question
Given data
Find the coefficient of concordance and test the agreements
between 3 sets of ranks.
Solution:
- Read More: Multicolinearity
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