RELATIONSHIP BETWEEN SPEARMAN’S r ’s AND KENDALL’S W - Research Methodology

As stated above, W is an appropriate measure of studying the degree of association among three or more sets of ranks, but we can as well determine the degree of association among k sets of rankings by averaging the Spearman’s correlation coefficients (r’s) between all possible pairs (i.e., kC2 or k (k – 1)/2) of rankings keeping in view that W bears a linear relation to the average r’s taken over all possible pairs. The relationship between the average of Spearman’s r’s and Kendall’s W can be put in the following form:
average of r’s = (kW – 1)/(k – 1)
But the method of finding W using average of Spearman’s r’s between all possible pairs is quite tedious, particularly when k happens to be a big figure and as such this method is rarely used in practice for finding W.

Illustration

Using data of illustration No. 9 above, find W using average of Spearman’s r’s.
Solution: As k = 4 in the given question, the possible pairs are equal to k(k – 1)/2 = 4(4 – 1)/2 = 6 and we work out Spearman’s r for each of these pairs as shown in Table 12.10. Now we can find W using the following relationship formula between r’s average and W

relationship formula between r’s average and W

[Note: This value of W is exactly the same as we had worked out using the formula:
W = s/[(1/12) (k2) (N3 – N)]


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