Research Methodology - Dr. Krishan K. Pandey

Testing of Hypotheses-II 311
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 11
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
Average of r’s = (kW – 1)/(k – 1)
or 0.655 = (4W – 1)/(4 – 1)
or (0.655) (3) = 4W – 1
b gbg
0. 655 3 + 1 2. 965
or W =
= = 0. 741
4 4
[Note: This value of W is exactly the same as we had worked out using the formula:
W = s/[(1/12) (k2 ) (N3 – N)]
CHARACTERISTICS OF DISTRIBUTION-FREE OR NON-PARAMETRIC TESTS
From what has been stated above in respect of important non-parametric tests, we can say that
these tests share in main the following characteristics:
1. They do not suppose any particular distribution and the consequential assumptions.
2. They are rather quick and easy to use i.e., they do not require laborious computations since
in many cases the observations are replaced by their rank order and in many others we
simply use signs.
3. They are often not as efficient or ‘sharp’ as tests of significance or the parametric tests.
An interval estimate with 95% confidence may be twice as large with the use of nonparametric
tests as with regular standard methods. The reason being that these tests do not
use all the available information but rather use groupings or rankings and the price we pay
is a loss in efficiency. In fact, when we use non-parametric tests, we make a trade-off: we
loose sharpness in estimating intervals, but we gain the ability to use less information and to
calculate faster.
4. When our measurements are not as accurate as is necessary for standard tests of
significance, then non-parametric methods come to our rescue which can be used fairly
satisfactorily.
5. Parametric tests cannot apply to ordinal or nominal scale data but non-parametric tests do
not suffer from any such limitation.
6. The parametric tests of difference like ‘t’ or ‘F’ make assumption about the homogeneity
of the variances whereas this is not necessary for non-parametric tests of difference.