Applied Statistics and Probability for Engineers

15-5 NONPARAMETRIC METHODS IN THE ANALYSIS OF VARIANCE 591

Since there is a fairly large number of ties, we use Equation 15-12 as the test statistic.

From Equation 15-13 we find

and the test statistic is

Since h

  13.28, we would reject the null hypothesis and conclude that treatments

differ. This same conclusion is given by the usual analysis of variance F-test.

15-5.2 Rank Transformation

The procedure used in the previous section whereby the observations are replaced by their

ranks is called the rank transformation.It is a very powerful and widely useful technique.

If we were to apply the ordinary F-test to the ranks rather than to the original data, we would

obtain

as the test statistic. Note that as the Kruskal-Wallis statistic Hincreases or decreases, F 0 also

increases or decreases. Now, since the distribution of F 0 is approximated by the F-distribution,

F 0 

H 1 a 12

1 N 1 H 21 Na 2

0.01,4^2

19.06



1

53.54

c5245.7

2512622

4

d

h 

1

s^2

ca

a

i 1

ri^2.

ni

N 1 N 122

4

d

53.54



1

24

c 5510 

2512622

2

d

s^2 

1

N 1

ca

a

i 1 a

ni

j 1

rij^2 

N 1 N 122

4

d

Table 15-4 Data and Ranks for the Tensile Testing Experiment

Percentage

of Cotton ri.

15 y 1 j 7 7 15 11 9

ranks r 1 j 2.0 2.0 12.5 7.0 4.0 27.5

20 y 2 j 12 17 12 18 18

ranks r 2 j 9.5 14.0 9.5 16.5 16.5 66.0

25 y 3 j 14 18 18 19 19

ranks r 3 j 11.0 16.5 16.5 20.5 20.5 85.0

30 y 4 j 19 25 22 19 23

ranks r 4 j 20.5 25.0 23.0 20.5 24.0 113.0

35 y 5 j 710111511

ranks r 5 j 2.0 5.0 7.0 12.5 7.0 33.5

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