Applied Statistics and Probability for Engineers

590 CHAPTER 15 NONPARAMETRIC STATISTICS

An alternative computing formula that is occasionally more convenient is

H (15-11)

12

N 1 N 12

(^) a
a
i 1
R^2 i.
ni ^31 N^12
We would usually prefer Equation 15-11 to Equation 15-10 because it involves the rank totals
rather than the averages.
The null hypothesis H 0 should be rejected if the sample data generate a large value for
H. The null distribution for Hhas been obtained by using the fact that under H 0 each pos-
sible assignment of ranks to the atreatments is equally likely. Thus, we could enumerate
all possible assignments and count the number of times each value of Hoccurs. This has
led to tables of the critical values of H, although most tables are restricted to small sample
sizes ni. In practice, we usually employ the following large-sample approximation.
Whenever H 0 is true and either
a 3 and ni 6 for i1, 2, 3
a
3 and ni 5 for i1, 2,... , a
Hhas approximately a chi-square distribution with a1 degrees of freedom. Since large val-
ues of Himply that H 0 is false, we will reject H 0 if the observed value
The test has approximate significance level .
Ties in the Kruskal-Wallis Test
When observations are tied, assign an average rank to each of the tied observations. When
there are ties, we should replace the test statistic in Equation 15-11 by
(15-12)
where niis the number of observations in the ith treatment, Nis the total number of observa-
tions, and
(15-13)
Note that S^2 is just the variance of the ranks. When the number of ties is moderate, there will
be little difference between Equations 15-11 and 15-12 and the simpler form (Equation 15-11)
may be used.
EXAMPLE 15-7 Montgomery (2001) presented data from an experiment in which five different levels of
cotton content in a synthetic fiber were tested to determine whether cotton content has any
effect on fiber tensile strength. The sample data and ranks from this experiment are shown in
Table 15-4. We will apply the Kruskal-Wallis test to these data, using   0.01.
S^2 
1
N 1
ca
a
i 1 a
ni
j 1
R^2 ij
N 1 N 122
4
d
H 
1
S^2
ca
a
i 1
R^2 i.
ni 
N 1 N 122
4
d
h^2 ,a 1
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