Page 231 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

212 5 Non-Parametric Tests of Hypotheses

5.4 Inference on More Than Two Populations

In the present section, we describe non-parametric tests that have parametric
counterparts already described in section 4.5. Note that some of the tests described
for contingency tables also apply to more than two independent samples.

5.4.1 The Kruskal-Wallis Test for Independent Samples

The Kruskal-Wallis test is the non-parametric counterpart of the one-way ANOVA
test described in section 4.5.2. The test assesses whether c independent samples are
from the same population or from populations with continuous distribution and the
same median for the variable being tested. The variable being tested must be at
least of ordinal type. The test procedure is a direct generalisation of the Mann-
Whitney rank sum test described in section 5.3.1.2. Thus, one starts by assigning
natural ordered ranks to the sample values, from the smallest to the largest. Tied
ranks are substituted by their average.

Commands 5.10. SPSS, STATISTICA, MATLAB and R commands used to
perform the Kruskal-Wallis test.

SPSS Analyze; Nonparametric Tests; K
Independent Samples
STATISTICA Statistics; Nonparametrics; Comparing
multiple indep. samples (groups)

MATLAB p=kruskalwallis(x)

R kruskal.test(X~CLASS)

Let R i denote the sum of ranks for sample i, with n i cases. Under the null
hypothesis, we expect that each R i will exhibit a small deviation from the average
of all R i, R . The test statistic is:

12 c
2
KW = ∑ n ( R − R) , 5.38
n( n + ) = i 1 i i
1

which, under the null hypothesis, has an asymptotic chi-square distribution with
df = c – 1 degrees of freedom (when the number of observations in each group
exceeds 5).
When there are tied ranks, a correction is inserted in formula 5.38, dividing the
KW value by: