Statistical Analysis for Education and Psychology Researchers

As in the Wilcoxon signed ranks test, the variance of the sampling distribution of the test
statistic is influenced by ties among the scores regardless of to which group the tied
scores belong. The correction for ties is given by:

Correction

for ties in K-

W H-

statistic—7.5

where g=the number of groups of different tied values, ti is the number of tied ranks in
each of the groups of tied values and N is the total number of observations in the
combined sample. The Kruskal-Wallis H statistic is then divided by this correction factor.

The value of H corrected for ties is therefore 6.965/0.989=7.04

Interpretation

When any of the independent groups has fewer than five observations we can describe
this as a small sample design. In this situation we use the exact sampling distribution of
the test statistic H. This is shown in Table 6, Appendix A4 (for three independent groups
only). The rejection region for the H-test statistic is H>tabled critical value, which is
based on the sample size in each of three groups and on the chosen alpha level.
In this example we have selected alpha of 0.05. The critical value given in Table 6,
Appendix A4 for alpha=0.05 and n 1 =n 2 =n 3 =5 is 5.78. The rejection region for the test
includes all values of H>5.78. Since the observed H is greater than the critical value,
7.04>5.78, the probability of obtaining an H-value as large as 7.04, when the null
hypothesis is true, is equal to or less than p=0.05. We reject the null hypothesis at the 5
per cent level and conclude there is sufficient evidence of a difference(s) among the three
groups of middle-class, working-class and Asian families in the way they correct reading
errors.
When H is found to be statistically significant it indicates that the k-samples do not
come from the same population, that is at least one of the samples has a different median
to at least one of the others. The H-statistic is not informative about which or how many
of the samples are significantly different from each other. When the number of
comparisons are small the Wilcoxon Mann-Whitney test may be used for post hoc
analysis following a significant Kruskal-Wallis H test. A more detailed procedure which

Inferences involving rank data 235