Statistical Analysis for Education and Psychology Researchers

Statistical Inference and Null Hypothesis

The Kruskal-Wallis test statistic, H, is sensitive to location shifts and under the null
hypothesis (equal populations) is asymptotically (large sample) distributed as Chi-square
with k−1 degrees of freedom. The null hypothesis is that the independent samples come
from the same population or from populations which have the same median. The non-
directional alternative hypothesis is that at least one sample has a different median to the
others. A large value of the test statistic leads to rejection of the null hypothesis.

Test Assumptions

The test assumptions are as follows:

• Data consists of observations which have been selected at random from an infinitely
  large population.


• The population(s) have an underlying continuous distribution but the response variable
  is a rank measurement.


• It is preferable that there are at least 4–5 subjects in each sample (independent group)
  because of the use of the Chi-square approximation for the H-test statistic. It is not
  necessary to have a balanced design (equal numbers in each independent group).

With heterogeneity of variance, different variances for the independent samples, it is
possible with this test procedure to reject the null hypothesis (equality of medians), when
means are in fact equal. A significant test statistic value, H, is therefore no assurance of
differences in treatment means.

Example from the Literature

Elliott and Hewison (1994) investigated type of reading help given by different
(independent) groups of parents. In their study there were four groups of parents/ other
family members: 24 middle-class families; 26 working-class families, 17 families who
had been involved in a Paired Reading Project; and 24 families of Asian origin. Four
response variables were analyzed separately, rapid corrections (maintaining flow in
reading), Semantic-based corrections; phonic-based corrections; and anomalies
(deliberate non-correction or missed correction). Kruskal-Wallis non-parametric analysis
of variance was performed on each response variable separately. The authors reported
differences between groups for all four response variables: rapid correction (H=32.34,
p<0.00001); semantic-based corrections (H=30.33, p<0.00001); phonic corrections
(H=11.39, p<0.009); and anomalies/non-corrections (H=20.10, p<0.0002). Post hoc tests
were not reported.

Worked Example

Small sample Kruskal-Wallis test procedure

Data shown in Table 7.5 is similar to that obtained by Elliott and Hewison (1994), who
investigated types of reading help given by different groups of parents. In their study

Inferences involving rank data 233