Statistical Analysis for Education and Psychology Researchers

Interpretation of Computer Output

Unlike the worked example, where the exact sampling distribution of H was used, SAS
output automatically provides a Chi-square approximation. The computed value of H
(corrected for ties) in the worked example is the same as the Chi-square value shown in
this output. The only difference is in the interpretation of the test statistic. In the worked
example the significance of H was evaluated using the exact sampling distribution and
tabled critical values (Table 6, Appendix A4). The SAS output refers to the Chi-square
approximation, (primarily intended for large samples studies) which is based on k− 1
degrees of freedom (k is the number of independent samples or groups). The rejection

region for the Kruskal-Wallis test includes all values of Chi-square larger than
(means Chi-square with 2 df) with p=0.05, that is 5.99147. Since the calculated value,
7.0404 exceeds the critical value (falls in the rejection region) we know that the observed
value has an associated probability under the null hypothesis that lies between p=0.05 and
p=0.02, (see Table 2, Appendix A4) which may be expressed as 0.05>p>0.02. In fact the
SAS output shows that the associated probability is 0.0296. This probability is small, less
than 5 per cent so we can reject the null-hypothesis of no difference and we arrive at the
same conclusion as in the worked example. If an investigator was concerned about using
the Chi-square approximation with small samples, less than 5 observations in any of the
groups, H can be evaluated using the exact sampling distribution (Table 6, Appendix A4).
For discussion on the adequacy of this Chi-square approximation for small samples the
reader is referred to Gabriel and Lachenbruch (1969).

Pairwise Multiple-comparisons for post-hoc Analysis

When an obtained H-statistic is significant, an investigator may wish to determine which
of the groups differ. A post hoc pairwise multiple-comparison procedure and
computational formula is described by Siegel and Castellan (1988), the SAS programme
Krusk-W1 (see Figure 11, Appendix A3) performs this procedure for all pairwise
comparisons in a design. An investigator should note that to control for experiment-wise
error, that is, to adjust for many non-independent pairwise comparisons, the initial alpha
level should be set to a liberal level (possibly 10 per cent). For a two-tailed test the
effective pairwise alpha level will be the original alpha level used for the Kruskal-Wallis
test divided by the number of possible comparisons, c, multiplied by two (α/(c×2)). The
number of comparisons is given by (k(k−1)/2) where k refers to the number of groups. So,
if an initial alpha of 10 per cent was selected with 3 groups, the effective alpha level for
the pairwise comparisons would be 0.1/6=0.0167. Output from the SAS programme
Krusk-W1 using data from Table 7.5 is shown in Figure 7.8.

Inferences involving rank data 237