The Kruskal-Wallis test

There are times when the required assumptions for ANOVA, a parametric test, are not met. One example would be if the underlying distributions are non-normal. In these cases, nonparametric tests are very useful and informative. For example, we saw in Section 11.3 that a nonparametric analog to the two-sample ttest, Wilcoxon s rank sum test, makes use of the ranks of observations rather than the scores themselves. When a one-factor ANOVA is not appropriate in a particular case a corresponding nonparametric approach called the Kruskal-Wallis test can be used. This test is a hypothesis test of the location of (more than) two distributions. 

All that is required for this test to be employed is that the observations classified into k groups are independently sampled from populations and the random variable is continuous with the same variability across the populations represented by the samples. Importantly, no assumption about the shape of the underlying distribution is required, making this test suitable for non-normal underlying distributions. 

In the Kruskal-Wallis test the original scores are first ranked and an ANOVA analysis is then carried out on the ranks. As with Wilcoxon s rank sum test, ranking of the observations must deal with ties. The sums of squares are based on 

All observations, x, are assigned ranks, r.., and therefore the usual sums of squares can be calculated for the rank scores, q.. For brevity, the expressions for each are provided in Table 11.8, a general one-way ANOVA table, on the basis of ranks. 

The quantities in the ANOVA table based on ranks represent similar quantities as the ANOVA table based on the original scores  

---

Analyses were conducted with SPSS version 13.0, and data satisfied the requirements of the statistical tests used non-normally distributed data were analysed with the Kruskal-Wallis test. We compared how highly the males were rated in terms of their attractiveness by the women in each of the three experimental conditions. 

A table of data is set up with each of the two variables being ranked separately. Tied ranks are assigned as demonstrated earlier under the Kruskall Wallis test. From... 

For statistical analysis, fetal abnormality values belong to two types those where at least 50% of litters have one or more fetuses affected, and those where most litters have no affected fetuses. For the first type, the incidences (percentage of affected fetuses within that litter) are analyzed by the Kruskal-Wallis test (13) for the second type, the number of litters with affected fetuses is compared with the number with no affected fetuses by Fisher s Exact test (14). 

Having calculated the level of significance can be obtained from appropriate tables. The Wilcoxon signed rank test is the non-parametric equivalent of the paired t-test. The Kruskal-Wallis test is another rank sums test that is used to test the null hypothesis that k independent samples come from identical populations against the alternative that the means of the populations are unequal. It provides a non-parametric alternative to the one-way analysis of variance. 

Non-parametric comparisons of location for three or more samples include the Kruskal-Wallis //-test. Here, the two data sets can be unequal in size, but again the underlying distributions are assumed to be similar. 

For the sake of this example, we use the data from the parametric ANOVA example to illustrate the Kruskal-Wallis test. If it seems at all strange to use the same data for both examples, a parametric analysis and a nonparametric analysis, it is worth noting that a nonparametric analysis is always appropriate for a given dataset meeting the requirements at the start of the chapter. Parametric analyses are not always appropriate for all datasets. 

In what situations would the Kruskal-Wallis test be appropriate ... 

The data obtained were subjected to the Kruskal-Wallis test. The test revealed significant between-month differences (P<0.05) in the three metals contained both in the honey and bee samples, while as regards the between-station differences, the results were statistically signihcant only for lead in bees. The comparison with the control stations (Tl and T2) revealed a significant difference only for chromium in honey and lead in bees. 

The Kruskal-Wallis test is a nonparametric alternative to the ANOVA F-test, described above, for multiple conditions. That is, it is an extension of the Wilcoxon rank-sum test to multiple conditions. Expression values are replaced with their ranks to form the test statistic without requiring an assumption of the form of the distribution. For example, the Kruskal-Wallis statistic is 7.2 and the p-value is less than 0.05 for the probe set used for illustration of the ANOVA F-test. 

According to the Multiple Comparison Test statistically significant differences are seen for AP between UFI or UFII and UF III there is no statistically significant difference for AP between UF I and UF II. According to the Kruskal-Wallis test there is no statistically significant difference between the values for FIR between the three resins tested. 

These include the Wilcoxon test, the Kruskal-Wallis test or the Kolmogoroff-Smirnov test. 

General guidelines for statistical analysis were presented previously (see section 5.1.3). Where continuous variables, such as consumption, are measured, means are compared using parametric tests (e.g., -test, ANOVA) or nonparamet-ric tests (e.g., Wilcoxon two-sample test or the Kruskal-Wallis test) as appropriate. For simple choice tests, the G-test or Fisher s exact test (Sokal Rohlf 1995) are often used to test for deviations of the observed pattern of choices from a random pattern. 

In order to analyze differences in personal values across groups (first comparison), the Kruskal-Wallis test is used. To measure the changes in moral values from the beginning to the end of the course (second comparison), the Wilcoxon signed-ranked test is used. 

The Kruskal-Wallis test is a non-parametric test that compares three or more independent groups, and does not need to assume that the sources come from a normal distribution. It is robust for groups with different sample sizes, and can work with ordinal data (rating-scale data). The nuU hypothesis for this test is that there is no significant difference among the medians of the k-groups . A small p-value rejects the null hypothesis, which means that at least one group s median differs from one of the others. 

To further identify which group is different, a Dunn s post-test is performed. This test compares the difference in the median for each group pair. If the statistic of the pair is greater than the critical value, then the null hypothesis that there is no significant difference between the medians of the pair is rejected. With smaller sample sizes the power of the Kruskal-Wallis test decreases, so these results need to be analyzed carefully. 

The tests performed were the same used for the analysis of individual motivational values. The Kruskal-Wallis test and the Dunn s post-test are used for the first comparison, and the Wilcoxon signed-ranked test is used for the second comparison. 

---