Kruskal-Wallis Nonparametric ANOVA

Data were expressed as the mean standard error of the mean (SEM). Differences between means were determined using one-way analysis of variance (ANOVA) followed by the Tukey-Kramer post hoc comparison and two-sided t test. For comparing percentages, nonparametric tests were also applied (Mann-Whitney, Kruskal-Wallis). Differences were considered significant when p < 0.05. 

To check the assumptions of the model, Bartlett s or Levene s tests can be used to assess the assumption of equality of variance, and the normal probability plot of the residuals (etj = Xij - Xj) to assess the assumption of normality. If either equality or normality are inappropriate, we can transform the data, or we can use the nonparametric Kruskal-Wallis test to compare the k groups. In any case, the ANOVA procedure is insensitive to moderate departures from the assumptions (Massart et al. 1990). 

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. 

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. 

The Kruskal-Wallis model is the nonparametric analog of a one-factor ANOVA model. It is used to compare multiple groups of one factor. For example, suppose one wants to evaluate the antimicrobial effects of five different hand soaps the Kruskal-Wallis model could be employed for this evaluation. 

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. 

Kruskal-Wallis test is the technique tests the null hypothesis that several populations have the same median. It is the nonparametric equivalent of the one-factor ANOVA. The test statistic is ... 

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