![]() The expected mean rank depends only on the total number of observations (for n observations, the expected mean rank in each group is ( n+1)/2), so it is not a very useful description of the data it's not something you would plot on a graph. The null hypothesis of the Kruskal–Wallis test is that the mean ranks of the groups are the same. ![]() I think calling the Kruskal–Wallis test an anova is confusing, and I recommend that you just call it the Kruskal–Wallis test. The Kruskal–Wallis test is sometimes called Kruskal–Wallis one-way anova or non-parametric one-way anova. For simplicity, I will only refer to Kruskal–Wallis on the rest of this web page, but everything also applies to the Mann–Whitney U-test. It uses a different test statistic ( U instead of the H of the Kruskal–Wallis test), but the P value is mathematically identical to that of a Kruskal–Wallis test. The Mann–Whitney U-test (also known as the Mann–Whitney–Wilcoxon test, the Wilcoxon rank-sum test, or the Wilcoxon two-sample test) is limited to nominal variables with only two values it is the non-parametric analogue to two-sample t–test. Dominance hierarchies (in behavioral biology) and developmental stages are the only ranked variables I can think of that are common in biology. The only time I recommend using Kruskal-Wallis is when your original data set actually consists of one nominal variable and one ranked variable in this case, you cannot do a one-way anova and must use the Kruskal–Wallis test. Instead, you should use Welch's anova for heteoscedastic data. If your data are heteroscedastic, Kruskal–Wallis is no better than one-way anova, and may be worse. ![]() While Kruskal-Wallis does not assume that the data are normal, it does assume that the different groups have the same distribution, and groups with different standard deviations have different distributions. The other assumption of one-way anova is that the variation within the groups is equal ( homoscedasticity). You lose information when you substitute ranks for the original values, which can make this a somewhat less powerful test than a one-way anova this is another reason to prefer one-way anova. Like most non-parametric tests, you perform it on ranked data, so you convert the measurement observations to their ranks in the overall data set: the smallest value gets a rank of 1, the next smallest gets a rank of 2, and so on. The Kruskal-Wallis test is a non-parametric test, which means that it does not assume that the data come from a distribution that can be completely described by two parameters, mean and standard deviation (the way a normal distribution can). Because many people use it, you should be familiar with it even if I convince you that it's overused. For this reason, I don't recommend the Kruskal-Wallis test as an alternative to one-way anova. I've done simulations with a variety of non-normal distributions, including flat, highly peaked, highly skewed, and bimodal, and the proportion of false positives is always around 5% or a little lower, just as it should be. However, one-way anova is not very sensitive to deviations from normality. Some people have the attitude that unless you have a large sample size and can clearly demonstrate that your data are normal, you should routinely use Kruskal–Wallis they think it is dangerous to use one-way anova, which assumes normality, when you don't know for sure that your data are normal. The most common use of the Kruskal–Wallis test is when you have one nominal variable and one measurement variable, an experiment that you would usually analyze using one-way anova, but the measurement variable does not meet the normality assumption of a one-way anova. It tests whether the mean ranks are the same in all the groups. Use the Kruskal–Wallis test when you have one nominal variable and one ranked variable. ![]()
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