What is the mean and variance of the statistic of Kruskal-Wallis test?
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. 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. However, one-way anova is not very sensitive to deviations from normality. 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. For this reason, I don't recommend the Kruskal-Wallis test as an alternative to one-way anova. Because many people use it, you should be familiar with it even if I convince you that it's overused.
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 (Square root of variance) (the way a normal distribution can). 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. 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.
Dear Raid,
Thanks a lot. I know that the KW test can be estimated by Chi-square distribution for a large number of treatments. Therefore, Your guess may be right. However, I want to know the proof of it.
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