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Let's assume the group variable is the dependent variable and can have the values [A,B,C,D]. It is nominal. The independent variable is ordinal and can have numbers between 0 and 5.

I want to test for differences between the 4 groups in the independent variable.

Is it possible to test this with the Kruskal-Wallis Test. I´m confused because literature says that for the Kruskal-Wallis Test the dependent variable needs to be ordinal. So this would be the other way around. Can I still use it?

I´m just interested in differences, not in an effect.

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You can just reverse it. Having different distributions of the categories at different levels of the ordinal variable is the same as having different distributions of the ordinal variable within the different categories.

This would then just be the regular Kruskal-Wallis test.

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