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Typically before running a statistical test, I would check the assumptions of independence, normality etc. before running the test. However some of the assumptions of ANOVA are with regards to the residuals (normality of residuals, independence of residuals).

So my question is, for the ANOVA, do I run the assumption checks after running the statistical test?

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Yes. Fit the model first, then check the assumptions. As Tim points out, you have no residuals before fitting the model, so you cannot make inferences about them

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You can check the normality of the data first. In case that the data are not normally distributed, then you cannot use ANOVA and use a nonparametric test such as Kruskal-Wallis test.

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