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I see that the theory says that if I want to test for difference in mean between 3+ groups I should use ANOVA first, and if ANOVA rejects the null hypothesis then I might want to look into Tukey as a post-hoc test to find which groups' means really differ.

My question then is: why not running Tukey directly? What really happens if I run Tukey without running ANOVA first? Which are the risks/implications of that?

The intuition tells me that if Tukey finds some significant differences, then ANOVA should have found it, too. When is that not the case?

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  • $\begingroup$ I don't have the answer to your question, but it is certainly possible to find significant pairwise differences after a non-significant ANOVA. See e.g. this thread $\endgroup$ Commented Mar 12, 2024 at 9:44
  • $\begingroup$ Running the F test before Turkey is actually a bad idea. See a book about multiple testing/comparisons. $\endgroup$ Commented Mar 12, 2024 at 13:46

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