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QUESTION: Should we do multiple comparisons correction for the number of effects (main effects and interactions) in multi-way ANOVA?

I might have failed to find a relevant question, but mostly questions related to multiple comparisons and ANOVA are about the posthoc pairwise comparisons.

I wonder when we use multi-way ANOVA, for exmaple, 3-way ANOVA, we actually test for 3 main effects, 2 two-way interactions, and 1 one-way interaction. I feel like we are doing 6 F-tests simultaneously, thus inflating type-I errors. But I have not heard of any remarks on this yet.

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A better question is to correct for the number of contrasts you test. One way to do so is the Holm-Bonferroni method (https://en.wikipedia.org/wiki/Holm%E2%80%93Bonferroni_method). See Cramer et al. (2016) https://doi.org/10.3758/s13423-015-0913-5.

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