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When conducting a 2-way fixed effects ANOVA, there are essentially four null hypothesis statistical tests (NHSTs) that can reasonably conducted with just the ANOVA tables. The first is the omnibus test: ¿is the overall model statistically significant? The next three are about the effects: ¿are the two main effects and the interaction effect statistically significant?

Most reputable textbooks will mention that you cannot entertain the last three if you fail to reach significance with the omnibus NHST.

However, I have yet to locate a textbook that suggests we need to do some type of alpha adjustment (say a Bonferroni adjustment) when looking at the significance of the three effects. Yet, these books will mention that you do need to account for Type-I inflation when doing any follow-up (post hoc) comparisons.

So, if the effect NHST is the omnibus test prior to the MCPs which require alpha adjustment, ¿why is it not the same for the omnibus NHST prior to the analysis of the effects?

As an addition to this, I have the same question for general multiple regression models as well...but I thought it would be easier to present in the 2-way ANOVA context as a question on this forum.

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  • $\begingroup$ Omnibus tests are not formally needed for multiple testing. That sounds like "Fisher's protected LSD" which has been pretty well trashed in the literature. The omnibus tests are important in the context of formal FWER-controlling multiple testing only if you are using closed testing procedures. $\endgroup$
    – BigBendRegion
    Commented May 9, 2023 at 14:24

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