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ANOVA analysis can be used to determine the significance of multiple groups. Why do we need Bonferroni correction to re-compute the alpha values?

ANOVA solves the problem of multiple testing by comparing multiple groups together.

Bonferroni correction does the same thing.

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An omnibus ANOVA is not multiple testing. It's a single test of all the groups at once. This rarely provides a satisfactory answer the research question, since you typically want to know which specific groups are different, not just whether "something" is different from "something else."

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  • $\begingroup$ interesting - so in that case ANOVA seems useless to me. $\endgroup$
    – user46925
    Commented Mar 25, 2016 at 12:01
  • $\begingroup$ +1 but your answer would improve if you elaborated on what exactly "a single test of all the groups at once" means. What is the null hypothesis of ANOVA? What are the null hypotheses if one performs all pairwise comparisons? Etc. $\endgroup$
    – amoeba
    Commented Mar 25, 2016 at 12:38
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    $\begingroup$ ANOVA is the first step. It has a perfect multiplicity adjustment and provides evidence for any difference. If the ANOVA is not significant you are on shaky ground to look at individual differences. But instead of Bonferroni for the individual contrasts I would use simultaneous confidence intervals. $\endgroup$
    – Frank Harrell
    Commented Mar 25, 2016 at 12:39
  • $\begingroup$ I disagree with basically all of Frank Harrell's statements above. First of all, ANOVA is not a multiplicity adjustment, as I stated in my answer. Rather, it avoids multiplicity by conducting only one test. Second, looking at individual differences does not typically require conducting an omnibus ANOVA. Lastly, the Bonferroni procedure can be used to produce simultaneous confidence intervals, so to say you should use simultaneous confidence intervals "instead" of Bonferroni doesn't appear to make sense. $\endgroup$
    – Bonferroni
    Commented Aug 6, 2016 at 18:29

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