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To compare multiple groups, one usually runs ANOVA. If ANOVA rejects the null hypothesis, one goes through the multiple comparisons.

The multiple comparisons are necessary to establish which group is different. Multiple comparisons are performed applying Bonferroni correction (or more recent types of correction, such as Holm's or Hommel's).

Such corrections guarantee the family wise error rate (FWER) to be at most alpha. FWER is the probability to return at least one wrong significance statement when comparing populations whose mean is actually the same.

I wonder why is the preceeding ANOVA necessary at all. I think one should run directly the multiple comparison with adjusted p-values.

In fact: a) if all means are equal: the multiple comparison will have the same FWER of ANOVA, because of the correction.

b)Some means are different. Then multiple comparisons are anyway necessary.

The preceeding ANOVA is often done, but I do not see a compelling reason for it.

Thanks.

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    $\begingroup$ The preceeding ANOVA usually has more power to detect that there is a statistically significant difference among groups. It often happens that you perform ANOVA and find a p-value < 0.05 and then when you perform the multiple comparisons all p-values are non-significant $\endgroup$ – Aghila May 12 '14 at 12:49
  • $\begingroup$ Thanks Aghila. I see your point. Thanks also to the users who pointed me to an exhaustive discussion already developed on the topic. $\endgroup$ – user45420 May 14 '14 at 9:17