Doing multiple comparisons increases the number of false positive findings. In the Bonferroni method, the adjusted critical level is $\hat{\alpha} = \frac{\large\alpha}{N}$ where $N$ is the number of tests. However, demanding this from each test increases the number of false negatives.

In the Holm-Bonferroni and FDR methods, the smallest p-values have to be likewise below $\frac{\large\alpha}{N}$. Is this some general property?

In order to control the number of false positives, when doing $N$ tests, do the smallest $p$ have to be smaller than $\alpha/N$? It seems the Holm-Bonferroni and FDR differ only with respect to the following values.

• Other exceptions (just to mention a FWER controlling one) would be Dunnett's test (and related test such as the one by Dunnett and Tamahane), where the threshold is a little bit higher than $\alpha/N$, weighted Bonferroni(-Holm) tests, various closed testing procedures etc. Just in case the list above may have given the impression that that the smallest p-value must be smaller than $\alpha/N$ for FWER controlling methods. Mar 1 '17 at 13:13