How many comparisons should I use when applying the Bonferroni Correction?

Question

I encountered a mixed-design experiment that consisted of two independent groups and three repeated conditions. The authors conducted an analysis of variance followed by post-hoc paired-samples t-tests. Three paired-samples t-tests were computed to compare the conditions within each group (i.e. six tests in total).

In applying the appropriate Bonferroni correction, should the authors correct alpha (0.05) by dividing by three (i.e. the number of comparisons within each group) or six (i.e. the total number of post-hoc comparisons)?

Answer 1

The reason to correct $\alpha$ is chance inflation, that is the increase of the actual Type I error rate over the nominal. The overall error rate (probability of making at least one Type I error) is given by:

$p_{E} = 1-(1-\alpha)^c$

where $c$ is the total number of tests conducted.

The Bonferroni inequality states that:

$p_E \leq c\frac{\alpha}{c}=\alpha$

This means that using $\frac{\alpha}{c}$ each of the $c$ tests will ensure that $p_E \leq \alpha$.

Therefore, you correct by the total number of tests and not some other factor.