The reasoning behind Fisher's LSD can be extended to cases beyond N=3.
I'll discuss the case of four groups in detail. To keep the familywise Type-I error rate at 0.05 or below, a multiple-comparison correction factor of 3 (i.e. a per-comparison alpha of 0.05/3) suffices, although there are six post-hoc comparisons among the four groups. This is because:
- in case all four true means are equal, the omnibus Anova over the four groups limits the familywise error rate to 0.05;
- in case three of the true means are equal and the fourth differs from them, there are only three comparisons that could potentially yield a Type-I error;
- in case two of the true means are equal and differ from the other two, which are equal to each other, there are only two comparisons that could potentially yield a Type-I error.
This exhausts the possibilities. In all cases, the probability of finding one or more p-values below 0.05 for groups whose true means are equal, stays at or below 0.05 if the correction factor for multiple comparisons is 3, and this is the definition of the familywise error rate.
This reasoning for four groups is a generalization from Fisher's explanation for his three-group Least Significant Difference method. For N groups, the correction factor, if the omnibus Anova test is significant, is (N-1)(N-2)/2. So the Bonferroni correction, by a factor of N(N-1)/2, is too strong. It suffices to use an alpha correction factor of 1 for N=3 (this is why Fisher's LSD works for N=3), a factor of 3 for N=4, a factor of 6 for N=5, a factor of 10 for N=6, and so on.