Bonferroni-Holm Correction for Multiple Comparisons

Bonferroni-Holm (1979) correction for multiple comparisons. This is a sequentially rejective version of the simple Bonferroni correction for multiple comparisons and strongly controls the family-wise error rate at level alpha.

It works as follows:
1) All p-values are sorted in order of smallest to largest. m is the number p-values.
2) If the 1st p-value is greater than or equal to alpha/m, the procedure is stopped and no p-values are significant. Otherwise, go on.
3) The 1st p-value is declared significant and now the second p-value is compared to alpha/(m-1). If the 2nd p-value is greater than or equal to alpha/(m-1), the procedure is stopped and no further p-values are significant. Otherwise, go on.
4) Et cetera.

As stated by Holm (1979) "Except in trivial non-interesting cases the sequentially rejective Bonferroni test has strictly larger probability of rejecting false hypotheses and thus it ought to replace the classical Bonferroni test at all instants where the latter usually is applied."

Reference:
Holm, S. (1979) A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics. 6, 65-70.