Bonferroni Correction

The Bonferroni correction is a multiple-comparison correction used when several dependent or independent statistical tests are being performed simultaneously (since while a given alpha value  may be appropriate for each individual comparison, it is not for the set of all comparisons). In order to avoid a lot of spurious positives, the alpha value needs to be lowered to account for the number of comparisons being performed.

The simplest and most conservative approach is the Bonferroni correction, which sets the alpha value for the entire set of  comparisons equal to  by taking the alpha value for each comparison equal to . Explicitly, given  tests  for hypotheses  () under the assumption  that all hypotheses  are false, and if the individual test critical values are , then the experiment-wide critical value is . In equation form, if

for , then

which follows from the Bonferroni inequalities.

REFERENCES:

Bonferroni, C. E. "Il calcolo delle assicurazioni su gruppi di teste." In Studi in Onore del Professore Salvatore Ortu Carboni. Rome: Italy, pp. 13-60, 1935.

Bonferroni, C. E. "Teoria statistica delle classi e calcolo delle probabilità." Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze 8, 3-62, 1936.

Dewey, M. "Carlo Emilio Bonferroni: Life and Works." http://www.aghmed.fsnet.co.uk/bonf/bonf.html.

Miller, R. G. Jr. Simultaneous Statistical Inference. New York: Springer-Verlag, 1991.

Perneger, T. V. "What's Wrong with Bonferroni Adjustments." Brit. Med. J. 316, 1236-1238, 1998.

Shaffer, J. P. "Multiple Hypothesis Testing." Ann. Rev. Psych. 46, 561-584, 1995.