Bonferroni Correction

The Bonferroni correction is a multiple-comparison correction used when several independent Statistical Tests are being performed simultaneously (since while a given Alpha Value $\alpha$ 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 $n$ comparisons equal to $\alpha$ by taking the Alpha Value for each comparison equal to $\alpha/n$. Explicitly, given $n$ tests $T_i$ for hypotheses $H_i$ ($1\leq i\leq n$) under the assumption $H_0$ that all hypotheses $H_i$ are false, and if the individual test critical values are $\leq\alpha/n$, then the experiment-wide critical value is $\leq\alpha$. In equation form, if

$$P(T_i {\rm\ passes\ } \vert H_0) \leq {\alpha\over n}$$

for $1\leq i\leq n$, then

$$P({\rm some\ } T_i{\rm\ passes\ } \vert H_0) \leq \alpha,$$

which follows from Bonferroni's Inequality.

Another correction instead uses $1-(1-\alpha)^{1/n}$. While this choice is applicable for two-sided hypotheses, multivariate normal statistics, and positive orthant dependent statistics, it is not, in general, correct (Shaffer 1995).

See also Alpha Value, Hypothesis Testing, Statistical Test

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.nottingham.ac.uk/~mhzmd/life.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.