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Hervé Abdi:

The Bonferonni and Šidák Corrections

beforehand. It should be emphasized, however, that the (subjec- tive) importance of the tests and the unequal allocation of the in- dividual α[PT ] should be decided a priori for this approach to be statistically valid. An unequal allocation of the α[PT ] can also be achieved using the Šidàk inequality, but it is more computationally involved.

3 Alternatives to Bonferonni

The Šidàk-Bonferonni approach becomes very conservative when the number of comparisons becomes large and when the tests are not independent (e.g., as in brain imaging). Recently, some al- ternative approaches have been proposed (see Shaffer, 1995, for a review) to make the correction less stringent (e.g., Holm 1979, Hochberg, 1988). A more recent approach redefines the problem by replacing the notion of α[PF ] by the false discovery rate (FDR) which is defined as the ratio of the number of Type I errors by the number of significant tests (Benjamini & Hochberg, 1995).


[1] Benjamini & Hochberg, (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. Journal of the Royal Statistical Societ , Serie B, 57, 289–300. [2] Games, P.A. (1977). An improved t table for simultaneous control on g contrasts. Journal of the American Statistical Association, 72, 531–534. [3] Hochberg Y. (1988). A sharper Bonferonni procedure for multi- ple tests of significance. Biometrika, 75, 800–803. [4] Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6, 65–70. [5] Rosenthal, R. & Rosnow, R.L. (1985). Contrast analysis: focused comparisons. Boston: Cambridge University Press. [6] Shaffer, J.P. (1995). Multiple Hypothesis Testing Annual Review of Psychology, 46, 561–584.


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