I was asked for an alternative procedure to multiple comparisons which does not require independence of the multiple tests being applied. I´ve found some readings where Bonferroni´s correction is said to not require independence, I was presented Bonferroni´s correction before but not informed of this neither able to check.

So, could anyone give me an example of such procedure and/or clarify on the independence assumptions for multiple comparisson corrections.

Thanks in advance.

PS. Sorry, in case this is a duplicated question.

**EDIT:** The answer to the possible duplicate isn't what I needed. In fact, the question there is "…what is positive regression dependency and what is non-positive regression dependency?" and the OP is, basically, concerned about the Benjamini–Hochberg procedure while I'm looking for a more general answer.

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#### Best Answer

The Bonferroni, Bonferroni-Holm and more generally closed testing procedures constructed using Bonferroni tests do not require an independence of the test statistics. Bonferroni-Holm and suitably constructured closed testing procedures will be uniformly more powerful than a simple Bonferroni procedure so that I would normally prefer them. The Simes test / Hochberg procedure require non-negative correlation between the test statistics in order to preserve one-sided familwise type I error rates. Various other procedures e.g. Dunnett or Dunnett-Tamahane require specific known correlations, which would be the case e.g. in case multiple groups are compared to a single reference group. Depending on what kind of application we are talking about, it may also be possible to estimate the correlation from the data or to do some kind of bootstrapping procedure that accounts for correlations.

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