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The two-tailed version of the Mann Whitney U test calculates the $U_{stat}$ value for both samples and then selects the lowest one. This is just performing two one-tailed tests but only keeping the best result. Also all the critical value tables double the alpha value for the two-tailed version of the test, which is what would be done if Bonferroni correction was used on two tests. Am I correct in saying that a two-tailed Mann Whitney U test equal to two one-tailed U tests with Bonferroni correction? Or have I made a mistake?

crit value table

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It's not explicitly a Bonferroni correction, but it's a related notion.

Bonferroni is an attempt to bound familywise error when we don't know what the dependence between the tests is; it's roughly correct for independent tests and low significance levels.

In this case it's is not a bound but an exact computation, and it's more productive to think of it not as multiple testing but as simply a convenient way of exploiting the symmetry of the null distribution of the test statistic.

With a symmetric two-tailed test, you can flip about the center (map both tails to either an upper or lower tail) and then test only one tail at half the tail probability.

This is just the same idea as with a two-tailed t-test where you look up $|t|$ in the upper tail rather than needing to have tables for both tails.

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  • $\begingroup$ "In this case it's is not a bound but an exact computation" what does this mean? $\endgroup$
    – Nathan
    May 14, 2018 at 7:25
  • $\begingroup$ With Bonferroni correction for multiple testing, you have separate tests; each test might reject or not reject at the same time as another. So the Bonferroni calculation is approximate; it places a bound on the overall type I error rate, but the true overall type I error rate will generally be lower. With the two tails of a Wilcoxon-Mann-Whitney, you can only be in one tail at a time so adding the two tail probabilities gives the exact significance level of the two tailed test. $\endgroup$
    – Glen_b
    May 14, 2018 at 17:42

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