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I'm doing a Bonferroni correction, after which our result is still significant at the 10% level.

Since my multiple outcomes are highly correlated, for which the Bonferroni is too conservative, I want to say along the line of "the statistical significance of our result is probably stronger / higher than the Bonferroni correction indicates."

However, since a result is either significant or not, it isn't 100% kosher to say it's more or less significant. So how to express the idea above correctly?

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    $\begingroup$ Just say it is significant at at least the 10% level. $\endgroup$ – user75138 Apr 27 '16 at 1:53
  • $\begingroup$ "Even worth a conservative Bonferroni correction..." is another option. $\endgroup$ – Björn Apr 27 '16 at 5:50
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Because you claim Bonferroni is too conservative, I suggest using the Holm modification. See:

Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6, 65-70.

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