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Let's say I have the done the following biological experiment:

I have 10 genetically different bacterial clones, which I have competed against a common reference clone (ancestral). I can then calculate a ratio, which tells me the competitive ability of the test clone. If this ratio is significantly above or below zero, then the test clone is competitively superior or inferior (respectively), relative to the reference.

Now, I am then carrying wilcoxon tests to ask, for each clone, whether this ratio is significantly different from 0. I have obtained my 10 p-values and I would like to know whether people think that in such a case, one should be correcting for multiple comparisons or not?

If I was comparing the ratio of all clones against each other, I would be ok with doing multiple comparisons, but in this case I am unsure.

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Depends on what you want to claim, e.g. if you want to claim that you have shown that out of the ones considered X was statistically significantly (e.g. at the 0.05 significance level) different from the control, then, yes, you need multiplicity control using some kind of testing procedure (e.g. Bonferroni-Holm, or Hochberg or some closed testing procedure using Simes etc.). Alternatively, if you might only want to say you have a finding for a false discovery rate of X%. Or you might just want to say that here is an interesting trend that may need further investigation.

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You should always correct for multiple comparisons if you do more than one comparison on the same sample. Plus, in your case, your samples are possibly paired.

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