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In a study in which I analyze several biomarkers through logistic and linear regressions, should I discuss in the discussion of the paper the results with p <0.05 (they only have a nominal significance) that are not significant after the Bonferroni correction?

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I don't think so, because Bonferroni Correction is just trying to solve this exact issue. In multiple experiments, you might possibly encounter with rare events and correcting your significance level corrects your interpretation as well. While making an experiment, we are assuming that we could have rare events with $0.05$ probability; but if we are making $100$ experiments, then probability of no rare events become $0.95^{100}$, which is very small. But, if we assume a significance level $\frac{0.05}{100}$ and find $0.9995^{0.0005} \approx 0.9512$, we see that we made nearly the same decision for a cohort of experiments. So, as a result, your cases that are still significant after adjusting the significance threshold as $\frac{\alpha}{m}$ are actually important, not the others.

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This depends on the underlying scientific questions for each biomarker and how they are related. If they are of very different kinds and give different information there is no need for any correction for multiplicity. Suppose for example that you were looking at HBA1C, middle upper arm circumference and self-reported quality of life (using biomarker in a wide sense) then you would not adjust. If the biomarkers could potentially be just different ways of saying the same thing then you probably would adjust.

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