# Are p values produced by p.adjust(method="fdr") actually probabilities?

I've been trying to clarify for myself how to interpret the p values produced by p.adjust with method="fdr/BH".

I'm aware of this question: https://stackoverflow.com/questions/10323817/r-unexpected-results-from-p-adjust-fdr and Multiple hypothesis testing with FDR in R - FDRtool and p.adjust and other similar discussions online, e.g. https://support.bioconductor.org/p/49864/

However, from this it's not clear how to interpret the values produced. It seems like the values this method produces are not really probabilities anymore - as are the case with other 'adjusted' p values, e.g. via Holm method. Rather, it's that if choose to reject null hypotheses where the p_fdr < X, then we are maintaining a false discovery rate < X. Is this the correct interpretation?

• Could you help us understand the implicit distinction you make between a (presumably long term, hypothetical) rate, as in "false discovery rate," and a probability?
– whuber
Dec 12, 2017 at 15:12
• Hmmm. I suppose I'm getting confused because this en.wikipedia.org/wiki/False_discovery_rate#Benjamini–Hochberg_procedure implies the BH procedure controls the familywise FDR rate provided you reject all tests which don't meet the criterion. But I wasn't clear that the values produced then related to a probability for the specific null hypothesis.
– bjw
Dec 13, 2017 at 16:40