# If one performs FDR corrections for multiple (independent) comparisons - what is the overall FDR?

Suppose that I perform two different comparisons on the same genomic dataset (with say 1000 genes, and three conditions).

For each comparison, I adjust the pvalues using Benjamini Hochberg, controlling the FDR. So, in each comparison, I would have say, 0.1 false positives on expectation.

Suppose that I do not control in any way for the fact that I perform two comparisons and not just one.

What would be the False Discovery Rate for the whole list of significantly changed genes for the two comparisons?

I think that if I set the threshold to be 0.1 then (for a single comparisons)

P(false positive) = 0.1

P(true positive) = 1 - 0.1

For two comparisons:

P(true positive) = (1 - 0.1)^2
P(false positive) = 1 - (1-0.1)^2

And thus the the FDR would be 1-(1-alpha)^n

But I would like some verification of that.

The Benjamini & Hochberg expected FDR does not however obey such a scaling rule exactly. The limma package (which you are probably familiar with) is specifically designed for the sort of genomic analysis you are refering to. The limma::decideTests function implements several different methods for extending FDR control across several contrasts simultaneously. Apart from the "global" method, the results are approximate and there is no exact theory supporting these calculations.