False discovery rate calculation in target-decoy matching context

Question

A common strategy in mass spectrometry of biological molecules is to upload observed spectra to a server so that they can be matched to a LARGE database of theoretical spectra of known molecules (a.k.a. target database). In order to control for false positives, a decoy database consisting of incorrect/irrelevant spectra is used.

I have been reading more into this subject and have come up some questions regarding the calculation of the FDR measure from this target-decoy strategy. The basic idea of the FDR value is very intuitive:

$FDR = \frac{FP}{FP + TP}$

where FP and TP stands for false and true positives respectively. This makes perfect sense to me; if I'm trying to guess some peoples' names out of a phone book, and get 8 right and 2 wrong, I would have 2 false out of 10 total guesses, and thus my false discovery rate would be 20%.

However reading this tutorial on how this is done in large scale on the servers, I got introduced to two different calculations, depending on whether or not the target and decoy databases are concatenated (page 2).

I don't think that this is a typo as I found other occurrences ^* of the mysterious factor 2 in front of FP in scientific literature. However the motivation behind this is never explained (at least I couldn't find it).

I would appreciate some insight on where this doubling comes from. Likewise I wonder whether or not FDR calculation this way assumes that the error rate for each spectra match is the same for the target database and decoy database (i.e. assuming that getting 25 decoy hits implies 25 target hits are also false positives). It's not really clear for me why the error rate has to be the same for the two databases. Any comments on this subject is also appreciated.

_{* one such reference is Elias et al Nature Methods - 2, 667 - 675 (2005)}

Answer 1

I believe you multiply by 2 because you need to control for your database being twice as large. There are other ways to do this calculation such as

decoy spectra identified / target spectra identified

The use of the term FDR for this calculation is totally confusing and is why people have started calling it target/decoy rate in the last year or so. It's also doubly confusing as people often fail to specify if they are using a spectra target/decoy or a peptide target/decoy...and there is no way to consistently calculate a protein target/decoy as different programs will weigh peptide --> protein evidence differently...It's a mess...Having said that I will always make this calculation just to double check I or the software has not done something stupid. For that it is very useful.