Combining ranked lists of true and false positives gives strange result. Is there a general principle here? Apologies if this question is too general. I'm happy to edit it as needed. That said, I've come across this counterintuitive result twice now, and I'd like to know if there's a known, general solution to it.
When I work with lots of biology data, I often do some combination of:


*

*scoring a list of potential hits, e.g. "what are the most probable proteins in my sample, from most to least likely?"

*collecting multiple replicates

*combining the scored lists across replicates

*applying a multiple comparison correction, e.g. an FDR cutoff. 


The problem is that when I combine lists from multiple replicates, I sometimes find fewer hits after the multiple comparison correction. That is, counterintuitively, more data makes my results worse. I believe this is because the true positives tend to overlap between lists (they're real) while the false positives don't (they're random, spurious hits). Therefore, when combining, the false positives accumulate faster than the true positives. Here's an example:

List 1 and List 2 have three TPs and two FPs (precision = 3 / 5 = 60%), and, if we applied a 50% FDR cutoff, we'd return all five hits from each list. The combined list on the other hand has a total precision of 50%; applying a 50% FDR cutoff gives four hits. By both measures the combined list is worse. And it just looks worse, e.g. there are fewer TPs near the top.
The effect is obviously dependent on how I chose to combine lists. For one, the combined list is non-redundant (e.g. A only appears once), and two, I combined scores by taking the maximum value (instead of e.g. the average). From what I've seen both of these are common choices, although I'm pretty sure this is where my error is. My solutions so far are ad hoc, e.g. different rules for combining scores.
Question 1: Is there a general principle at work here? Like I said I've come across this twice... (there must be a thesis on it!)
Question 2: Could I combine scores in a "Bayesian way"? I don't know what that means in practice, although "accumulating evidence" makes me think "Bayesian".
 A: FDR is designed to keep the false discovery rate under the threshold you require - and I see it does, in both separate and combined lists. That the rate is slightly better in separate lists (40 instead of 50 %) could just be because of low number of total discoveries (5), I believe.
The total number of claimed hits does not seem like a good property for me, and it is certainly not the goal of FDR-based corrections. You could relax the $q$ threshold to 100 %, and all your proteins will be claimed as hits then!..
As to combining FDR qs in a more informed way: I am not qualified to comment much on this, maybe there is some ad hoc way allowing that - but I don't see the point of first applying a decision rule (filtering on FDR, p-val or whatever), and then combining the decisions. Why not combine the actual test statistics, as in the standard meta-analysis approach? Decision rules don't add any information, unless, say, the computational cost of combining results is high, and you would prefer to reduce the separate lists initially.
A: I think the "general principle" you are looking for is set theory: https://en.wikipedia.org/wiki/Set_theory
You have a set of replicates S = {A,B,C,D,E,F,G,H} which is your combined list. From S, you have two subsets L1 = {A,B,C,D,E} which is List 1 and L2 = {A,B,F,G,H} which is List 2. 
Combining List 1 and List 2, without duplicates, is called a union. This is represented as L1 ∪ L2. An intersection, L1 ∩ L2, is the set of all common elements between L1 and L2. In this example L1 ∩ L2 = {A,B}. The formula for union is (L1 ∪ L2) = L1 + L2 - (L1 ∩ L2). This means you are adding the elements from each list together, and then removing the duplicates, so that you aren't double counting. In general, when lists overlap, L1 ∪ L2 < L1 + L2. If the two lists are mutually exclusive, then L1 ∪ L2 = L1 + L2.
