# Multiple Test Correction: robust against many insignificant p-values

I work in bioinformatics so I've seen my fair share of null multiple tests (figure 1) As well as a clear signal in p-value distributions (figure2)

But I've also occasionally seen a type of p-value distribution with a high and low peak.

I suspect this is a characterized event but I don't know what it is called so I cant look it up. Thinking about it, it seems to confound the False Discovery Rate's (FDR) ability to estimate the background uniform distribution causing no values to survive the multiple test correction though there clearly appears to be a significant signal.

So.

1) Is there a name for this phenomenon? What causes it? Is it enough to say that there are several inappropriate tests and can for that reason be ignored? Am I justified in trimming this long insignificant tail since these tests were clearly inappropriate?

2) Is there a multiple test correction robust to this phenomenon?

3) Is a large number of extremely insignificant p-values the same as a large number of moderately insignificant p-values:

Is this a different phenomenon? Does it require a different response and/or multiple test correction in order to salvage the signal?

P-values are uniformly distributed under the null. In other words, if there is no true effect, you are equally likely to observe any p-value in the range $[0, 1]$.