I am repeating a test on a large amount of data and FDR-correcting the p-values afterwards for multiple testing. Yet I still do not have enough power. However, I feel like it is not necessary to test a lot of cases, because I know they will not be significant (let's assume I'm right, because I'm risk-averse). Is it ok to discard these data points before testing in order to improve the overall power, or should the number of data points I dropped still be included when I do the FDR-correction?
Extension: This simple example seems obviously wrong, because we look case by case whether to discard the data or not. Other cases however might be less clear-cut, because they rely on criteria set before, but still use the data.
For example: in the case of observables that are counts, dropping all counts lower than, say 2, because we believe such counts could have happened by chance, and are unrelated to the modeling. Another example: performing the exact same test only once. Say during the multiple testing, we test whether a count of 5 and another one of 4 could have come from the same poisson distribution. Subsequent tests of 5 and 4 have a known answer so we don't include them in the FDR correction.
What I am looking for, is an open-minded yet professional statistician's answer to what is common practice in exploratory analysis of big data, namely play around with the data until we find what is the interesting signal in there, and then design some tests specific to this dataset that give the expected answer (I am of course being very simplistic here). What are accepted practices? What is a no-go? References appreciated.