Is it correct to do pre-filtering before multiple tests? I often see that authors of large-scale studies that theoretically have to perform multiple test correction for enormous amount of tests do pre-filtering, in other words - throw away all tests that does not look good from "the first sight" (eg, comparison of 10000 features between two groups - researchers may throw away all features with big variances or small variances before doing any tests).
Eg, one good and surely correct example of pre-filtering: Tarone's trick, eg on 7th slide here: https://www.ethz.ch/content/dam/ethz/special-interest/bsse/borgwardt-lab/documents/talk_slides/KBorgwardt_UniBasel_20160421.pdf . 
But the correctness of pre-filtering is sometimes not so obvious in other studies. Sometimes it literally looks like "we filter p-values that are greater than 0.5" or close to it, "majority of examples we filter out most probably have large p-values" - but in other words, they may use "effect size" instead of "p-value" (while applying t-test), and so on.
The question is: when pre-filtering of some tests before FDR control procedure is correct, and when it is cheating? 
UPD: I found this paper that obviously address the issue, but only started to read it.
 A: You've misunderstood the recommendation of Tarone's method. Read his paper yourself. Tarone pointed out that the $p$-value for discrete analyses has a theoretical lower bound. For instance, if one measures the following $2 \times 2$ contingency table relating an allele pattern to a disease status:
$$ \begin{array}{rcc} 
 & \text{Cancer} & \text{Disease free} \\ \hline
\text{aa} & 4 & 0 \\
\text{aA or AA} & 0 & 4 \\
\end{array}$$
The table demonstrates the largest possible association that this design can achieve: an odds ratio of $\infty$ but a two-sided p-value of 0.02857 = 2/70 where 70 is the number of permutations of tables 8 patients having 4 diseased and 4 disease free and 4 homozygous recessive genotypes and 4 heterozygous or homozygous dominant genotypes. $p = 0.02857$ is the smallest achievable $p$-value in this case. If two or more comparisons are made, the significance level of 0.025 is theoretically impossible to achieve. 
Tarone's method immediately discards such tests not on the basis of being statistically non-significant but on the basis of lacking precision. This decision was based only on the actual marginal frequencies (4/4 by 4/4). This was sufficient to discredit the comparison.
