Comparing scientific studies (p-values, q-values) I am comparing two studies. Each study consists of an affected (cancer) and control groups of patients, for which a number of biomarkers (gut microbial species) were studied. For each biomarker univariate testing is performed (Mann-Whitney or Kruskal-Wallis) and the p-values are then adjusted using the Benjamini-Hochberg procedure (resulting in q-values).
The difficulty arises when comparing the results of the two experiments, since they contain different number of biomarkers, which only partially overlap. It seems that one cannot compare significance of the results based on the q-values, since these depend on all the biomarkers tested in each experiment.
I see two possible solutions

*

*Comparing significance of results (for common biomarkers) based on the p-values.

*Recalculating the BH correction only for the common biomarkers.

I will appreciate the input on whether these constitute valid procedures (and more generally, on the ways of carrying a comparison between two experiments).
Update
To put it in a more formal way: in an experiment $\nu$ we study a collection of biomarkers $\mathcal{A}^{(\nu)}=\{a_i^{(\nu)},i=1..M^{(\nu)}\}$. Every experiment consists of two groups of subjects, (affected and control), producing the results $x_{ij}$, $y_{ij}$ (the subjects are not paired and the sizes of the affected and control groups are not necessarily the same). Pairwise testing is then performed for every marker, so test if it is expressed differently in the two groups (null hypothesis: no difference), resulting in a vector of p-values $p_i^{(\nu)}$. The p-values are then adjusted using the Benjamini-Hochberg procedure, $\mathbf{q}^{(\nu)}=BH(\mathbf{p}^{(\nu)})$, and the decision of the faulure of the null hypothesis for each biomarker is made, respective to some threshold $\alpha$: $q_i{(\nu)}<\alpha$.
Biomarkers used in different experiments are not identical, but overlap, that is the set of common biomarkers is $\mathcal{B} = \mathcal{A}^{(1)}\cap\mathcal{A}^{(2)}$.
How can we compare the results of the two studies? (Note that the results may come from different scientific groups, different equipment, somewhat different procedures, so the data are not necessarily comparable. However, it is likely that the differences are reduced to a proportionality coefficient.)
Remark
I am not talking about comparing the p-values themselves, but rather how to judge which results are significant in both experiments, and which are not, and how to adjust for multiple testing in this context.
 A: 
rather how to judge which results are significant in both experiments, and which are not,

If one study finds significance and the other does not, it means that either one makes an alpha error or the other a beta error. The alpha error should not exceed 5% probability whilst very littls is known about the beta error except that it's probablity will depend on sample size. So if one study yields significance and the other not, that is not necessarily contradictory. p-values are random numbers and there is not much point in comparing them.

However, it is likely that the differences are reduced to a proportionality coefficient


you may assume that I have all the data used to obtain the p-values

Wouldn't it be much more worthwhile to take the original data, estimate the "proportionality" constant with it's confidence intervall for both studies and compare, how well those go together?
As far as I understand the question I'd compare the estimates of the common "proportionality factor" and not the p-(or q-)values
