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I have a three dimensional table of size $6\times6\times81$. Each cell of the table is a hypothesis test. Slicing the table on the third dimension produces $81$ sets of hypothesis tests which are independent between sets but dependent within sets. Originally I was thinking that I could just control the false discovery rate using the Benjamini-Hochberg procedure on all hypothesis tests simultaneously. Is that a reasonable way to attack this problem? My second thought is to control the false discovery rate within each slice along the third dimension of the table, then apply some other sort of correction after that. Does anyone have more information on this sort of procedure?

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There is no universal answer to your question. Global B-H would control the FDR over all the 24516 hypotheses. B-H within each of the 81 sets, will give you FDR control within each slice, but no overall guarantees. If you want both within slice and overall FDR control, have a look at this paper.

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