Suppose we have a population of patients. For each patient, we measure 1000 features and observe if they suffer from any of 10 diseases. We wish to determine which features, if any, are significant in predicting each disease.
My question is: do we need to treat this as a single multiple hypothesis problem (1000 features $\times$ 10 diseases = 10,000 hypotheses), or can we treat it as 10 individual problems (each with 1000 hypotheses)? I'd certainly prefer the latter, since the $p$-value correction will suppress fewer terms.
It's pretty clear that to compute the family-wise error rate, we'd need to combine all the hypotheses. I was hoping that the FDR might behave differently, and allow me to analyze each disease independently.
More broadly, I would appreciate any pointers towards different statistical tool in case this general approach is off-base.
I'm treating Michael Lew's thoughtful answer below as correct, but I subsequently stumbled across a more (statistically) powerful tool for handling FDR. It seems to be very relevant for my problem. Anyone interested in this problem may find these papers helpful: