Is there an optimal way to correct for multiple comparisons where tests depend on the significance of other tests? Suppose I have a set of tests, represented by their (uncorrected) p-values: $T = \{p_1, \dots,p_m \}$.
These could be, for example, p-values from $m$ ANOVAs, performed on linear regressions with $d$ coefficients each.
The underlying linear regressions also have, each, a set of $d$ p-values attached to them (if I want to test for the coefficients, for example): $S_i=\{p'_1,...,p'_d\}$. These are the effects of interest.
To keep the number of test for a minimum (and optimize power), I would like to only assess significance of the sets $S_i$ that had significant $p_i$.
How can I correctly perform the correction for multiple comparisons in this scenario, for either FWER or FDR?
It's reminiscent of step-down procedures I guess, where correction of further p's depends on the significance of earlier ones. I also see relations to post-hoc pairwise-testing procedures, but I couldn't find anything related to this.

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*Should I correct for multiple comparisons the initial, omnibus tests, $T$, and then correct only the corresponding $S_i$'s? (this increases power, but I'm not sure it keeps FWER/FDR under control, because I'm not taking into account the $T$'s when correcting $S$'s)

*Should I jointly correct $S$ and $T$ before assessing significance? (this surely decreases power though, because the number of tests increases this way)

*Should I not correct $T$, and then correct subtests in the $S_i$'s corresponding to the significant $T$'s? (This is similar to #1)

*Any other alternatives?

 A: It came to my attention this type of procedure has been described elsewhere.
You'll find it under two-step, hierarchical, hypothesis testing.
I'll reproduce the method described in Li and Ghosh, 2014.
Given $m$ sets of hypotheses $\left\{H(1),\ldots,H(m)\right\}$ which we want to test simultaneously.
In each set, there's a screening hypothesis, denoted by the $H_0(i)$ null-hypothesis.
The other hypothesis in the set will only be tested if the first one is rejected $H_0(i)$.
The procedure consists of:

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*Correct for multiple comparisons (Benjamini-Hochberg FDR, in the study) the p-values associated with all screening tests and store the number of rejections $R$.

*For each set where a rejection occurred, control for the family-wise error rate at the $R\alpha/m$ level.

Authors posit that this simple procedure controls the Overall FDR (OFDR) across sets of hypothesis.
However, the procedure requires that individual non-screening hypothesis in each set are independent of all screening hypothesis in other sets.

Li, Yihan, and Debashis Ghosh. “A Two-Step Hierarchical Hypothesis Set Testing Framework, with Applications to Gene Expression Data on Ordered Categories.” BMC Bioinformatics 15, no. 1 (2014). https://doi.org/10.1186/1471-2105-15-108.
