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What is different about the q-value and local FDR when both are defined as posterior probabilities of the null being true?

For example in Storey (2010),

Under these modeling assumptions, it follows that q-value(pi) = mint≥pi Pr(Hi = 0|Pi ≤ t), which is a Bayesian analogue of the p-value – or rather a “Bayesian posterior Type I error rate.”

But the definition of the local FDR given just a few lines later seems quite the same

This connects the pFDR to the posterior error probability Pr(Hi = 0|Pi = pi), making this latter quantity sometimes interpreted as a local false discovery rate (Efron et al. 2001, Storey 2001).

I can see the mathematical definitions are different, but what's different conceptually?

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q-value is a tail probability as you can see by your definition, it is the minimum pFDR in which you will still reject the corresponding hypothesis.

pFDR is defined as $E(V / R | R > 0)$, which I encountered also as Bayesian FDR.

The local FDR ($fdr$) is the probability that the hypothesis comes from the null at a specific value of the statistic.

The connection between the two is the following:

$pFDR(z) = P(H_i = 0| R_i = 1) = E_{Z|R_i=1}P(H_i=0|Z)=E_{Z|R_i=1}fdr(Z_i)$

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  • $\begingroup$ Keep in mind that any frequentist-based method that uses null hypotheses is going to suffer from higher type I and type II errors when multiplicity adjusting than can be achieved with simple likelihood ratios. Most multiplicity errors come from using null hypotheses instead of demanding that differences be non-trivial (i.e, using null hypotheses where something greater than zero is hypothesized). See statisticalevidence.com $\endgroup$ – Frank Harrell Feb 4 '18 at 13:08
  • $\begingroup$ @FrankHarrell I agree. I find a parameter estimation approach and aiming to reduce Type M and Type S errors --- the underlying cause of classical Type I and Type II errors. I find Bayesian modeling to be ideal for this, and if I do a hypothesis test, Bayes Factors are preferable. Nevertheless I like to understand other perspectives and FDR seems far more agreeable within a frequentist context than FWER. $\endgroup$ – BKV Feb 5 '18 at 2:16
  • $\begingroup$ Just note that likelihood ratios are much simpler than Bayes factors here, and work extremely well. But I prefer global Bayesian modeling if computational issues not too severe. $\endgroup$ – Frank Harrell Feb 5 '18 at 12:42

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