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