Last updated: 2018-06-08
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Modified: analysis/gd_w.rmd
| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | 6c001e8 | LSun | 2018-06-08 | wflow_publish(“analysis/fre_bay_fdr.Rmd”) |
Prof. Chao Gao asked for a more detailed explanation of the connection between frequentist and Bayesian interpretations of FDR.
In Storey 2003, Storey defined a new concept \[ pFDR := E\left[\frac VR \mid R > 0\right] \] and argued that it’s a more relevant quantity than Benjamini and Hochberg 1995’s FDR, which can be written as \[ FDR := E\left[\frac VR \mid R > 0\right]P(R > 0) \]
In Storey’s Theorem 1, they gave a Bayesian interpretation of pFDR which is \[ pFDR(\Gamma) = Pr(H_T = H_0|T \in \Gamma) \] where \(\Gamma\) is the rejection region and \(T\) a test statistic.
In Efron’s series of papers, he proposed a concept “Local FDR” as \[ lfdr(t) = Pr(H_T = H_0 | T = t) \] which can be called more simply as “posterior null probability” or else.
Efron in his unpublished manuscript (pp 4-5) provided the connection between lfdr and pFDR which can be derived this way \[ \begin{aligned} pFDR(\Gamma) &= Pr(H_T = H_0|T \in \Gamma) \\ &= \frac{Pr(H_T = H_0 \quad \& \quad T \in \Gamma)}{Pr(T \in \Gamma)}\\ &= \frac{ \int_\Gamma Pr(H_T= H_0 \quad \& \quad T = t) dt }{ Pr(T \in \Gamma) }\\ &= \frac{ \int_\Gamma lfdr(t) f_T(t)dt }{ Pr(T \in \Gamma) }\\ &= \int_\Gamma lfdr(t) f(t|T \in \Gamma) dt\\ \end{aligned} \] which is essentially an average of all lfdr in the rejection region, so pFDR, as a tail quantity, can be seen as an average version of lfdr, as a local quantity, which makes sense.
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