Towards "simultaneous selective inference": post-hoc bounds on the false discovery proportion
arXiv: Statistics Theory(2018)
摘要
The false discovery rate (FDR) has become a popular Type-I error criterion for multiple testing, but it is not without its flaws. Indeed, (a) controlling the mean of the false discovery proportion (FDP) does not preclude large FDP variability, and (b) committing to an error level $q$ before observing the data limits its use in exploratory data analysis. We take a step towards addressing both of the above drawbacks by proving uniform FDP bounds for a variety of existing FDR procedures. In particular, many such procedures proceed by examining a $textit{path}$ of potential rejection sets $varnothing = mathcal R_0 subseteq mathcal R_1 subseteq cdots subseteq mathcal R_n subseteq [n]$, assigning an estimate $widehat{text{FDP}}(mathcal R_k)$ to each one, and choosing the final rejection set $mathcal R_{k^*}$ via $k^* = max{k: widehat{text{FDP}}(mathcal R_k) leq q}$. We prove that for a wide variety of such procedures (including Benjamini-Hochberg), under independent p-values, $widehat{text{FDP}}$ bounds the FDP to within a small explicit constant factor $c_{text{alg}}(alpha)$, uniformly across the entire path, with probability $1-alpha$. This constant is close to 2 for several procedures at the 95% confidence level. These bounds imply that existing FDR procedures also have FDP bounded with high probability by a small constant multiple of the target FDR level $q$. Our bounds also open up a middle ground between fully simultaneous inference and fully selective inference. They allow the scientist to $textit{spot}$ one or more suitable rejection sets (Select Post-hoc On the algorithmu0027s Trajectory) by picking data-dependent sizes or error-levels, after examining the entire path of $widehat{text{FDP}}(mathcal R_k)$ and the uniform upper band on FDP.
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