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Suppose we have a set $S$ consisting of $p$ features, and a subset $S_+$ of the features are positive. If $Q$ is any subset of $S$, define the false positive rate as the proportion of features in $Q$ which are not positive:

$$FPR[Q] = 1 - \frac{|Q \cap S_+|}{|Q|}$$

where $|\cdot|$ denotes cardinality. If $Q$ is a function of the data, $d$, then we can define the false discovery rate as the expected false positive rate:

$$FDR[Q(d)] = \mathbb{E}_d[FPR[Q(d)]].$$

Now suppose that I have a method for ranking the features in $S$ by likelihood of significance. I will report the top $r$ features most likely to be significant, based on my data, $Q_r(d)$. Formally, I have a family of set-valued functions

$$Q_1(d) \subset Q_2(d) \subset Q_3(d) \subset \cdots \subset Q_p(d)$$

where $|Q_r(d)| = r.$

What I want to know is the maximum $r$ such that the set $Q_r$ has a false discovery rate less than a certain critical value, $q_{crit}$. That is, I want to know what is the value

$$IFDR_{q_{crit}} = \max_r \{r \in \{1,...,d\}: FDR[Q_r] \leq q_{crit}\}$$

Is there a name for this 'inverse false discovery rate' function? If not, can you suggest a name better than 'inverse false discovery rate'?

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  • $\begingroup$ I'm confused. Did you mean to write $|Q_r(d)| = r$ instead of $= d$? $\endgroup$ – shabbychef May 28 '11 at 21:18
  • $\begingroup$ It is not obvious that $FDR[Q_r]$ is monotone. It might be for specific choices of ranking procedures, but it does not follow from the set inclusions alone. Thus for the value of $r$ you are asking about there is no general guarantee that FDR is smaller than $q_{\mathit{crit}}$ for smaller values of $r$. Are you really interested in this theoretical quantity? $\endgroup$ – NRH May 29 '11 at 17:50
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You are trying to find the 'rejection region' for a given $q_{crit}$. ($q_{crit}$ is typically referred to as $\alpha$ in the literature.)

Further Reading:
Storey, JD "A direct approach to false discovery rates" J. R. Statist. Soc. (2002) www.genomine.org/papers/directfdr.pdf

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