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'?

  • $\begingroup$ I'm confused. Did you mean to write $|Q_r(d)| = r$ instead of $= d$? $\endgroup$
    – shabbychef
    Commented May 28, 2011 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
    Commented May 29, 2011 at 17:50

1 Answer 1


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.