A list can have a controlled expected fraction of false discoveries while a shorter…

A list can have a controlled expected fraction of false discoveries while a shorter list extracted from it does not. Removing discoveries changes the denominator as well as the scientific story being reported. There are four null hypotheses. Under the configuration studied here, \(H_1,H_2\) are false and have p-values identically zero. The hypotheses \(H_3,H_4\) are true and have mutually independent \(\operatorname{Uniform}(0,1)\) p-values \(U,V\). These true-null p-values are also independent of the first two values. Apply the Benjamini–Hochberg procedure at a fixed level \(q\in(0,1)\) to all four p-values: if the ordered values are \(P_{(1)}\le\cdots\le P_{(4)}\), set \[ \widehat k=\max\left\{k:P_{(k)}\le\frac{kq}{4}\right\} \] and reject the first \(\widehat k\) hypotheses. The two zero p-values ensure that \(\widehat k\ge2\). For a reported set \(R\), let \[ \operatorname{FDP}(R)=\frac{|R\cap\{3,4\}|}{\max\{|R|,1\}}, \qquad \operatorname{FDR}(R)=E\{\operatorname{FDP}(R)\}. \] Derive the probabilities that the full procedure rejects zero, one or two true nulls, and calculate its FDR directly. A report then retains only rejected hypotheses whose labels belong to the prespecified set \(\{3,4\}\). This label filter is fixed before observing the p-values; it does not consult which hypotheses are actually true. Calculate the filtered report's FDR. Evaluate both the full and filtered FDRs at \(q=0.10\). Compare this filtering operation with applying a new two-hypothesis Benjamini–Hochberg procedure, at level \(q\), directly to \(U,V\). Calculate the FDR of that procedure. Explain what the example establishes about inheriting a full-list guarantee, and why the fixed-label description does not require an analyst who knows the truth of the hypotheses.

The two zero p-values occupy the first two ranks. Write the remaining step-up inequalities in terms of the smaller and larger uniform p-values.