Benjamini-Hochberg leave-two-out identity ($R^{(ij)}=R_{-ij}+2$)

Replace two p-values $P_i,P_j$ by zeros in the Benjamini-Hochberg procedure at level $\alpha$, keeping the original denominator $m$, and let $R^{(ij)}$ be the resulting rejection count. Equivalently, remove the two values and apply the step-up critical values $\alpha (k+2)/m$ to the remaining ordered p-values; their rejection count is $R_{-ij}=R^{(ij)}-2$. For $r\ge 2$,If both original values are rejected, lowering them changes no ordered rank greater than $r$, so the rejection count stays $r$. Conversely, if the modified count is $r$ and both original values meet the rank-$r$ threshold, reinserting them leaves at least $r$ qualifying values. Monotonicity under lowering gives at most $r$ original rejections. This identity is deterministic and does not require independence or a distributional assumption.

For independent p-values, with $m_0$ true-null values independent and uniform on $[0,1]$, let $R^{(i)}$ denote the Benjamini-Hochberg procedure rejection count after replacing one true-null value by zero. The Benjamini-Hochberg leave-one-out identity givesFor distinct true-null indices, the Benjamini-Hochberg leave-two-out identity givesIndeed, conditional on the remaining values and $R^{(ij)}=r$, the two independent uniform values both meet their threshold with probability $(\alpha r/m{)}^2$, cancelling the denominator $r^2$. Expanding the square of the false rejection count into its diagonal terms and ordered distinct pairs, and using symmetry among the true-null values, provesIf no null is true, both sides are zero; if exactly one is true, the pair contribution is absent. This formula complements the exact false discovery rate $\alpha m_0/m$ by describing second-moment variability.