【转载请注明出处】http://www.cnblogs.com/mashiqi
Today let's talk about a intuitive explanation of Benjamini-Hochberg Procedure. My teacher Can told me this explanation.
Suppose there are $M$ hypothesis:$$H_1,H_2,cdots,H_M$$and corresponding $M$ p-values:$$p_1,p_2,cdots,p_M$$Let's suppose $p_i$ are in ascending order: $p_1 leq p_2 leq cdots leq p_M$ for convenience. Now we want to let the FDR to be a positive scale, say $alpha$, then what is the threshold value $p$ that can be used to reject hypotheses.
We know that the Benjamini-Hochberg Procedure is like this: let $k$ be the largest i for which $p_i leq frac{i}{M} alpha$, then reject all $H_i,~i=1,2,cdots,k$.
We wants to ask why this above gives the FDR at $alpha$? Let's consider a probability $p$, the threshold value. If we reject all $H_i$ thich satisfy corresponding $p_i leq p$, then the FDR is at $alpha$. But how do we get the value of $p$? Let's take a look at the exact definition of False Discovery Rate:$$FDR = E[frac{ sharp{falsely~say~significant} }{sharp{say~significant}}]$$
The $$sharp{say~significant} = sharp{p_i leq p}$$. If the $H_i$ is null, then $p_i$ will be uniformly distributed,so $$sharp{falsely~say~significant} = pi_0 imes p imes M$$, where $pi_0$ is the non-hypothesis probability. Then we get$$frac{pi_0 imes p imes M}{sharp{p_i leq p}}=alpha$$
This gives a explanation.