False Discovery Rate, a intuitive explanation

 

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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 $$\sharp\{falsely~say~significant\} = \pi_0 \times p \times M$$, where $\pi_0$ is the non-hypothesis probability. Then we get$$\frac{\pi_0 \times p \times M}{\sharp\{p_i \leq p\}}=\alpha$$

This gives a explanation.