Why is controlling FDR less stringent than controlling FWER?

Question

I have read that controlling FDR is less stringent than controlling FWER, such as in Wikipedia:

FDR controlling procedures exert a less stringent control over false discovery compared to familywise error rate (FWER) procedures (such as the Bonferroni correction). This increases power at the cost of increasing the rate of type I errors, i.e., rejecting the null hypothesis of no effect when it should be accepted.

But I was wondering how it is shown to be true mathematically?

Is there some relation between FDR and FWER?

Answer 1

Indeed, @cardinal is quite right that the paper is as clear as it gets. So, for what it's worth, in case you do not have access to the paper, here's a slightly elaborated version of how Benjamini–Hochberg argue:

The FDR $Q_e$ is the expected value of the proportion of false rejections $v$ to all rejections $r$. Now, $r$ is, obviously, the sum of false and correct rejections; call the latter $s$.

In summary, (using capital letters for random variables and lowercase letters for realized values),

$$Q_e=E\left(\frac{V}{R}\right)=E\left(\frac{V}{V+S}\right)=:E\left(Q\right).$$

One takes $Q=0$ if $R=0$.

Now, there are two possibilities: either all $m$ nulls are true or just $m_0<m$ of them are true. In the first case, there cannot be correct rejections, so $r=v$. Thus, if there are any rejections ($r\geq 1$), $q=1$, otherwise $q=0$. Hence,

$$\newcommand{\FDR}{\mathrm{FDR}}\newcommand{\FWER}{\mathrm{FWER}}\FDR=E(Q)=1\cdot P(Q=1)+0\cdot P(Q=0)=P(Q=1)=P(V \geq 1)=\FWER$$

So, $\FDR=\FWER$ in this case, such that any procedure that controls the $\FDR$ trivially also controls the $\FWER$ and vice versa.

In the second case in which $m_0<m$, if $v>0$ (so if there is at least one false rejection), we obviously have (this being a fraction with also $v$ in the denominator) that $v/r\leq 1$. This implies that the indicator function that takes the value 1 if there is at least one false rejection, $\mathbf 1_{V\geq 1}$ will never be less than $Q$, $\mathbf 1_{V\geq 1}\geq Q$. Now, take expectation on either side of the inequality, which by monotonicity of $E$ leaves the inequality intact,

$$E(\mathbf 1_{V\geq 1})\geq E(Q)=\FDR$$

The expected value of an indicator function being the probability of the event in the indicator, we have $E(\mathbf 1_{V\geq 1})=P(V\geq 1)$, which again is the $\FWER$.

Thus, when we have a procedure that controls the $\FWER$ in the sense that $\FWER\leq \alpha$, we must have that $\FDR\leq\alpha$.

Conversely, having $\FDR$ control at some $\alpha$ may come with a substantially larger $\FWER$. Intuitively, accepting a nonzero expected fraction of false rejections ($\FDR$) out of a potentially large total of hypotheses tested may imply a very high probability of at least one false rejection ($\FWER$).

So, a procedure has to be less strict when only $\FDR$ control is desired, which is also good for power. This is the same idea as in any basic hypothesis test: when you test at the 5% level you reject more frequently (both correct and false nulls) than when testing at the 1% level simply because you have a smaller critical value.