Family-Wise Error Rate

Definition

Consider a result of Multiple Testing

fact \ test result	do not reject $H_0$	reject $H_0$	total
$H_0$ is true	$U$	$V$	$m_0$
$H_1$ is true	$T$	$S$	$m_1$
total	$m-R$	$R$	$m$
where $R=V+S$ is the number of rejected null hypothesis, and $m$ is the total number of hypotheses tested.

$FWER=P(V\ge 1|m_0>0)$

The family-wise error rate (FWER) is the probability of making at least one type 1 error in the family Therefore, by assuring $FWER=P(V\ge 1|m_0>0)\le \alpha$, the probability of making one or more type 1 errors in the family is controlled at level $\alpha$