FWER control is one strand of attack on the [[Multiple Testing]] problem. Recall the setup of multiple testing, with ![[Multiple Testing#^e1f677]] FWER is then one of the metrics we may want to control in testing: > [!definition|*] Family-wise Error Rate (FWER) > > The **family-wise error rate (FWER)** is given by $\begin{align*} \mathrm{FWER}&:= \mathbb{P}[\text{reject }H_{0} ~|~ H_{0}]\\[0.4em] &= \mathbb{P}[\text{reject any of }H_{0i} ~|~ \text{all of }H_{0i} \text{ are true}]. \end{align*}$ ### Controlling FWER The most fundamental way of controlling FWER is through the Bonferroni correction: > [!definition|*] Bonferroni's Correction > Given a size $\alpha$, **Bonferroni's global test** rejects any individual $H_{0i}$ with level $\alpha / n$ instead of $\alpha$. Under Bonferroni's correction, the FWER is $\mathrm{FWER}=\mathbb{P}_{0}\left[ \bigcup_{i}\text{reject }H_{0i} \right] \le \sum_{i}\mathbb{P}_{0}[\text{reject }H_{0i}]\le \sum_{i}\alpha / n=\alpha.$ Holm's procedure ^cc1173 ### Adjusted p-Values In a multiple testing context, consider a global test $T_{\alpha}:\mathcal{X} \mapsto \{ 0,1 \}^{n}$ (where the outcome's $i$th entry is $1$ iff $H_{0i}$ is rejected) of size $\alpha$. > [!definition|*] Adjusted p-value > The **adjusted p-value** of the test (on the $i$th hypothesis) is $\tilde{p}_{i}:=\inf \{ \alpha ~|~ T_{\alpha}(\mathbf{x}) \text{ rejects }H_{0i}\},$i.e. the lowest size at which $H_{0i}$ is not rejected.