= Solution
A procedure controls the <familywise error rate> at level $\alpha$ when, for every configuration of true nulls $I$,
$$
\mathbb P(\text{at least one }H_i, i\in I,
\text{ is rejected})\leq\alpha.
$$
Let $i_*=\min I$. The step-down procedure can reject any true null only if it reaches and rejects $H_{i_*}$, which requires $p_{i_*}\leq\alpha$. Since a valid <p-value> under its null satisfies $\mathbb P(p_{i_*}\leq\alpha)\leq\alpha$, the procedure controls FWER without any assumption on dependence among the p-values.
A distribution $P$ has the <global Markov property for a directed acyclic graph> $G$ when every <D-separation> statement $A\mathrel{\perp_G}B\mid C$ implies the corresponding <conditional independence> $Z_A\perp\!\!\!\perp Z_B\mid Z_C$ under $P$.
If $G\in\mathcal S$ has nonadjacent vertices $j,k$, choose a <topological ordering>. Suppose $j$ precedes $k$. Then $j$ is a non-descendant and non-parent of $k$, so the directed local Markov property gives
$$
Z_j\perp\!\!\!\perp Z_k\mid Z_{\operatorname{pa}_G(k)}.
$$
The reversed ordering case is analogous. Hence
$$
H_0\subseteq\bigcup_{S\subseteq[p]\setminus\{j,k\}}H_S.
$$
Use the <intersection-union test>: reject $H_0$ exactly when $p_S\leq\alpha$ for every $S$. Under $H_0$, at least one $H_{S_*}$ is true, so
$$
\mathbb P(\text{false rejection of }H_0)
\leq\mathbb P(p_{S_*}\leq\alpha)\leq\alpha.
$$
This is nontrivial because it rejects whenever every tested conditional independence is rejected.
Back to article page
