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For nonempty index sets forming a laminar family of sets, reject an intersection hypothesis when the displayed adjusted value is at most $\alpha$. The inclusion-maximal true sets are disjoint and have total size at most $m$. Any false rejection forces some maximal true $J$ to have $p_J\le \alpha |J|/m$. Validity and the union bound control the familywise error rate by $\alpha$, without independence or a complete closure.

The familywise error rate is the probability of rejecting at least one true null:A valid p-value is super-uniform under its null: $\mathbb{P}(p_i\le t)\le t$ for $0\le t\le 1$. The Bonferroni correction rejects $H_i$ if $p_i\le \alpha /m$. The union bound givesThis controls the familywise error rate for every configuration of true and false nulls, without any independence assumption on the p-values.

For a nonempty index set $I\subseteq \{1,\dots ,m\}$, the intersection hypothesis $H_I={\bigcap }_{i\in I}{H}_i$ asserts that every null indexed by $I$ holds. The closure of a family of statistical hypotheses consists of all these nonempty intersections, including the original hypotheses. For each intersection choose a local test ${\varphi }_I$ of size at most $\alpha$ under $H_I$. Such tests always exist: for example, within $I$ use the Bonferroni correction, rejecting $H_I$ if ${\min }_{i\in I}{p}_i\le \alpha /|I|$. More powerful local intersection tests can be substituted whenever valid.

The closed testing procedure rejects an elementary hypothesis $H_i$ only when every local test ${\varphi }_I$ with $i\in I$ rejects its intersection hypothesis. If $I_0\ne \varnothing$, the intersection $H_{I_0}$ is true. Rejecting any true $H_i$ requires rejection by its local test ${\varphi }_{I_0}$, soIf $I_0=\varnothing$, there is no possible false rejection. This proves closed-testing control of the familywise error rate. The same argument applies if intersection hypotheses themselves are reported as rejected: requiring all their supersets to reject includes $I_0$ whenever the reported intersection is true.

Finally, consider the given hierarchical family as a laminar family of sets. Call a set $I\in \mathcal{I}$ true when $I\subseteq I_0$, so its entire intersection hypothesis is true. Let $\mathcal{M}$ be the inclusion-maximal true sets in $\mathcal{I}$. Distinct members of $\mathcal{M}$ are disjoint: if they met, laminarity would make one contain the other, contradicting maximality. Consequently,Every true set $I$ lies in a maximal true $J$. If $H_I$ is rejected by the prescribed adjusted p-value rule, the maximum over its ancestors includes $J$, henceThus the event of any false rejection is contained in the union of these events over the disjoint maximal true sets. Using validity of each $p_J$ under its true intersection and the union bound,Equivalently, with probability at least $1-\alpha$ every rejection is correct. This proves familywise error control for a laminar hypothesis family. It requires neither independence nor the presence of all possible intersections or singletons in $\mathcal{I}$. Empty index sets are excluded, as required by the denominator $|J|$; if there are no true sets the result is immediate. Adjusted values exceeding one can be truncated at one without changing decisions for $\alpha <1$.