**Holm-Bonferroni corrections** are applied **within comparison families** in the paper's **supplementary tables**, controlling the **family-wise error rate** across the multiple paired comparisons reported per family. The main paper text reports per-comparison results without folding the correction into the headline numbers, so the supplement is where multiple-testing-adjusted significance is read off.

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The **Holm-Bonferroni step-down procedure**, introduced by Sture Holm in 1979, is a sequentially rejective multiple-testing method that strongly controls the family-wise error rate at level $$\alpha$$ while being uniformly more powerful than the single-step Bonferroni correction. Given $$m$$ hypotheses with p-values $$p_1, \dots, p_m$$, sort them in ascending order as $$p_{(1)} \leq p_{(2)} \leq \dots \leq p_{(m)}$$ with corresponding hypotheses $$H_{(1)}, \dots, H_{(m)}$$. For $$i = 1, 2, \dots, m$$ in turn, compare $$p_{(i)} \leq \frac{\alpha}{m - i + 1}$$. If the inequality holds, reject $$H_{(i)}$$ and proceed to step $$i+1$$. At the first index $$i$$ where the inequality fails, stop and retain $$H_{(i)}, \dots, H_{(m)}$$. The thresholds increase from $$\alpha/m$$ at step 1 to $$\alpha$$ at step $$m$$, so any hypothesis rejected by Bonferroni is also rejected by Holm, but Holm can reject additional hypotheses once earlier ones have been rejected. The procedure controls $$\mathrm{FWER} \leq \alpha$$ strongly under any joint distribution of the test statistics.

Holm-Bonferroni Step-Down Procedure

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Canonical paper URL: https://1iman.com/assets/working-papers/pdf/emnlp-auditable-strict-parity-evaluation-of-prerequisite-graph-retrieval-for-rag-under-leakage-controls.pdf
Portfolio PDF URL: https://1iman.com/assets/working-papers/pdf/emnlp-auditable-strict-parity-evaluation-of-prerequisite-graph-retrieval-for-rag-under-leakage-controls.pdf
Paper title: Auditable Strict-Parity Evaluation of Prerequisite-Graph Retrieval for RAG under Leakage Controls
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