Optimal multiple testing under family-wise error control: elementary symmetric polynomials and a scalable algorithm
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| Format: | Preprint |
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2026
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| _version_ | 1866908957862264832 |
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| author | Dubey, Prasanjit Huo, Xiaoming |
| author_facet | Dubey, Prasanjit Huo, Xiaoming |
| contents | Simultaneously testing $K$ hypotheses while controlling the family-wise error rate is a fundamental problem in statistics. Existing procedures (Bonferroni, Holm, Hochberg, Hommel) provide valid control but sacrifice power, increasingly so as $K$ grows, because they base decisions on marginal $p$-value ranks rather than the joint likelihood. Rosset et al. (2022) formulated the most powerful family-wise-error-rate-controlling test as a dual program and proved the existence of an optimal dual vector $μ^*$, but left its computation as an open problem. We solve this problem for $K$ exchangeable hypotheses. The key insight is that the family-wise error rate constraint coefficients $b_{l,k}(\vec{u})$ admit closed-form expressions through elementary symmetric polynomials of the likelihood-ratio values $g(u_1), \ldots, g(u_K)$. This algebraic structure implies a global monotonicity theorem: the target functions $F_γ(μ) = {\rm FWER}_γ(\vec{D}^μ)$ are simultaneously non-increasing in every component of $μ$, for arbitrary $K$, which guarantees unique coordinate-wise roots and enables a bisection-based coordinate-descent algorithm with $O(\log \varepsilon^{-1})$ convergence rate. The relative power gain over Hommel's method grows from 15\% at $K{=}3$ to 84\% at $K{=}12$. Applications to replication studies, a clinical trial, and a replicability assessment illustrate both the power gains and the role of the exchangeability assumption. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_10986 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal multiple testing under family-wise error control: elementary symmetric polynomials and a scalable algorithm Dubey, Prasanjit Huo, Xiaoming Methodology Computation Simultaneously testing $K$ hypotheses while controlling the family-wise error rate is a fundamental problem in statistics. Existing procedures (Bonferroni, Holm, Hochberg, Hommel) provide valid control but sacrifice power, increasingly so as $K$ grows, because they base decisions on marginal $p$-value ranks rather than the joint likelihood. Rosset et al. (2022) formulated the most powerful family-wise-error-rate-controlling test as a dual program and proved the existence of an optimal dual vector $μ^*$, but left its computation as an open problem. We solve this problem for $K$ exchangeable hypotheses. The key insight is that the family-wise error rate constraint coefficients $b_{l,k}(\vec{u})$ admit closed-form expressions through elementary symmetric polynomials of the likelihood-ratio values $g(u_1), \ldots, g(u_K)$. This algebraic structure implies a global monotonicity theorem: the target functions $F_γ(μ) = {\rm FWER}_γ(\vec{D}^μ)$ are simultaneously non-increasing in every component of $μ$, for arbitrary $K$, which guarantees unique coordinate-wise roots and enables a bisection-based coordinate-descent algorithm with $O(\log \varepsilon^{-1})$ convergence rate. The relative power gain over Hommel's method grows from 15\% at $K{=}3$ to 84\% at $K{=}12$. Applications to replication studies, a clinical trial, and a replicability assessment illustrate both the power gains and the role of the exchangeability assumption. |
| title | Optimal multiple testing under family-wise error control: elementary symmetric polynomials and a scalable algorithm |
| topic | Methodology Computation |
| url | https://arxiv.org/abs/2604.10986 |