We introduce a novel meta-analysis framework to combine dependent tests under a general setting, and utilize it to synthesize various microbiome association tests that are calculated from the same dataset. Our development builds upon the classical meta-analysis methods of aggregating $p$-values and also a more recent general method of combining confidence distributions, but makes generalizations to handle dependent tests. The proposed framework ensures rigorous statistical guarantees, and we provide a comprehensive study and compare it with various existing dependent combination methods. Notably, we demonstrate that the widely used Cauchy combination method for dependent tests, referred to as the vanilla Cauchy combination in this article, can be viewed as a special case within our framework. Moreover, the proposed framework provides a way to address the problem when the distributional assumptions underlying the vanilla Cauchy combination are violated. Our numerical results demonstrate that ignoring the dependence among the to-be-combined components may lead to a severe size distortion phenomenon. Compared to the existing $p$-value combination methods, including the vanilla Cauchy combination method, the proposed combination framework can handle the dependence accurately and utilizes the information efficiently to construct tests with accurate size and enhanced power. The development is applied to Microbiome Association Studies, where we aggregate information from multiple existing tests using the same dataset. The combined tests harness the strengths of each individual test across a wide range of alternative spaces, %resulting in a significant enhancement of testing power across a wide range of alternative spaces, enabling more efficient and meaningful discoveries of vital microbiome associations.

翻译：本文提出了一种新颖的元分析框架，用于在一般设置下组合依赖检验，并将其应用于合成基于同一数据集计算的多种微生物组关联检验。我们的工作建立在经典的聚合$p$值的元分析方法以及一种较新的置信分布组合通用方法之上，但对其进行了推广以处理依赖检验。所提出的框架确保了严格的统计保证，我们对其进行了全面研究，并与现有的多种依赖组合方法进行了比较。值得注意的是，我们证明了广泛使用的依赖检验柯西组合方法（本文中称为朴素柯西组合）可被视为我们框架中的一个特例。此外，该框架还提供了一种解决朴素柯西组合所基于的分布假设被违反时的问题的方法。数值结果表明，忽略待组合成分之间的依赖性可能导致严重的规模失真现象。与现有的$p$值组合方法（包括朴素柯西组合方法）相比，所提出的组合框架能够准确处理依赖性，并有效利用信息来构建具有准确规模和增强功效的检验。该发展被应用于微生物组关联研究，其中我们使用同一数据集聚合了来自多种现有检验的信息。组合后的检验发挥了每种检验在广泛备择空间中的优势，从而显著提升了检验功效，使得更高效、更有意义地发现重要的微生物组关联成为可能。