Combining test statistics from independent trials or experiments is a popular method of meta-analysis. However, there is very limited theoretical understanding of the power of the combined test, especially in high-dimensional models considering composite hypotheses tests. We derive a mathematical framework to study standard {meta-analysis} testing approaches in the context of the many normal means model, which serves as the platform to investigate more complex models. We introduce a natural and mild restriction on the meta-level combination functions of the local trials. This allows us to mathematically quantify the cost of compressing $m$ trials into real-valued test statistics and combining these. We then derive minimax lower and matching upper bounds for the separation rates of standard combination methods for e.g. p-values and e-values, quantifying the loss relative to using the full, pooled data. We observe an elbow effect, revealing that in certain cases combining the locally optimal tests in each trial results in a sub-optimal {meta-analysis} method and develop approaches to achieve the global optima. We also explore the possible gains of allowing limited coordination between the trial designs. Our results connect meta-analysis with bandwidth constraint distributed inference and build on recent information theoretic developments in the latter field.

翻译：将来自独立试验或实验的检验统计量进行组合是元分析中常用的一种方法。然而，对于组合检验的检验功效，特别是针对高维模型中的复合假设检验，目前理论认识非常有限。我们建立了一个数学框架，以研究在众多正态均值模型背景下的标准元分析检验方法，该模型作为探索更复杂模型的平台。我们提出了对局部试验的元级组合函数的一种自然且温和的限制条件。这使得我们能够从数学上量化将m个试验压缩为实值检验统计量并加以组合的成本。随后，我们推导了标准组合方法（例如基于p值和e值）在分离速率上的极小极大下界和匹配上界，量化了相对于使用完整合并数据的损失。我们观察到一种肘形效应，揭示了在某些情况下，组合每个试验中的局部最优检验会导致次优的元分析方法，并提出了实现全局最优的方法。我们还探讨了允许试验设计之间进行有限协调可能带来的收益。我们的研究将元分析与带宽约束分布式推断联系起来，并借鉴了后一领域近期信息论的发展成果。