A common approach to evaluating the significance of a collection of $p$-values combines them with a pooling function, in particular when the original data are not available. These pooled $p$-values convert a sample of $p$-values into a single number which behaves like a univariate $p$-value. To clarify discussion of these functions, a telescoping series of alternative hypotheses are introduced that communicate the strength and prevalence of non-null evidence in the $p$-values before general pooling formulae are discussed. A pattern noticed in the UMP pooled $p$-value for a particular alternative motivates the definition and discussion of central and marginal rejection levels at $\alpha$. It is proven that central rejection is always greater than or equal to marginal rejection, motivating a quotient to measure the balance between the two for pooled $p$-values. A combining function based on the $\chi^2_{\kappa}$ quantile transformation is proposed to control this quotient and shown to be robust to mis-specified parameters relative to the UMP. Different powers for different parameter settings motivate a map of plausible alternatives based on where this pooled $p$-value is minimized.

翻译：在评估一组$p$值的显著性时，常用方法是通过合并函数对其进行组合，尤其是在原始数据不可得的情况下。这些合并后的$p$值将一组$p$值转化为一个单一数值，其行为类似于单变量$p$值。为清晰讨论这些函数，引入了一系列伸缩性备择假设，以传达$p$值中非零证据的强度与普遍性，随后讨论通用合并公式。对特定备择假设下UMP合并$p$值中观察到的模式，激发了在$\alpha$水平上定义并讨论中心拒绝水平与边际拒绝水平。本文证明中心拒绝始终大于或等于边际拒绝，从而提出以商来度量合并$p$值中两者的平衡。基于$\chi^2_{\kappa}$分位数变换的合并函数被提出以控制该商，并证明其相对于UMP针对错误指定参数具有稳健性。不同参数设置下的不同功效，启发了一个基于合并$p$值最小化位置的可能备择假设映射图。