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，常见方法是通过合并函数将其组合，尤其是在原始数据不可获取的情况下。这些合并后的 $p$-值将 $p$-值样本转化为单一数值，其行为类似于单变量 $p$-值。为清晰讨论这些函数，引入一系列嵌套替代假设，用以传达 $p$-值中非零证据的强度和普遍性，随后讨论一般合并公式。在特定替代假设下，UMP合并 $p$-值中观察到的模式，激发了对 $\alpha$ 水平下中心拒绝与边缘拒绝水平的定义与讨论。证明中心拒绝始终大于或等于边缘拒绝，由此引入商数以衡量二者在合并 $p$-值中的平衡。基于 $\chi^2_{\kappa}$ 分位数变换的合并函数被提出以控制该商数，并证明其相对于UMP对误设参数具有稳健性。不同参数设置下的不同幂次，催生了基于该合并 $p$-值最小化位置的合理替代假设图谱。