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A common approach to evaluating the significance of a collection of$p$-values combines them with a pooling function, in particular when theoriginal 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. Toclarify discussion of these functions, a telescoping series of alternativehypotheses are introduced that communicate the strength and prevalence ofnon-null evidence in the $p$-values before general pooling formulae arediscussed. A pattern noticed in the UMP pooled $p$-value for a particularalternative motivates the definition and discussion of central and marginalrejection levels at $\alpha$. It is proven that central rejection is alwaysgreater than or equal to marginal rejection, motivating a quotient to measurethe balance between the two for pooled $p$-values. A combining function basedon the $\chi^2_{\kappa}$ quantile transformation is proposed to control thisquotient and shown to be robust to mis-specified parameters relative to theUMP. Different powers for different parameter settings motivate a map ofplausible alternatives based on where this pooled $p$-value is minimized.

Balancing central and marginal rejection when combining independent significance tests