Combining independent, arbitrarily weighted P-values: a new solution to an old problem using a novel expansion with controllable accuracy

TL;DR

This paper introduces a new, accurate, and numerically stable method for combining independent P-values with arbitrary weights, addressing issues with nearly degenerate weights and unifying existing formulas.

Contribution

The authors present a novel derivation of a generalized formula for combining P-values, including a controlled expansion for nearly degenerate weights to improve stability and accuracy.

Findings

Provides a unified formula reducing to Good's and Fisher's methods

Develops a controlled expansion for nearly degenerate weights

Ensures stable and accurate numerical computation

Abstract

Good's formula and Fisher's method are frequently used for combining independent P-values. Interestingly, the equivalent of Good's formula already emerged in 1910 and mathematical expressions relevant to even more general situations have been repeatedly derived, albeit in different context. We provide here a novel derivation and show how the analytic formula obtained reduces to the two aforementioned ones as special cases. The main novelty of this paper, however, is the explicit treatment of nearly degenerate weights, which are known to cause numerical instabilities. We derive a controlled expansion, in powers of differences in inverse weights, that provides both accurate statistics and stable numerics.