Karl Pearson's meta-analysis revisited

TL;DR

This paper revisits Pearson's 1934 meta-analysis method, clarifies its admissibility, compares its power to Fisher's test, and introduces an FFT-based approach for bounding the CDF of sums of nonnegative variables.

Contribution

It corrects the historical misunderstanding about Pearson's method, establishes its admissibility, and demonstrates its practical advantages and applications.

Findings

Pearson's method is admissible and has better power in certain scenarios.

It is particularly effective when nonzero parameters share the same sign.

An FFT-based technique for bounding the CDF of sums of nonnegative variables is introduced.

Abstract

This paper revisits a meta-analysis method proposed by Pearson [Biometrika 26 (1934) 425--442] and first used by David [Biometrika 26 (1934) 1--11]. It was thought to be inadmissible for over fifty years, dating back to a paper of Birnbaum [J. Amer. Statist. Assoc. 49 (1954) 559--574]. It turns out that the method Birnbaum analyzed is not the one that Pearson proposed. We show that Pearson's proposal is admissible. Because it is admissible, it has better power than the standard test of Fisher [Statistical Methods for Research Workers (1932) Oliver and Boyd] at some alternatives, and worse power at others. Pearson's method has the advantage when all or most of the nonzero parameters share the same sign. Pearson's test has proved useful in a genomic setting, screening for age-related genes. This paper also presents an FFT-based method for getting hard upper and lower bounds on the CDF of a sum of nonnegative random variables.