Abstract

Combining $p$-values to integrate multiple effects is of long-standing interest in social science and biomedical research. In this paper, we revisit a classical scenario closely related to meta-analysis with unknown heterogeneity, {which combines finite and fixed number of $p$-values while the sample size for generating each $p$-value can go to infinity.} Many modified Fisher's methods have been developed in the past decade for this purpose but their asymptotic properties and finite-sample numerical performance have not been developed, which will be pursued in this paper. The result concludes that Fisher and {adaptive rank truncated product} method have top performance and complementary advantages across different proportions of true signals. Consequently, we propose an ensemble method, namely Fisher ensemble, to combine the two top-performing Fisher-related methods using a robust harmonic mean ensemble approach. We show that Fisher ensemble achieves asymptotic Bahadur optimality and integrates strengths of the two methods in simulations. We subsequently extend Fisher ensemble to a variation with emphasized power for concordant effect size directions. A transcriptomic meta-analysis application confirms the theoretical and simulation conclusions, generates intriguing biomarker and pathway findings, and demonstrates the strengths and strategy of using the proposed Fisher ensemble methods.