The harmonic mean p-value for combining dependent tests
Analysis of `big data' frequently involves statistical comparison of millions of competing hypotheses to discover hidden processes underlying observed patterns of data, for example in the search for genetic determinants of disease in genome-wide association studies (GWAS). Controlling the family-wise error rate (FWER) is considered the strongest protection against false positives, but makes it difficult to reach the multiple testing-corrected significance threshold. Here I introduce the harmonic mean p-value (HMP) which controls the FWER while greatly improving statistical power by combining dependent tests using generalized central limit theorem. I show that the HMP easily combines information to detect statistically significant signals among groups of individually nonsignificant hypotheses in examples of a human GWAS for neuroticism and a joint human-pathogen GWAS for hepatitis C viral load. The HMP simultaneously tests all combinations of hypotheses, allowing the smallest groups of hypotheses that retain significance to be sought. The power of the HMP to detect significant hypothesis groups is greater than the power of the Benjamini-Hochberg procedure to detect significant hypotheses, even though the latter only controls the weaker false discovery rate (FDR). The HMP has broad implications for the analysis of large datasets because it enhances the potential for scientific discovery.