The Generalized Harmonic Mean for p‐Values: Combining Dependent and Independent Tests

In medical research, particularly in fields such as genomics, multi-center clinical trials, and meta-analysis, effectively combining the p-values from multiple related hypothesis tests has always been a challenging statistical issue. To address this problem and enhance the statistical power of comprehensive analysis, this study proposes a generalized harmonic mean for p-values (GHMP( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow> <mml:annotation>$$ \xi $$</mml:annotation></mml:semantics> </mml:math> )) combination method and builds two kinds of combination tests based on this framework. The first kind of test is designed for applications with small significance levels and has more lenient conditions for adapting to correlations, making it suitable for the complex dependency structures commonly found in actual research. The second kind of test introduces a novel high-order tail approximation technique based on stable distribution theory, which can more accurately estimate the extreme tail probabilities at large significance levels under independent or weakly correlated conditions. Extensive simulation experiments show that both kinds of tests perform robustly across various configurations, with statistical power not inferior to the traditional Cauchy combination test (CCT) and minimum p-value (MinP) methods, and demonstrate superior detection capabilities in several scenarios. Additionally, GHMP( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow> <mml:annotation>$$ \xi $$</mml:annotation></mml:semantics> </mml:math> ) has high computational efficiency and has been empirically validated in real genetic data. These characteristics make it a reliable and practical analytical tool for high-dimensional medical research, such as genome-wide association studies (GWAS) and large-scale meta-analysis.