  The issue of combining individual $p$-values to aggregate multiple small
effects is prevalent in many scientific investigations and is a long-standing
statistical topic. Many classical methods are designed for combining
independent and frequent signals in a traditional meta-analysis sense using the
sum of transformed $p$-values with the transformation of light-tailed
distributions, in which Fisher's method and Stouffer's method are the most
well-known. Since the early 2000, advances in big data promoted methods to
aggregate independent, sparse and weak signals, such as the renowned higher
criticism and Berk-Jones tests. Recently, Liu and Xie(2020) and Wilson(2019)
independently proposed Cauchy and harmonic mean combination tests to robustly
combine $p$-values under "arbitrary" dependency structure, where a notable
application is to combine $p$-values from a set of often correlated SNPs in
genome-wide association studies. The proposed tests are the transformation of
heavy-tailed distributions for improved power with the sparse signal. It calls
for a natural question to investigate heavy-tailed distribution transformation,
to understand the connection among existing methods, and to explore the
conditions for a method to possess robustness to dependency. In this paper, we
investigate the regularly varying distribution, which is a rich family of
heavy-tailed distribution and includes Pareto distribution as a special case.
We show that only an equivalent class of Cauchy and harmonic mean tests have
sufficient robustness to dependency in a practical sense. We also show an issue
caused by large negative penalty in the Cauchy method and propose a simple, yet
practical modification. Finally, we present simulations and apply to a
neuroticism GWAS application to verify the discovered theoretical insights and
provide practical guidance.
