  Aggregating multiple effects is often encountered in large-scale data
analysis where the fraction of significant effects is generally small. Many
existing methods cannot handle it effectively because of lack of computational
accuracy for small p-values. The Cauchy combination test (abbreviated as CCT) (
J Am Statist Assoc, 2020, 115(529):393-402) is a powerful and computational
effective test to aggregate individual $p$-values under arbitrary correlation
structures. This work revisits CCT and shows three key contributions including
that (i) the tail probability of CCT can be well approximated by a standard
Cauchy distribution under much more relaxed conditions placed on individual
p-values instead of the original test statistics; (ii) the relaxation
conditions are shown to be satisfied for many popular copulas formulating
bivariate distributions; (iii) the power of CCT is no less than that of the
minimum-type test as the number of tests goes to infinity with some regular
conditions. These results further broaden the theories and applications of CCT.
The simulation results verify the theoretic results and the performance of CCT
is further evaluated with data from a prostate cancer study.
