  Combining dependent tests of significance has broad applications but the
$p$-value calculation is challenging. Current moment-matching methods (e.g.,
Brown's approximation) for Fisher's combination test tend to significantly
inflate the type I error rate at the level less than 0.05. It could lead to
significant false discoveries in big data analyses. This paper provides several
more accurate and computationally efficient $p$-value calculation methods for a
general family of Fisher type statistics, referred as the GFisher. The GFisher
covers Fisher's combination, Good's statistic, Lancaster's statistic, weighted
Z-score combination, etc. It allows a flexible weighting scheme, as well as an
omnibus procedure that automatically adapts proper weights and degrees of
freedom to a given data. The new $p$-value calculation methods are based on
novel ideas of moment-ratio matching and joint-distribution surrogating.
Systematic simulations show that they are accurate under multivariate Gaussian,
and robust under the generalized linear model and the multivariate
$t$-distribution, down to at least $10^{-6}$ level. We illustrate the
usefulness of the GFisher and the new $p$-value calculation methods in
analyzing both simulated and real data of gene-based SNP-set association
studies in genetics. Relevant computation has been implemented into R package
$GFisher$.
