For each gene g, let
$$N_g = \sum_{s=1}^S \omega_s \Phi^{-1}(1-p_{gs}),$$
where \(p_{gs}\) corresponds to the raw p-value obtained for gene g in a differential
analysis for study s (assumed to be uniformly distributed under the null hypothesis), \(\Phi\) the
cumulative distribution function of the standard normal distribution, and \(\omega_s\) a set of weights.
We define the weights \(\omega_s\) as in Marot and Mayer (2009):
$$\omega_s = \sqrt{\frac{\sum_c R_{cs}}{\sum_\ell \sum_c R_{c\ell}}},$$
where \(\sum_c R_{cs}\) is the total number of biological replicates in study s. This allows
studies with large numbers of biological replicates to be attributed a larger weight than smaller studies.
Under the null hypothesis, the test statistic \(N_g\) follows a N(0,1) distribution. A unilateral
test on the righthand tail of the distribution may then be performed, and classical procedures for the
correction of multiple testing, such as that of Benjamini and Hochberg (1995), may subsequently be applied to
the obtained p-values to control the false discovery rate at a desired level \(\alpha\).