  In applications such as clinical safety analysis, the data of the experiments
usually consists of frequency counts. In the analysis of such data, researchers
often face the problem of multiple testing based on discrete test statistics,
aimed at controlling family-wise error rate (FWER). Most existing FWER
controlling procedures are developed for continuous data, which are often
conservative when analyzing discrete data. By using minimal attainable
$p$-values, several FWER controlling procedures have been specifically
developed for discrete data in the literature. In this paper, by utilizing
known marginal distributions of true null $p$-values, three more powerful
stepwise procedures are developed, which are modified versions of the
conventional Bonferroni, Holm and Hochberg procedures, respectively. It is
shown that the first two procedures strongly control the FWER under arbitrary
dependence and are more powerful than the existing Tarone-type procedures,
while the last one only ensures control of the FWER in special settings.
Through extensive simulation studies, we provide numerical evidence of superior
performance of the proposed procedures in terms of the FWER control and minimal
power. A real clinical safety data is used to demonstrate applications of our
proposed procedures. An R package "MHTdiscrete" and a web application are
developed for implementing the proposed procedures.
