  Large scale discrete uniform and homogeneous $P$-values often arise in
applications with multiple testing. For example, this occurs in genome wide
association studies whenever a nonparametric one-sample (or two-sample) test is
applied throughout the gene loci. In this paper we consider $q$-values for such
scenarios based on several existing estimators for the proportion of true null
hypothesis, $\pi_0$, which take the discreteness of the $P$-values into
account. The theoretical guarantees of the several approaches with respect to
the estimation of $\pi_0$ and the false discovery rate control are reviewed.
The performance of the discrete $q$-values is investigated through intensive
Monte Carlo simulations, including location, scale and omnibus nonparametric
tests, and possibly dependent $P$-values. The methods are applied to genetic
and financial data for illustration purposes too. Since the particular
estimator of $\pi_0$ used to compute the $q$-values may influence the power,
relative advantages and disadvantages of the reviewed procedures are discussed.
Practical recommendations are given.
