  The multiple testing literature has primarily dealt with three types of
dependence assumptions between p-values: independence, positive regression
dependence, and arbitrary dependence. In this paper, we provide what we believe
are the first theoretical results under various notions of negative dependence
(negative Gaussian dependence, negative regression dependence, negative
association, negative orthant dependence and weak negative dependence). These
include the Simes global null test and the Benjamini-Hochberg procedure, which
are known experimentally to be anti-conservative under negative dependence. The
anti-conservativeness of these procedures is bounded by factors smaller than
that under arbitrary dependence (in particular, by factors independent of the
number of hypotheses). We also provide new results about negatively dependent
e-values, and provide several examples as to when negative dependence may
arise. Our proofs are elementary and short, thus amenable to extensions.
