  The partial conjunction null hypothesis is tested in order to discover a
signal that is present in multiple studies. The standard approach of carrying
out a multiple test procedure on the partial conjunction (PC) $p$-values can be
extremely conservative. We suggest alleviating this conservativeness, by
eliminating many of the conservative PC $p$-values prior to the application of
a multiple test procedure. This leads to the following two step procedure:
first, select the set with PC $p$-values below a selection threshold; second,
within the selected set only, apply a family-wise error rate or false discovery
rate controlling procedure on the conditional PC $p$-values. The conditional PC
$p$-values are valid if the null p-values are uniform and the combining method
is Fisher. The proof of their validity is based on a novel inequality in hazard
rate order of partial sums of order statistics which may be of independent
interest. We also provide the conditions for which the false discovery rate
controlling procedures considered will be below the nominal level. We
demonstrate the potential usefulness of our novel method, CoFilter (conditional
testing after filtering), for analyzing multiple genome wide association
studies of Crohn's disease.
