  This paper continues the line of research initiated in Liu et. al. (2016) on
developing a novel framework for multiple testing of hypotheses grouped in a
one-way classified form using hypothesis-specific local false discovery rates
(Lfdr's). It is built on an extension of the standard two-class mixture model
from single to multiple groups, defining hypothesis-specific Lfdr as a function
of the conditional Lfdr for the hypothesis given that it is within an important
group and the Lfdr for the group itself and involving a new parameter that
measures grouping effect. This definition captures the underlying group
structure for the hypotheses belonging to a group more effectively than the
standard two-class mixture model. Two new Lfdr based methods, possessing
meaningful optimalities, are produced in their oracle forms. One, designed to
control false discoveries across the entire collection of hypotheses, is
proposed as a powerful alternative to simply pooling all the hypotheses into a
single group and using commonly used Lfdr based method under the standard
single-group two-class mixture model. The other is proposed as an Lfdr analog
of the method of Benjamini and Bogomolov (2014) for selective inference. It
controls Lfdr based measure of false discoveries associated with selecting
groups concurrently with controlling the average of within-group false
discovery proportions across the selected groups. Simulation studies and
real-data application show that our proposed methods are often more powerful
than their relevant competitors.
