  Causal mediation analysis, pleiotropy analysis, and replication analysis are
three highly popular genetic study designs. Although these analyses address
different scientific questions, the underlying inference problems all involve
large-scale testing of composite null hypotheses. The goal is to determine
whether all null hypotheses - as opposed to at least one - in a set of
individual tests should simultaneously be rejected. Various recent methodology
has been proposed for the aforementioned situations, and an appealing empirical
Bayes strategy is to apply the popular two-group model, calculating local false
discovery rates (lfdr) for each set of hypotheses. However, such a strategy is
difficult due to the need for multivariate density estimation. Furthermore, the
multiple testing rules for the empirical Bayes lfdr approach and conventional
frequentist z-statistics can disagree, which is troubling for a field that
ubiquitously utilizes summary statistics. This work proposes a framework to
unify two-group testing in genetic association composite null settings, the
conditionally symmetric multidimensional Gaussian mixture model (csmGmm). The
csmGmm is shown to demonstrate more robust operating characteristics than
recently-proposed alternatives. Crucially, the csmGmm also offers strong
interpretability guarantees by harmonizing lfdr and z-statistic testing rules.
We extend the base csmGmm to cover each of the mediation, pleiotropy, and
replication settings, and we prove that the lfdr z-statistic agreement holds in
each situation. We apply the model to a collection of translational lung cancer
genetic association studies that motivated this work.
