  Modern data sets in various domains often include units that were sampled
non-randomly from the population and have a latent correlation structure. Here
we investigate a common form of this setting, where every unit is associated
with a latent variable, all latent variables are correlated, and the
probability of sampling a unit depends on its response. Such settings often
arise in case-control studies, where the sampled units are correlated due to
spatial proximity, family relations, or other sources of relatedness. Maximum
likelihood estimation in such settings is challenging from both a computational
and statistical perspective, necessitating approximations that take the
sampling scheme into account. We propose a family of approximate likelihood
approaches which combine composite likelihood and expectation propagation. We
demonstrate the efficacy of our solutions via extensive simulations. We utilize
them to investigate the genetic architecture of several complex disorders
collected in case-control genetic association studies, where hundreds of
thousands of genetic variants are measured for every individual, and the
underlying disease liabilities of individuals are correlated due to genetic
similarity. Our work is the first to provide a tractable likelihood-based
solution for case-control data with complex dependency structures.
