  Researchers often have to deal with heterogeneous population with mixed
regression relationships, increasingly so in the era of data explosion. In such
problems, when there are many candidate predictors, it is not only of interest
to identify the predictors that are associated with the outcome, but also to
distinguish the true sources of heterogeneity, i.e., to identify the predictors
that have different effects among the clusters and thus are the true
contributors to the formation of the clusters. We clarify the concepts of the
source of heterogeneity that account for potential scale differences of the
clusters and propose a regularized finite mixture effects regression to achieve
heterogeneity pursuit and feature selection simultaneously. As the name
suggests, the problem is formulated under an effects-model parameterization, in
which the cluster labels are missing and the effect of each predictor on the
outcome is decomposed to a common effect term and a set of cluster-specific
terms. A constrained sparse estimation of these effects leads to the
identification of both the variables with common effects and those with
heterogeneous effects. We propose an efficient algorithm and show that our
approach can achieve both estimation and selection consistency. Simulation
studies further demonstrate the effectiveness of our method under various
practical scenarios. Three applications are presented, namely, an imaging
genetics study for linking genetic factors and brain neuroimaging traits in
Alzheimer's disease, a public health study for exploring the association
between suicide risk among adolescents and their school district
characteristics, and a sport analytics study for understanding how the salary
levels of baseball players are associated with their performance and
contractual status.
