  Clustered data is ubiquitous in a variety of scientific fields. In this
paper, we propose a flexible and interpretable modeling approach, called
grouped heterogenous mixture modeling, for clustered data, which models
cluster-wise conditional distributions by mixtures of latent conditional
distributions common to all the clusters. In the model, we assume that clusters
are divided into a finite number of groups and mixing proportions are the same
within the same group. We provide a simple generalized EM algorithm for
computing the maximum likelihood estimator, and an information criterion to
select the numbers of groups and latent distributions. We also propose
structured grouping strategies by introducing penalties on grouping parameters
in the likelihood function. Under the settings where both the number of
clusters and cluster sizes tend to infinity, we present asymptotic properties
of the maximum likelihood estimator and the information criterion. We
demonstrate the proposed method through simulation studies and an application
to crime risk modeling in Tokyo.
