  Model-based clustering methods for continuous data are well established and
commonly used in a wide range of applications. However, model-based clustering
methods for categorical data are less standard. Latent class analysis is a
commonly used method for model-based clustering of binary data and/or
categorical data, but due to an assumed local independence structure there may
not be a correspondence between the estimated latent classes and groups in the
population of interest. The mixture of latent trait analyzers model extends
latent class analysis by assuming a model for the categorical response
variables that depends on both a categorical latent class and a continuous
latent trait variable; the discrete latent class accommodates group structure
and the continuous latent trait accommodates dependence within these groups.
Fitting the mixture of latent trait analyzers model is potentially difficult
because the likelihood function involves an integral that cannot be evaluated
analytically. We develop a variational approach for fitting the mixture of
latent trait models and this provides an efficient model fitting strategy. The
mixture of latent trait analyzers model is demonstrated on the analysis of data
from the National Long Term Care Survey (NLTCS) and voting in the U.S.
Congress. The model is shown to yield intuitive clustering results and it gives
a much better fit than either latent class analysis or latent trait analysis
alone.
