  Mixtures of linear mixed models are widely used for modelling longitudinal
data for which observation times differ between subjects. In typical
applications, temporal trends are described using a basis expansion, with basis
coefficients treated as random effects varying by subject. Additional random
effects can describe variation between mixture components, or other known
sources of variation in complex experimental designs. A key advantage of these
models is that they provide a natural mechanism for clustering, which can be
helpful for interpretation in many applications. Current versions of mixtures
of linear mixed models are not specifically designed for the case where there
are many observations per subject and a complex temporal trend, which requires
a large number of basis functions to capture. In this case, the
subject-specific basis coefficients are a high-dimensional random effects
vector, for which the covariance matrix is hard to specify and estimate,
especially if it varies between mixture components. To address this issue, we
consider the use of recently-developed deep mixture of factor analyzers models
as the prior for the random effects. The resulting deep mixture of linear mixed
models is well-suited to high-dimensional settings, and we describe an
efficient variational inference approach to posterior computation. The efficacy
of the method is demonstrated on both real and simulated data.
