  Growth mixture models (GMMs) incorporate both conventional random effects
growth modeling and latent trajectory classes as in finite mixture modeling;
therefore, they offer a way to handle the unobserved heterogeneity between
subjects in their development. GMMs with Gaussian random effects dominate the
literature. When the data are asymmetric and/or have heavier tails, more than
one latent class is required to capture the observed variable distribution.
Therefore, a GMM with continuous non-elliptical distributions is proposed to
capture skewness and heavier tails in the data set. Specifically, multivariate
skew-t distributions and generalized hyperbolic distributions are introduced to
extend GMMs. When extending GMMs, four statistical models are considered with
differing distributions of measurement errors and random effects. The
mathematical development of GMMs with non-elliptical distributions relies on
their expression as normal variance-mean mixtures and the resultant
relationship with the generalized inverse Gaussian distribution. Parameter
estimation is outlined within the expectation-maximization framework before the
performance of our GMMs with non-elliptical distributions is illustrated on
simulated and real data.
