Extending Growth Mixture Models Using Continuous Non-Elliptical Distributions

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.