Growth mixture modeling with non‐normal distributions

Abstract

A limiting feature of previous work on growth mixture modeling is the assumption of normally distributed variables within each latent class. With strongly non-normal outcomes, this means that several latent classes are required to capture the observed variable distributions. Being able to relax the assumption of within-class normality has the advantage that a non-normal observed distribution does not necessitate using more than one class to fit the distribution. It is valuable to add parameters representing the skewness and the thickness of the tails. A new growth mixture model of this kind is proposed drawing on recent work in a series of papers using the skew-t distribution. The new method is illustrated using the longitudinal development of body mass index in two data sets. The first data set is from the National Longitudinal Survey of Youth covering ages 12-23 years. Here, the development is related to an antecedent measuring socioeconomic background. The second data set is from the Framingham Heart Study covering ages 25-65 years. Here, the development is related to the concurrent event of treatment for hypertension using a joint growth mixture-survival model.

Bayesian Inference on Bivariate Semi-continuous Mixed-effects Models with Application to Longitudinal Substance Use Data

“…Following the studies by Sahu et a. [21] and Muthen and Asparouhov [22], it can be shown that, by introducing a random vector w ij = (w…”

“…Different versions of multivariate skew distributions have been proposed and used in the literature [1,[19][20][21][22]. A new class of distributions by introducing skewness in multivariate elliptically distributions, referred to as skew-elliptical (SE) distributions, were developed in the literature [20,21].…”

“…For completeness, this Appendix briefly summarizes the multivariate ST and SN distributions introduced by Sahu et al [21] and Muthen and Asparouhov [22] to be suitable for a Bayesian inference since it is built using the conditional method; see references by Sahu et al [21] and Muthen and Asparouhov [22] for detailed discussions on properties of ST and SN distributions.…”

“…We relax the normality assumption in BLMM for the model errors with skew distributions including skew-t (ST) and skew-normal (SN) distributions in the family of skew-elliptical (SE) distribution [1,18]. A detailed review and related propoerties on different versions of multivariate skew distributions can be found in the literature [1,[19][20][21][22]. In particular, Sahu et al [21] developed Bayesian inference for skewed data using SN and ST distributions followed by Jara et al [1].…”

“…The skew distribution family, which accommodates asymmetry in a more flexible manner, can model both positively and negatively skewed data (depending on the sign of the estimated skewness parameter from data), and includes normal and t distributions as special cases. We consider a multivariate ST distribution introduced by Sahu et al [21] and Muthen and Asparouhov [22], which is suitable for a Bayesian inference. It is noted that the ST distribution is approximate to the SN distribution when its degrees of freedom approach infinity, the ST distribution reduces to a t distribution if skewness parameter is zero and the SN distribution reduces to a normal (N) distribution if skewness parameter is zero.…”