A Comparison of Different Nonnormal Distributions in Growth Mixture Models

Abstract

The purpose of the present study is to compare nonnormal distributions (i.e., t, skew-normal, skew-t with equal skew and skew-t with unequal skew) in growth mixture models (GMMs) based on diverse conditions of a number of time points, sample sizes, and skewness for intercepts. To carry out this research, two simulation studies were conducted with two different models: an unconditional GMM and a GMM with a continuous distal outcome variable. For the simulation, data were generated under the conditions of a different number of time points (4, 8), sample size (300, 800, 1,500), and skewness for intercept (1.2, 2, 4). Results demonstrate that it is not appropriate to fit nonnormal data to normal, t, or skew-normal distributions other than the skew-t distribution. It was also found that if there is skewness over time, it is necessary to model skewness in the slope as well.

Growth Mixture Modeling With Nonnormal Distributions: Implications for Data Transformation

“…Since nonnormal distributions with various degrees of skewness and kurtosis are readily found in real data, the within-class normality assumption of GMM is often considered a limiting feature (Son et al, 2019). Because of this normality assumption, GMM tends to prefer models that include spurious classes to explain the observed variable distributions to a true model, particularly when the outcomes strongly follow nonnormal distributions (Bauer & Curran, 2004;Muthén & Asparouhov, 2015).…”

“…To deal with nonnormality, recent articles have investigated the use of skew-t distributions for structural equation models (SEMs) and mixture models (e.g., Asparouhov & Muthén, 2016;Frühwirth-Schnatter & Pyne, 2010;Lee & McLachlan, 2014;Lin et al, 2007). GMM with distributions generated from multiple levels of nonnormality have also been examined in previous simulation studies (e.g., Bauer & Curran, 2003;Guerra-Peña & Steinley, 2016;Jung & Wickrama, 2008;Muthén & Asparouhov, 2015;Son et al, 2019). However, most of this research has been based around the performance of fit indices, such as the Akaike information criterion (AIC; Akaike, 1974), the Bayesian information criterion (BIC; Schwartz, 1978), and sample-size adjusted BIC (SBIC; Sclove, 1987), and likelihood ratio tests, such as the Lo-Mendell-Rubin adjusted likelihood ratio test (LMR-LRT; Lo et al, 2001) and the bootstrap likelihood ratio test (BLRT; McLachlan & Peel, 2000), in terms of how to determine the number of latent classes.…”

“…However, most of this research has been based around the performance of fit indices, such as the Akaike information criterion (AIC; Akaike, 1974), the Bayesian information criterion (BIC; Schwartz, 1978), and sample-size adjusted BIC (SBIC; Sclove, 1987), and likelihood ratio tests, such as the Lo-Mendell-Rubin adjusted likelihood ratio test (LMR-LRT; Lo et al, 2001) and the bootstrap likelihood ratio test (BLRT; McLachlan & Peel, 2000), in terms of how to determine the number of latent classes. Only several studies (Depaoli et al, 2019;Muthén & Asparouhov, 2015;Son et al, 2019) have focused on the extent to which class-specific parameter estimates are biased by the within-class normality assumption in GMM.…”

“…But this study left several points to be developed when it comes to including various skewness or kurtosis levels of nonnormal distributions and investigating t distribution, which also belongs to skew-t family of continuous distributions. Son et al (2019) attempted to extend their model by considering more diverse conditions for skewness as well as four types of skew-t family distribution (i.e., the normal, t, skew-normal, and skew-t), but it remained limited in terms of kurtosis levels and class proportion within the population. They took into account only one level of kurtosis and class proportion (i.e., a balanced proportion between two latent classes), meaning that the model could not be used to further understanding of the efficiency of skew-t distributions in GMM under various nonnormal conditions that may reflect empirical cases where researchers can frequently come across in real data.…”

A Bayesian sensitivity analysis of the effect of different random effects distributions on growth curve models

“…Usually a marginal model is used for analysing growth curve repeated measurements. In this framework, a normality assumption is considered for within-subject errors, but some studies such as Louzada et al (2014) and Son et al (2019) are considered assumption of abnormality for the response variable. Another popular approach for analysing growth curve repeated measurements data is a non-linear mixed effects model.…”