A Bayesian Model For The Estimation Of Latent Interaction And Quadratic Effects When Latent Variables Are Non-Normally Distributed

Abstract

Structural equation models with interaction and quadratic effects have become a standard tool for testing nonlinear hypotheses in the social sciences. Most of the current approaches assume normally distributed latent predictor variables. In this article, we present a Bayesian model for the estimation of latent nonlinear effects when the latent predictor variables are nonnormally distributed. The nonnormal predictor distribution is approximated by a finite mixture distribution. We conduct a simulation study that demonstrates the advantages of the proposed Bayesian model over contemporary approaches (Latent Moderated Structural Equations [LMS], Quasi-Maximum-Likelihood [QML], and the extended unconstrained approach) when the latent predictor variables follow a nonnormal distribution. The conventional approaches show biased estimates of the nonlinear effects; the proposed Bayesian model provides unbiased estimates. We present an empirical example from work and stress research and provide syntax for substantive researchers. Advantages and limitations of the new model are discussed.

A Heterogeneous Growth Curve Model for Nonnormal Data

“…The extension can be viewed as an indirect application of a mixture model. For cross-sectional SEM with interaction and quadratic effects a similar model has been proposed by Kelava and Nagengast (2012) and Kelava et al (2014).…”

“…In this case, classspecific model parameters are not interpreted for each class separately, but only across classes. Mixture models are then only used as a statistical means to approximate, for example, nonnormal distributions (Kelava & Nagengast, 2012;Kelava, Nagengast, & Brandt, 2014;McLachlan & Peel, 2000;Wall, Guo, & Amemiya, 2012) or nonlinear relationships (Bauer, 2005;Pek, Sterba, Kok, & Bauer, 2009). Second, if the GMM is applied (indirectly) in order to model heterogeneity in the growth trajectories, the precision with which the heteroscedasticity can be approximated by the semi-parametric class model depends on the number of latent classes.…”

Fitting Nonlinear Structural Equation Models in R with Package nlsem

“…The nonlinear extensions of the SEMM framework (Kelava and Nagengast 2012; can also be specified in these software packages.…”

“…Bayesian approaches (Arminger and Stein 1997;Feng, Wang, Wang, and Song 2015;Kelava and Nagengast 2012;Lee 2007;Song and Lu 2010) can be feasibly implemented in Bayesian software packages like WinBUGS (Lunn, Thomas, Best, and Spiegelhalter 2000) or OpenBUGS (Lunn, Spiegelhalter, Thomas, and Best 2009).…”

“…Recently, a method was proposed that combines parametric and semiparametric models to estimate nonlinear effects (Kelava and Nagengast 2012;). These models allow for an analysis of specific functional forms of relationships among latent variables in the mixture modeling framework while relaxing the distributional assumptions that are needed for the parametric models.…”