title: The Standardization of Linear and Nonlinear Effects in Direct and Indirect Applications of Structural Equation Mixture Models for Normal and Nonnormal Data.
abstract: The application of mixture models to flexibly estimate linear and nonlinear effects in the SEM framework has received increasing attention (e.g., Jedidi et al., 1997b; Bauer, 2005; Muthén and Asparouhov, 2009; Wall et al., 2012; Kelava and Brandt, 2014; Muthén and Asparouhov, 2014). The advantage of mixture models is that unobserved subgroups with class-specific relationships can be extracted (direct application), or that the mixtures can be used as a statistical tool to approximate nonnormal (latent) distributions (indirect application). Here, we provide a general standardization procedure for linear and nonlinear interaction and quadratic effects in mixture models. The procedure can also be applied to multiple group models or to single class models with nonlinear effects like LMS (Klein and Moosbrugger, 2000). We show that it is necessary to take nonnormality of the data into account for a correct standardization. We present an empirical example from education science applying the proposed procedure.
authors.affiliation: Hector Research Institute of Education Sciences and Psychology, Eberhard Karls Universität Tübingen Tübingen, Germany.
authors.forename: Holger
authors.initials: H
authors.lastname: Brandt
authors.affiliation: Hector Research Institute of Education Sciences and Psychology, Eberhard Karls Universität Tübingen Tübingen, Germany.
authors.forename: Nora
authors.initials: N
authors.lastname: Umbach
authors.affiliation: Hector Research Institute of Education Sciences and Psychology, Eberhard Karls Universität Tübingen Tübingen, Germany.
authors.forename: Augustin
authors.initials: A
authors.lastname: Kelava
keywords: interaction effect; mixture model; nonlinear effect; nonnormality; quadratic effect; standardization