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.
Highlights
The formulas we present are general in the sense that they can be applied to all mixture models including nonlinear effects, or to multiple sample analyses with nonlinear effects when class membership is known a priori
We illustrate the standardization procedure for direct and indirect applications of nonlinear structural equation mixture models with an example based on data from the Program for International Student Assessment 2009 (PISA; Organisation for Economic Co-Operation and Development, 2010), which is publicly available under http://pisa2009.acer.edu.au/downloads. php
The larger fit indices for the single class solution indicated the necessity to account for nonnormality in the data when analyzing nonlinear effects
Summary
This generalization allows one to standardize (linear and nonlinear) group-specific regression coefficients for multiple group and latent class models in a direct application of mixture models. The standardized coefficients for the linear effects γ1,g and γ2,g depend on the class-specific means of the latent predictor variables (see Moosbrugger et al, 1997).
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