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. 2015 Nov 30:6:1813.
doi: 10.3389/fpsyg.2015.01813. eCollection 2015.

The Standardization of Linear and Nonlinear Effects in Direct and Indirect Applications of Structural Equation Mixture Models for Normal and Nonnormal Data

Holger Brandt  1 , Nora Umbach  1 , Augustin Kelava  1
Affiliations

The Standardization of Linear and Nonlinear Effects in Direct and Indirect Applications of Structural Equation Mixture Models for Normal and Nonnormal Data

Holger Brandt et al. Front Psychol. .

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.

Keywords: interaction effect; mixture model; nonlinear effect; nonnormality; quadratic effect; standardization.

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Figures

Figure 1
Figure 1
Scatter plots of the estimated factor scores based on the two class solutions of Model (a) (left panels) and Model (b) (right panels) for reading attitude and reading skills (upper panels) and online activities and reading skills (lower panels). The model based relationships between the variables are indicated with solid lines, class membership is indicated by black or gray dots.
Figure 2
Figure 2
Standardized multivariate relationship between reading attitude, online activities and reading skills.
Figure 3
Figure 3
Nonnormal bivariate distribution of the latent predictors.
Figure 4
Figure 4
Simple slopes for female (black lines) and male students (gray lines) based on a standardization using the pooled variances. The relationship for online activities and reading skills were estimated for low, average, and high reading attitudes.
See this image and copyright information in PMC

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