# Fitting structural equation mixture models

### Description

Estimation of structural equation models with nonlinear effects and underlying nonnormal distributions.

### Details

This is a package for estimating nonlinear structural equation mixture models using an expectation-maximization (EM) algorithm. Four different approaches are implemented. Firstly, the Latent Moderated Structural Equations (LMS) approach (Klein & Moosbrugger, 2000) and the Quasi-Maximum Likelihood (QML) approach (Klein & Muthen, 2007), which allow for two-way interaction and quadratic terms in the structural model. Due to the nonlinearity, the latent criterion variables cannot be assumed to be normally distributed. Therefore, the latent criterins's distribution is approximated with a mixture of normal distributions in LMS. Secondly, the Structural Equation finite Mixture Model (STEMM or SEMM) approach (Jedidi, Jagpal & DeSarbo, 1997), which uses mixtures to model latent classes. In this way it can deal with heterogeneity in the sample or nonlinearity and nonnormality of the latent variables and their indicators. And thirdly, a combination of these two approaches, the Nonlinear Structural Equation Mixture Model (NSEMM) approach (Kelava, Nagengast & Brandt, 2014). Here, interaction and quadratic terms as well as latent classes can be modeled.

The models can be specified with `specify_sem`

. Depending
on the specification of `interaction`

and the number of latent classes
(`num.classes`

) the returned object will be of class
`singleClass`

, `semm`

, or `nsemm`

. Each of these can be
estimated using `em`

and models of type `singleClass`

can additionally be fitted with the function `qml`

.

### Future Features

NSEMM, LMS and QML for more than one latent endogenous variable.

Parameter standardization.

### References

Jedidi, K., Jagpal, H. S., & DeSarbo, W. S. (1997). STEMM: A General
Finite Mixture Structural Equation Model, *Journal of
Classification, 14*, 23–50.

Kelava, A., Nagengast, B., & Brandt, H. (2014). A nonlinear structural
equation mixture modeling approach for non-normally distributed latent
predictor variables. *Structural Equation Modeling, 21*, 468-481.

Klein, A. &, Moosbrugger, H. (2000). Maximum likelihood estimation of
latent interaction effects with the LMS method. *Psychometrika, 65*,
457–474.

Klein, A. &, Muthen, B. O. (2007). Quasi-Maximum Likelihood Estimation
of Structural Equation Models With Multiple Interaction and Quadratic
Effects *Multivariate Behavioral Research, 42*, 647–673.