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## Contributions: Abstract## Problems of analyzing nonlinear structural equation models with latent product and quadratic terms
The problems of analyzing moderated structural equation models have received much attention in the recent literature. These problems are even aggravated when latent quadratic terms are added to the moderated equation and include 1) nonlinearity of parameters, 2) specification of mean structures, and, as the major characteristic of nonlinear models, 3) non-normally distributed variables. An appropriate specification of measurement models for the latent product and quadratic terms is needed because the multiplication of indicator variables leads to nonlinear terms of the factor loadings and to complex error terms. Methods for analyzing structural equation models usually assume that the observed and latent variables are centered. But even if the latent exogenous variables have zero expectations, the latent product variable does not have zero expectation in general, and latent quadratic variables always have expectations greater than zero. To avoid misspecification, the constant intercept term and the expectations of the nonlinear terms have to be estimated. But the main problem of analyzing latent nonlinear structural equation models is the distributional problem. The distributions resulting from nonlinear terms are known to be non-normal. Even if all indicators of the latent exogenous variables and the latent variables themselves are normally distributed, the distributions of the latent product variable, the latent quadratic variables, and the latent endogenous variable are definitely nonnormal. An estimation method that takes the non-normality of the variables explicitly into account and does not require the forming of any product variables is Latent Moderated Structural Equations (LMS). Based on a stochastic analysis of the multivariate density function of indicator variables, LMS implements an iterative maximum likelihood (ML) estimation for the model parameters. Using artificial data sets, the methodological problems of nonlinear models will be demonstrated and possibilities and limitations of dealing with these problems using LMS are discussed. |