Model selection and the estimation of statistical power in nonlinear structural equation modeling
Authors:   , Irmer Julien Patrick 
Published: December 2024

In the social sciences, relations among latent variables are of great interest. These associations are not necessarily linear, and selecting the correct model is of significant importance; otherwise, the inferences drawn from these models might be arbitrarily wrong. Latent variables cannot be measured directly but are operationalized indirectly via multiple indicators. Structural Equation Modeling (SEM) is used to examine the associations between latent variables through the correlational structure in multiple measurements (Bollen, 1989). The model selection process is guided by (robust) model fit tests, such as the (robust) χ2-test (e.g., Satorra & Bentler, 2010) in linear SEM analysis. Although there are extensions for specific quadratic and interaction SEM (QISEM, Büchner & Klein, 2020), these model fit tests are not generally applicable to nonlinear SEM (NLSEM) due to the lack of a saturated comparison model (Büchner & Klein, 2020; Mooijaart & Satorra, 2009). In regression analysis, non-parametric trend estimates and scatterplots can be used to examine the structural form and guide model selection. However, these are not suitable for NLSEM due to measurement errors in the observations. Latent variables cannot be observed directly, but estimates for these are available by using factor scores. Factor scores are transformations of the data that estimate the latent variables with approximation error. Still, factor scores are indeterminate (Grice, 2001), meaning that several approaches will result in different results. However, as the number of measurements increases, the approximation error diminishes, and factor scores become close estimates of the latent variables.
In this thesis in Manuscript A, a simple set of assumptions is derived to identify the trends in NLSEM using linear factor scores. These trends are described by the conditional expectation of endogenous variables given exogenous variables. Identification means that the trends can be estimated from observed data. Linear factor scores are simple linear transformations of the data and therefore are straightforward to compute. However, due to the approximation error in linear factor scores, these assumptions are asymptotic in nature, as they are only applicable for a large number of measurements per latent variable, as then the prediction error becomes negligible or can be explicitly modeled. In contrast to previous identification results (Kelava et al., 2017), these asymptotic assumptions used in this thesis allow for cross-relations between the measurements within the exogenous part of the model and within the endogenous part of the model. This implies that cross-loadings and residual covariances are permitted within measurements of the exogenous latent variables (ξ) and within measurements of the endogenous latent variables (η), while cross-relations between measurements of the exogenous part and the endogenous part of the model are disallowed. Independence assumptions on the measurement errors of the exogenous and endogenous parts of the model, along with independence from the latent variables, imply that the Bartlett (1937) factor score (BFS) has a specific structure that connects it to the literature on non-parametric regression with measurement error (Delaigle et al., 2009; Huang & Zhou, 2017). Although, in this literature, the distribution of the measurement error is assumed to be known, this similarity, combined with continuity assumptions on the nonlinear trend, can be used to identify the conditional expectation. This is shown mathematically in Manuscript A. Consequently, trends in NLSEM can be estimated using BFS under the given assumptions either by ignoring the approximation error or by explicitly modeling the approximation error. For a large number of measurements in the exogenous part of the model, the approximation error variance of the BFS on the exogenous part of the model is small. Therefore, this approximation error can be ignored by inputting BFS into non-parametric regression methods such as the LOESS (locally estimated scatterplot smoothing, Cleveland, 1979, 1981). Since BFS are linear transformations of the data, a central limits effect occurs on the approximation error.