Comparing the Robustness of the Structural after Measurement (SAM) Approach to Structural Equation Modeling (SEM) against Local Model Misspecifications with Alternative Estimation Approaches

Abstract

Structural equation models (SEM), or confirmatory factor analysis as a special case, contain model parameters at the measurement part and the structural part. In most social-science SEM applications, all parameters are simultaneously estimated in a one-step approach (e.g., with maximum likelihood estimation). In a recent article, Rosseel and Loh (2022, Psychol. Methods) proposed a two-step structural after measurement (SAM) approach to SEM that estimates the parameters of the measurement model in the first step and the parameters of the structural model in the second step. Rosseel and Loh claimed that SAM is more robust to local model misspecifications (i.e., cross loadings and residual correlations) than one-step maximum likelihood estimation. In this article, it is demonstrated with analytical derivations and simulation studies that SAM is generally not more robust to misspecifications than one-step estimation approaches. Alternative estimation methods are proposed that provide more robustness to misspecifications. SAM suffers from finite-sample bias that depends on the size of factor reliability and factor correlations. A bootstrap-bias-corrected LSAM estimate provides less biased estimates in finite samples. Nevertheless, we argue in the discussion section that applied researchers should nevertheless adopt SAM because robustness to local misspecifications is an irrelevant property when applying SAM. Parameter estimates in a structural model are of interest because intentionally misspecified SEMs frequently offer clearly interpretable factors. In contrast, SEMs with some empirically driven model modifications will result in biased estimates of the structural parameters because the meaning of factors is unintentionally changed.

Implementation Aspects in Regularized Structural Equation Models

“…Alternatively, the non-differentiable optimization function can be replaced by a differentiable one [12,[35][36][37][38]. The penalty function involves the non-differentiable absolute value function that can be replaced by |x| ≈ (x 2 + ε) 1/2 or more generally |x| p ≈ (x 2 + ε) p/2 (14) for a sufficiently small ε > 0, such as ε = 10 −3 or ε = 10 −4 .…”

Model-Robust Estimation of Multiple-Group Structural Equation Models

“…The L p loss function with p = 2 is the square loss function ρ(x) = x 2 and corresponds to ULS estimation. This loss function has been successively applied in linking methods (Kolen and Brennan 2014;Robitzsch 2022b), which can be considered an alternative approach to a joint estimation of SEM for multiple groups. The loss function with p = 0.5 or p = 0.25 has been proposed in the invariance alignment linking approach (Asparouhov and Muthén 2014;Muthén and Asparouhov 2014;Pokropek et al 2020).…”

“…In practice, the minimization of ( 17) is carried out on a discrete one-or two-dimensional grid of κ values, and the optimal regularization parameter is selected that minimizes the Bayesian information criterion (BIC). The optimization of the nondifferentiable fitting function can be carried out using gradient descent (Hastie et al 2015) approaches or by substituting the nondifferentiable optimization functions with differentiable approximating functions (Battauz 2020;Oelker and Tutz 2017;Robitzsch 2022a;Tutz and Gertheiss 2016). In our experience, the latter approach performs satisfactorily in applications.…”

“…As is true with all statistical models, these restrictions are unlikely to hold in practice, and model assumptions in SEM are only an approximation of a true data-generating model. In SEM, model deviations (i.e., model errors) in covariances emerge as a difference between a population covariance matrix Σ and a model-implied covariance matrix Σ(θ) (see Robitzsch 2022a andWu andBrowne 2015). Similarly, there can be model errors in the mean vector leading to a difference between the population mean vector µ and the model-implied mean vector µ(θ).…”

“…In this article, we discuss model-robust estimation of SEM when modeling means and covariance matrices in the more general case of multiple groups (Jöreskog et al 2016;Robitzsch 2022b). Model errors in the modeled covariance matrix can emerge to incorrectly specified loadings in the matrix Λ or unmodeled residual error correlations in the matrix Ψ.…”