Structural Equation Models (SEMs)

• Structural Equation Models are multivariate statistical models that integrate measurement models and regression-based structural paths to explore latent constructs and causal relationships.
• Recent developments extend SEMs to incorporate non-Gaussian, nonlinear, and high-dimensional data using regularization, Bayesian methods, and copula-based approaches, enhancing analytical flexibility.
• SEMs are widely applied for mediation analysis, causal inference, and network estimation across disciplines like psychology, genomics, and engineering, offering robust model diagnostics and feedback.

Structural Equation Models (SEMs) are a broad class of multivariate statistical models that integrate systems of regression equations, path diagrams, and latent variable measurement models within a formal mathematical structure. SEMs are widely used for causal inference, mediation analysis, measurement modeling, and the empirical interpretation of complex multivariate systems in fields ranging from psychology and political science to genomics and engineering. Recent methodological advances have further extended SEMs to accommodate non-Gaussian data, handle infinite-dimensional systems, support regularization and Bayesian inference, and model complex nonlinear and mixed data structures.

1. Mathematical Foundations and Model Structure

Classical SEMs consist of two primary components: a measurement model and a structural model. The measurement model links observed indicator variables to unobserved latent variables, while the structural model specifies the dependence structure (often causal) among the latent variables and exogenous observed variables via a system of equations.

Measurement model (Confirmatory Factor Analysis):

$y = \Lambda \eta + \epsilon$

where $y$ is a $p \times 1$ vector of observed indicators, $\eta$ is an $m \times 1$ vector of latent variables, $\Lambda$ is the $p \times m$ matrix of factor loadings, and $\epsilon$ is a measurement error vector.

Structural model:

$\eta = B \eta + \Gamma x + \zeta$

where $B$ encodes structural relations among the latent variables, $\Gamma$ are regressions from exogenous variables $x$, and $\zeta$ is a vector of structural disturbances.

Under Gaussianity, the model-implied covariance structure is

$$\Sigma(\theta) = \Lambda (I-B)^{-1} \Psi (I-B)^{-T} \Lambda^T + \Theta_\epsilon$$

with $\Psi$ the structural error covariances and $\Theta_\epsilon$ the measurement-error covariance.

Identification requires that the number of knowns ($p(p+1)/2$ covariances in the observed data) exceeds the number of free parameters and that each latent variable's scale is fixed (typically by constraining one loading to unity) (Jenatabadi, 2015, Zheng, 30 Mar 2025).

2. Model Variants: Classical, Composite, and Non-Gaussian SEMs

Reflective Latent Variable Models treat constructs as common factors producing covariation among indicators, with measurement errors capturing unique variances (Schamberger et al., 8 Aug 2025, Zheng, 30 Mar 2025). In contrast, Composite (Formative) Models view constructs as linear combinations of observed variables, dispensing with shared variance explanations and allowing error-free construction of certain composites (Schamberger et al., 8 Aug 2025). The combined latent-and-composite formulation extends the flexibility of SEMs without loss of analytic or computational tools from traditional frameworks.

Copula Structural Equation Models (CoSEMs) address technical limitations of linear/Gaussian SEMs, notably their inability to represent non-normal, discrete, or mixed data types. Using Sklar’s theorem, a joint distribution is specified as

$$f(x_1,\ldots,x_d) = c(F_1(x_1),\ldots,F_d(x_d);\theta) \prod_{j=1}^d f_j(x_j)$$

with marginals $f_j$, CDFs $F_j$, and a copula density $c$ governing dependence (Chen et al., 20 Oct 2025). This approach, including Gaussian copula SEMs and vine-copula decompositions, encodes DAG structures in the dependence parameters and supports flexible tail dependence modeling, essential for mediation pathway analysis in non-Gaussian settings.

Distributional SEMs further extend the classical model by letting both the mean and variance of latent variables be functions of other latents, enabling the explicit modeling of latent heteroscedasticity (Fazio et al., 2024).

3. Estimation, Regularization, and Bayesian Approaches

SEMs are typically estimated by minimizing a discrepancy between the observed sample covariance ($S$) and the model-implied covariance ($\Sigma(\theta)$). The most common estimation criteria are:

• Maximum Likelihood (ML): Under Gaussianity, minimize $$\log|\Sigma| + \mathrm{tr}(S\Sigma^{-1}) - \log|S| - p$$ (Jenatabadi, 2015).
• Generalized or Weighted Least Squares (GLS/WLS): Alternative loss functions for non-Gaussian or categorical indicators (Zheng, 30 Mar 2025).
• Full Information Maximum Likelihood (FIML) or Multiple Imputation: For handling missing data under MAR assumptions (Schamberger et al., 8 Aug 2025).

Regularized SEMs introduce sparsity and model selection into high-dimensional SEMs by penalizing the path matrix, often with an $\ell_1$ (LASSO) or group-LASSO penalty for edge selection. Convex semidefinite programming approaches yield globally solvable estimation procedures with sample complexity guarantees and practical ADMM/PPXA algorithms (Pruttiakaravanich et al., 2018, Kesteren et al., 2019, Shen et al., 2016).

Bayesian SEMs place priors on all parameters and latent scores, enabling incorporation of prior information and robust finite-sample inference. Packages such as blavaan use MCMC via JAGS and introduce parameter expansion for models with sparse covariance structure, automatically handling latent variables and missing data (Merkle et al., 2015). Fit indices such as DIC, WAIC, and LOO provide Bayesian model comparison.

Variational Bayes methods have been extended to mixtures of Gaussian SEMs, providing rapid, tractable inference in models with multimodal, skewed, and incomplete data (Dang et al., 2024).

4. Generalized and Infinite-Dimensional SEMs

Standard SEMs are limited to acyclic, finite-variable systems. Generalized SEMs (GSEMs) remove these constraints, enabling the representation of systems with infinitely many variables and infinite ranges—critical for modeling dynamical systems (e.g., ODEs, continuous-time processes) (Halpern et al., 2021, Peters et al., 2021). GSEMs are specified by direct mappings from exogenous contexts and allowed interventions to sets of outcome trajectories, bypassing well-founded recursions and supporting arbitrary interventions.

For temporally indexed variables, non-recursive (cyclic) SEMs can be reinterpreted as discrete-time dynamical systems with time-delayed feedback and analyzed using temporal logics such as CPLTL. This permits causal reasoning in models with feedback loops and supports efficient model checking procedures (Gladyshev et al., 17 Jan 2025). Identifiability in homoscedastic cyclic (feedback) SEMs can be characterized using algebraic matroids and projection-based graphical properties (Drton et al., 2023).

5. Model Fit, Diagnostics, and Robustness

Global model fit is assessed with the ML chi-squared test, RMSEA, CFI/TLI, and SRMR, but these indices can be misleading in small samples, non-normal data, or misspecified models (Jenatabadi, 2015, Zheng, 30 Mar 2025, Hertzog, 2018). Simulation studies demonstrate the need to combine multiple fit indices and iterative refitting; BIC tends to outperform AIC in model recovery for moderate to large models (Hertzog, 2018). For regularized models, information criteria such as AIC, BIC, KIC, and their variants guide the choice of penalty parameters.

Finite-sample bias in ML estimates can be pronounced when $n/p$ is low. Recent work applies reduced-bias M-estimation (RBM) to SEMs, offering analytic bias correction (of $O(1/n)$) with minor computational overhead and no loss in mean squared error (Jamil et al., 29 Sep 2025).

6. Extensions: Latent Variable Identification and Alternative SEM Formalisms

Partial identification of individual parameters in latent-variable SEMs—often necessary in complex settings or with limited data—can be achieved using model-implied instrumental variables (MIIV), local algebraic transformations, and trek-separation based graphical criteria. The latent-to-observed (L2O) transformation and its graphical formulation allow non-iterative, equation-wise parameter identification, subsuming prior MIIV approaches and enabling identification strategies for settings previously inaccessible to standard methods (Ankan et al., 2023).

Alternative SEM implementations include the piecewise SEM approach, which fits each path as a separate generalized linear or mixed-effects model and tests d-separation via conditional independence tests and Fisher’s $C$ statistic, beneficial for hierarchical, non-normal, and phylogenetically structured data but currently limited in handling latent variables and feedback loops (Lefcheck, 2015).

Computational frameworks such as tensorsem formulate SEMs as computation graphs in machine learning libraries (e.g., TensorFlow), facilitating robust optimizers, arbitrary loss functions, and gradient-based learning for both smooth and non-convex/penalized models (Kesteren et al., 2019).

7. Applications and Practical Recommendations

SEMs play a central role in mediation analysis, causal inference, multi-group comparison (measurement invariance), high-dimensional network estimation (e.g., genomics, fMRI), and in integrating information from multi-modal or mixed-type data sources. Extensions via copula and vine models, kernel-based regularized estimation, and distributional SEMs allow their application to non-Gaussian, nonlinear, and heterogeneous data (Chen et al., 20 Oct 2025, Shen et al., 2016, Fazio et al., 2024).

Best practices include meticulous theoretical specification, explicit scale fixation for latent constructs, rigorous fit assessment using a combination of indices, cautious respecification based on modification indices, and sensitivity analysis. Recommendations for robust inference are especially acute in small samples or with violations of normality, where RBM or Bayesian methods offer statistical advantages (Jamil et al., 29 Sep 2025, Merkle et al., 2015, Dang et al., 2024).

Ongoing research seeks to unify SEM theory for infinite-variable (GSEM) or dynamically evolving systems, enhance computational tools for flexible high-dimensional settings, and generalize identification theory via algebraic and graphical methods.