Multi-Group SEM

• MGSEM is a statistical framework that models and compares latent and observed relationships across multiple groups with partial invariance.
• It employs methods like Bayesian, maximum likelihood, and PLS-SEM to capture group-specific measurement and structural nuances.
• Practical applications include multi-cohort studies and customer segmentation, emphasizing confounding adjustment and model fit assessment.

Multi-Group Structural Equation Modeling (MGSEM) refers to a class of statistical modeling frameworks designed to capture and compare the structure of relationships among observed and latent variables across multiple predefined or data-driven groups. MGSEM extends standard SEM by explicitly modeling parameters and structural relationships that may vary across cohorts, clusters, or other meaningful subpopulations. This approach is particularly relevant in contexts involving diverse data sources, longitudinal multi-cohort studies, or segmentation problems where population homogeneity cannot be assumed.

1. Model Specification and Parameterization

MGSEM frameworks posit the existence of multiple groups ($c = 1, \dots, C$), each potentially differing in measurement instruments, population characteristics, or structural relationships. The standard parameterization includes a two-tier design: a measurement model within each group (often using confirmatory factor analysis, CFA) and a group-specific structural model linking latent factors to exogenous predictors. For instance, consider a cohort-indexed measurement model:

$$y_{c,i,t} = \nu_{c,t} + \sum_{k=1}^{d_c} \lambda_{c,tk} \xi_{c,i,k} + \varepsilon_{c,i,t}, \quad \varepsilon_{c,i,t} \sim N(0, \psi_{c,t})$$

Here, $y_{c,i,t}$ is the observed outcome for subject $i$ in cohort $c$ at measurement $t$; $\nu_{c,t}$ (intercept), $\lambda_{c,tk}$ (factor loading), and $\psi_{c,t}$ (residual variance) are all allowed to be group-specific, flexibly accommodating measurement heterogeneity. Subdomain-level latent variables $\xi_{c,i,k}$ are modeled hierarchically via:

$$\xi_{c,i,k} = \gamma_{c,k} \eta_{c,i} + u_{c,i,k}, \quad u_{c,i,k} \sim N(0, \phi_{c,k})$$

$$\eta_{c,i} = \beta_1 X_{c,i} + \beta_2 (X_{c,i} - X^{bp})_+ + \alpha_c P_{c,i} + e_{c,i}, \quad e_{c,i} \sim N(0, \sigma_c^2)$$

$X_{c,i}$ is a group-specific observed predictor; $P_{c,i}$ is an externally computed propensity score; $\beta_1, \beta_2$ are global structural parameters, with $\alpha_c$ and $\sigma_c^2$ potentially cohort-specific. This architecture permits coherent analysis across groups even with non-overlapping measurements, provided the modeling is appropriately hierarchical and partial invariance is addressed (Dang et al., 2020).

2. Measurement Invariance and Partial Invariance

Measurement invariance addresses whether latent constructs are measured equivalently across groups. MGSEM methods relax the assumption of full invariance, often using partial invariance, in which only minimal identification constraints are imposed. Parameters such as loadings, intercepts, and residual variances ($\lambda_{c,tk}, \nu_{c,t}, \psi_{c,t}$) are allowed to vary freely between groups, with the sole identifiability restriction typically being the scaling of one reference loading per cohort (e.g., $\gamma_{c,1} = 1$). Shared measurement items across groups may use the same notation but are estimated independently. Additional residual correlations can be specified where CFA misfit remains, preserving model flexibility. Partial invariance facilitates group-level comparisons and the integration of heterogeneous datasets, such as multi-cohort longitudinal studies with differing instruments (Dang et al., 2020).

3. Estimation Methodologies: Bayesian, Maximum Likelihood, and Partial Least Squares

MGSEM can be estimated via maximum likelihood, Bayesian methods, or algorithmic approaches such as partial least squares (PLS).

Bayesian Estimation

Bayesian MGSEM leverages hierarchical priors and Markov Chain Monte Carlo (MCMC), providing coherent uncertainty quantification in complex, high-dimensional multi-group settings. Parameters (including latent scores) are sampled using Hamiltonian Monte Carlo (HMC), with convergence monitored via diagnostics ($\widehat{R}<1.01$, effective sample size $>1{,}000$) and model fit assessed through WAIC and Bayes Factors. Priors are typically weakly informative to prevent pathological draws (e.g., negative variances), with rigorous marginalization strategies (e.g., integrating out subdomain factors) used for computational efficiency. Missing data are handled by FIML-style integration, restricted to observed outcomes (Dang et al., 2020).

PLS-SEM with Simultaneous Clustering

PLS-SEM with $K$-means (PLS-SEM-KM) extends MGSEM by enabling simultaneous non-hierarchical clustering and structural modeling, providing unsupervised segmentation consistent with latent structure. The methodology minimizes a joint least-squares criterion encompassing reflective measurement, structural paths, and cluster centroids in LV-space:

$$\min_{U,C,\Lambda,B,\Gamma,\Xi,H}\, \|X - Y\Lambda^T\|_F^2 + \|H - HB^T - \Xi\Gamma^T\|_F^2 + \|Y - UC\Lambda\|_F^2$$

Subject to orthonormality and binary cluster membership constraints. The iterative algorithm alternates between centroid computation, latent score updates, inner/outer approximations, and convergence checking. Group (cluster) number is selected via the gap-statistic or penalized $R^{2*}$ (Fordellone et al., 2018).

4. Confounding Adjustment and Covariate Balancing

MGSEM approaches incorporate propensity-score adjustment to address confounding. In Bayesian implementations, the propensity score ($P_{c,i}$) is externally computed via logistic or multinomial regression, estimating $\Pr(\text{treatment}|\text{covariates})$ (e.g., for prenatal alcohol exposure) and introduced linearly into the structural equation with cohort-specific coefficients ($\alpha_c$):

$$\eta_{c,i} = \beta_1 X_{c,i} + \beta_2 (X_{c,i} - X^{bp})_+ + \alpha_c P_{c,i} + e_{c,i}$$

This inclusion blocks observed confounders, ensuring unbiased estimation of exposure effects. The adequacy of confounding control is predicated on the quality of propensity-score estimation and the inclusion of relevant covariates (Dang et al., 2020).

5. Model Selection, Identification, and Goodness-of-Fit

Model selection in MGSEM involves both structural and measurement model considerations. In PLS-SEM-KM, outer model (block-to-LV mapping) and inner path matrices must be specified a priori to guarantee identifiability; reflective measurement blocks require unidimensionality. The number of groups is chosen via the gap-statistic or penalized global fit metrics. Fit indices encompass block communalities, Cronbach’s $\alpha$, $R^2$ for each endogenous latent variable, and a global GoF defined as $$\sqrt{\overline{\text{communality}} \times \overline{R^2}}$$. Classification quality (i.e., the accuracy of cluster assignments) is summarized by the adjusted Rand Index (ARI) and penalized $R^{2*}$ (Fordellone et al., 2018). Trace-plots, residual correlation checks, and posterior predictive densities further assess model adequacy in Bayesian frameworks.

Metric	Bayesian MGSEM (Dang et al., 2020)	PLS-SEM-KM (Fordellone et al., 2018)
Convergence	$\widehat{R}$, ESS, traces	Iterative difference, restart
Goodness-of-fit	WAIC, Bayes Factor	GoF, communality, $R^2, ARI$
Model selection	WAIC, Bayes Factor	Gap-statistic, $R^{2*}$

6. Applications and Empirical Evidence

MGSEM has been applied across diverse empirical domains. In (Dang et al., 2020), a Bayesian MGSEM was used to link prenatal alcohol exposure (PAE) to latent cognitive outcomes, integrating data from six longitudinal U.S. birth cohorts. The model handled varying measurement instruments across studies via partial invariance and used piecewise-linear structural relationships to probe for change points in PAE risk. Key posterior means for structural coefficients ($\beta_1=-2.54$ [CI: $-4.75, -0.32$], $\beta_2=-3.09$ [CI: $-7.93, 1.76$]) indicated that even low PAE levels negatively affect cognition, with the change-point itself being weakly identified. Goodness-of-fit statistics (WAIC) showed negligible evidence favoring the broken-stick specification. Sensitivity diagnostics (trace-plots, predictive residual checks) substantiated the model's adequacy.

In (Fordellone et al., 2018), simulation studies with the PLS-SEM-KM methodology demonstrated near-perfect cluster recovery (mean ARI $\approx0.98$ versus $\approx0.65$ for two-stage approaches), accurate identification of group number, and higher explained variance—especially in unbalanced scenarios. Application to the European Customer Satisfaction Index (ECSI) data revealed that group segmentation aligns with well-interpreted business logic, with fitted structural paths conforming to theoretical expectations.

7. Practical Recommendations and Limitations

MGSEM practitioners should attend to careful pre-specification of measurement and structural models, utilization of robust group determination criteria (gap-statistic, penalized $R^{2*}$), and confirm unidimensionality in reflective blocks. Multiple random initializations are advised to avoid suboptimal minima in clustering-based approaches, especially as noise increases. Analysts should monitor convergence via model-specific diagnostics and evaluate classification/fit using both traditional SEM indices and clustering-aware metrics.

MGSEM methods assume sufficient between-group separation and adequate sample sizes within clusters or cohorts to ensure identifiability and power; violations may compromise interpretability. Measurement non-invariance is both a challenge and opportunity, recoverable via the partial-invariance approach. Propensity-score adjustment effectiveness relies on the comprehensiveness of observed confounders.

Simultaneous clustering and SEM approaches, such as PLS-SEM-KM, outperform traditional sequential modeling when uncovering heterogeneous structural relationships, but are sensitive to initialization and reflectivity assumptions. Bayesian MGSEM offers rigorous uncertainty quantification and principled missing data handling but is computationally intensive, although analytical marginalization can offer substantial efficiency gains (Dang et al., 2020).

In sum, MGSEM provides a flexible platform for modeling and comparing complex, group-structured data, enabling rigorous investigation of both measurement and structural heterogeneity (Dang et al., 2020, Fordellone et al., 2018).