Factor- and Composite-Based Structural Equation Modeling -- A New Approach to Incorporate Composites in the Traditional SEM Framework

1. Problem Statement

Structural Equation Modeling (SEM) traditionally models constructs as common factors (reflective latent variables), where the latent variable causes the indicators. However, many constructs are better conceptualized as composites (formative constructs), which are linear combinations of their indicators (e.g., Socio-economic status, Firm performance).

Current methods for analyzing models containing both common factors and composites face significant limitations:

• Outside Traditional SEM Framework: Methods like Partial Least Squares Path Modeling (PLS-PM), Consistent PLS (PLSc), Generalized Structured Component Analysis (GSCA), and Singular Value Decomposition SEM (SVD-SEM) often lack flexibility in model specification, cannot easily handle missing data, rely on computationally intensive bootstrapping for inference (lacking analytic standard errors), or are restricted to standardized estimates only.
• Inside Traditional SEM Framework: Existing attempts to integrate composites into standard SEM (e.g., One-step approach, Pseudo-indicator approach, Henseler-Ogasawara specification) suffer from issues such as:
  • Inability to place composites in dependent positions (mediators/outcomes).
  • Inability to freely estimate composite weights.
  • Increased model complexity and convergence issues due to the introduction of artificial variables.
  • Difficulty in deriving actual parameters (weights, variances) from estimated model parameters.

The Gap: There is a lack of a unified framework that allows researchers to model common factors and composites simultaneously within the traditional SEM framework, thereby leveraging established estimators (like Maximum Likelihood), model fit indices, and missing data handling techniques without sacrificing model flexibility or identification.

2. Methodology: Factor- and Composite-Based SEM (FC-SEM)

The authors propose FC-SEM, a new specification that integrates composites directly into the traditional SEM variance-covariance structure.

A. Model Specification

The model is divided into three components:

1. Measurement Model for Common Factors:
   $$y_f = \Lambda_f \eta_f + \varepsilon_f$$
   Where $y_f$ are indicators, $\eta_f$ are common factors, $\Lambda_f$ are loadings, and $\varepsilon_f$ are measurement errors.
2. Measurement Model for Composites:
   $$\eta_c = W' y_c$$
   Where $y_c$ are composite indicators, $\eta_c$ are composites, and $W$ contains the composite weights. Crucially, the variances and covariances of the indicators forming a composite ($y_c$) are unrestricted and captured in a matrix $T$.
3. Structural Model:
   $$\eta = B\eta + \zeta$$
   Where $\eta$ contains both common factors and composites, $B$ represents structural paths, and $\zeta$ are structural errors.

B. Derivation of the Model-Implied Variance-Covariance Matrix

The core innovation is the derivation of the model-implied variance-covariance matrix ($\Sigma(\theta)$) for the observed indicators ($y = [y_f, y_c]'$).

• For common factors, the standard formula applies: $\Sigma_f = \Lambda_f V(\eta_f) \Lambda_f' + \Theta_f$.
• For composites, the authors derive composite loadings ($\Lambda_c$) via Ordinary Least Squares (OLS) regression of composites on their indicators:
  $$\Lambda_c = T W (W' T W)^{-1}$$
• The full matrix is derived as:
  $$\Sigma(\theta) = \Lambda V(\eta) \Lambda' + \Theta$$
  Where $\Theta$ is a block-diagonal matrix containing measurement error variances for factors and a specific term ($\Theta_c = T - \Lambda_c W' T W \Lambda_c'$) for composites.

This derivation allows the model to be treated mathematically like a standard SEM, enabling the use of traditional estimators.

C. Identification and Estimation

• Identification: The model requires adherence to the $t$-rule (degrees of freedom $\ge 0$). Specific rules include scaling common factors (fixing a loading or variance) and scaling composites (fixing a weight, fixing variance, or constraining the sum of weights). Composites must also be related to at least one other variable outside their own indicator set.
• Estimation: Because the model-implied covariance matrix is derived, standard estimators from the traditional SEM framework can be applied directly. The authors demonstrate the use of Maximum Likelihood (ML), Generalized Least Squares (GLS), and Weighted Least Squares (WLS).
• Software Implementation: The approach is implemented in the open-source R package lavaan (version 0.6-20). Composites are specified using the <~ operator (e.g., Composite <~ y1 + y2 + y3).

3. Key Contributions

1. Unified Framework: FC-SEM successfully bridges the gap between common factor models and composite models within the traditional SEM framework.
2. Analytic Inference: Unlike PLS-PM or PLSc, FC-SEM allows for the analytic derivation of standard errors and test statistics, removing the sole reliance on bootstrapping and reducing computational time.
3. Flexibility:
   • Allows composites to be in dependent positions (mediators/outcomes) in the structural model.
   • Allows for free estimation of composite weights.
   • Supports various scaling methods for composites.
   • Enables the use of advanced SEM features like missing data handling (FIML), multi-group analysis, and model fit assessment (e.g., $\chi^2$, RMSEA, SRMR).
4. Implementation: Provides a user-friendly implementation in lavaan, the most widely used SEM software, making the method accessible to a broad research audience.

4. Results

The authors validated the approach through three stages:

1. Scenario Analysis (Perfect Fit): Using a population model with 4 constructs (2 factors, 2 composites), they generated a sample that perfectly matched the population covariance matrix. The FC-SEM approach successfully retrieved all population parameters exactly, confirming the mathematical correctness of the specification and identification rules.
2. Finite Sample Simulation: A Monte Carlo simulation ($N=1,000$, sample size $n=500$) demonstrated that the ML estimator yields consistent and unbiased estimates close to population values. The average standard errors closely matched empirical standard deviations, confirming the validity of analytic inference.
3. Empirical Example: Applied to the American Customer Satisfaction Index (ACSI) dataset.
   • The model included 8 constructs (4 composites, 4 factors).
   • Results were compared against IGSCA, PLSc, and the H-O specification.
   • Finding: The standardized parameter estimates from FC-SEM were highly similar to those from other methods. Notably, the results were almost identical to the H-O specification but achieved with a much simpler model syntax and without convergence issues.

5. Significance

• Methodological Advancement: FC-SEM resolves the long-standing tension between factor-based and composite-based modeling, allowing researchers to choose the correct construct representation (reflective vs. formative) without abandoning the robust statistical tools of traditional SEM.
• Practical Utility: By integrating into lavaan, it lowers the barrier to entry for using composites in SEM. Researchers can now test complex theories involving mixed construct types while utilizing standard fit indices and handling real-world data issues like missing values.
• Future Directions: The paper highlights that while the estimation framework is established, future research is needed to refine model fit criteria specifically for composite models and to address the assumption that composite indicators are free from measurement error.

In summary, this paper presents a rigorous, flexible, and implementable solution for analyzing structural equation models that contain both reflective and formative constructs, significantly enhancing the applicability of SEM in diverse research disciplines.