Structural Equation Model

Concept

---

Theoretically, Structural Equation Model (SEM) is a multivariate statistical modelling framework that simultaneously estimates relationships among observed variables, latent variables, and measurement error. SEM integrates factor analysis and path analysis within a unified system of equations, allowing researchers to evaluate complex causal hypotheses involving direct, indirect, and total effects. It is founded on covariance structure analysis and is widely used to investigate latent constructs that cannot be measured directly.

Mathematically, SEM specifies separate measurement and structural components. The measurement model relates observed indicators to latent variables, whereas the structural model specifies relationships among latent and observed variables. Model parameters are typically estimated using maximum likelihood estimation, although alternative estimators such as weighted least squares and Bayesian estimation are available. Model adequacy is evaluated using recognised fit indices including the ?� statistic, RMSEA, CFI, TLI, and SRMR.

In practice, Structural Equation Models are estimated using specialised statistical software and are applied to investigate causal pathways, mediation, latent health constructs, patient-reported outcomes, health behaviours, and healthcare utilisation. In health economics, SEM is used to evaluate complex relationships between socioeconomic status, disease severity, quality of life, healthcare resource use, and economic outcomes while explicitly accounting for measurement error.

Purpose

---

Used to estimate complex systems of relationships among observed and latent variables, evaluate causal pathways, quantify direct and indirect effects, account for measurement error, and test theoretical models in health economic research.

---

Mathematical Formulae

Primary Formula

Structural model:

? = B? + �? + ?

where:

• ? = endogenous latent variables
• ? = exogenous latent variables
• B = matrix of relationships among endogenous variables
• � = matrix of effects of exogenous variables
• ? = structural disturbances

Supporting Formulae

Measurement model for endogenous variables:

y = ??? + �

Measurement model for exogenous variables:

x = ??? + �

Covariance structure:

�(?) = � implied by the structural equation model

Maximum likelihood fitting function:

FML = ln|�(?)| + tr(S�(?)??) ? ln|S| ? p

where:

• S = sample covariance matrix
• �(?) = model-implied covariance matrix
• p = number of observed variables

Related Mathematical Methods

• Confirmatory Factor Analysis
• Path Analysis
• Latent Variable Model
• Maximum Likelihood Estimation
• Covariance Structure Analysis
• Mediation Analysis
• Factor Analysis
• Model Fit Assessment

---

Example

A health economist investigates how socioeconomic status influences healthcare costs both directly and indirectly through health-related quality of life.

The SEM includes:

• Latent socioeconomic status measured by income, education, and occupation.
• Latent health status measured by EQ-5D, disease severity, and functional status.
• Annual healthcare costs as an observed outcome.

Estimated standardised effects are:

• Socioeconomic status ? Health status: ? = 0.58
• Health status ? Healthcare costs: ? = ?0.47
• Socioeconomic status ? Healthcare costs: ? = ?0.19

The model demonstrates that much of the relationship between socioeconomic status and healthcare costs operates indirectly through health status.

---

Excel Implementation

Function	Example Formula	Health Economics Application
MMULT	=MMULT(Matrix1,Matrix2)	Perform matrix multiplication used in covariance calculations.
MINVERSE	=MINVERSE(CovarianceMatrix)	Calculate inverse covariance matrices for SEM estimation.
TRANSPOSE	=TRANSPOSE(MatrixRange)	Construct loading and covariance matrices.
SUMPRODUCT	=SUMPRODUCT(Observed,Predicted)	Calculate covariance and model-fit components.
MDETERM	=MDETERM(CovarianceMatrix)	Calculate matrix determinants used in likelihood estimation.

VBA (Optional)

Automate covariance matrix preparation and export data for structural equation modelling software while generating model fit summaries.

---

Sources

• Bollen KA. Structural Equations with Latent Variables.
• Kline RB. Principles and Practice of Structural Equation Modeling. 5th ed.
• J�reskog KG. A General Method for Estimating a Linear Structural Equation System. In: Structural Equation Models in the Social Sciences. 1973.
• Byrne BM. Structural Equation Modeling with AMOS.
• Hair JF, Black WC, Babin BJ, Anderson RE. Multivariate Data Analysis.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
• Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.