8.6 Structural Equations
Linking measurement and relationships
Confirmatory factor analysis asks how constructs are measured. Structural equation modeling adds another layer:
How are the measured constructs and observed variables related?
In a broad statistical usage, CFA is already part of the SEM family. In the narrower teaching usage adopted here, SEM refers to models that combine a measurement model with directed regressions or paths.
8.6.1 Path Models as Systems of Regressions
A path model contains several linked regression equations. For a simple mediation model:
\[ M = \alpha_M + aX + \varepsilon_M, \]
\[ Y = \alpha_Y + c'X + bM + \varepsilon_Y. \]
The indirect effect is \(ab\), the direct effect is \(c'\), and the total effect is
\[ c=c'+ab. \]
When all variables are observed, this is usually called path analysis. SEM can replace some observed variables with latent factors measured by several indicators.
8.6.2 Measurement Model Plus Structural Model
A measurement model describes the indicators:
\[ \mathbf{x} = \boldsymbol{\nu} + \boldsymbol{\Lambda}\boldsymbol{\eta} + \boldsymbol{\varepsilon}. \]
A structural model describes relationships among latent and observed variables:
\[ \boldsymbol{\eta} = \boldsymbol{\alpha} + \mathbf{B}\boldsymbol{\eta} + \boldsymbol{\Gamma}\mathbf{x} + \boldsymbol{\zeta}. \]
The central advantage is that relationships can be estimated while the measurement model remains explicit. A latent predictor is not treated as a perfectly observed scale score, and a latent outcome retains its measurement uncertainty within the joint model.
8.6.3 A Simple Latent Mediation Model
Suppose motivation is proposed to mediate the relationship between educational resources and competence. Motivation and competence may each be measured by several indicators.
The model contains two claims:
- the indicators measure motivation and competence adequately;
- the structural paths among resources, motivation, and competence are substantively defensible.
A compact lavaan skeleton is:
The defined indirect effect is a model parameter. Its causal interpretation requires much more than statistical significance.
8.6.4 Measurement Comes First
A structural model does not rescue a weak measurement model. If indicators do not support the intended factors, the interpretation of regressions among those factors becomes unclear.
A defensible sequence is:
- define the constructs and indicators;
- evaluate the measurement models;
- establish sufficient comparability when groups or time points are compared;
- specify the structural relations;
- evaluate global and local fit of the combined model;
- examine alternative explanations and robustness.
The measurement and structural parts are estimated together in the final SEM, but they should remain conceptually distinguishable.
8.6.5 SEM Is Not Automatically Causal
A directed arrow expresses a model assumption. It does not create temporal order, random assignment, or control for confounding.
A causal interpretation requires evidence and assumptions about:
- temporal ordering;
- omitted common causes;
- selection and missingness;
- measurement validity;
- functional form;
- interference and treatment definition where relevant.
Different directed models can sometimes imply the same covariance structure. Good fit therefore does not prove the direction of causation.
A well-fitting SEM is a well-fitting system of statistical restrictions. It becomes a causal model only when the research design and causal assumptions justify that interpretation.
8.6.6 Where Higher-Order and Bifactor Models Belong
Second-order and bifactor models remain confirmatory measurement models in the organization of this book.
- A second-order model explains relationships among first-order factors with a broader factor.
- A bifactor model gives indicators direct loadings on a general factor and specific factors.
- A structural model adds regressions among constructs or between constructs and external variables.
All can be estimated in SEM software, but they answer different substantive questions. Software membership should not replace conceptual classification.
8.6.7 Model Fit Continues, but the Questions Multiply
The same evaluation principles continue:
- convergence and admissible estimates;
- global and local fit;
- comparison of defensible alternatives;
- validation beyond the original sample.
The structural part adds further questions:
- Are the path directions theoretically justified?
- Are indirect effects meaningful and sufficiently precise?
- Are important confounders omitted?
- Are coefficients stable across groups or time?
- Would another plausible model imply similar observed relationships?
8.6.8 What This Section Will Add Later
The current section establishes the bridge from CFA to SEM. Later development can add:
- observed path analysis;
- latent mediation and moderation;
- longitudinal and cross-lagged models;
- latent growth models;
- measurement invariance before mean comparisons;
- model comparison and equivalent models;
- connections to causal graphs and identification.
SEM extends regression by allowing several equations to be estimated together and extends CFA by placing measured latent constructs inside those equations.