Core Concepts

• Structural Equation Modeling (SEM) tests the relationships between non-ostensible (endogenous) factors
  • Much like a linear regression model tests relationships between ostensible (exogenous) variables / items
• But SEM:
  • Is more flexible
  • Can test more complex models
    • Including causal relationships
  • Can investigate endogenous variables as well as exogenous
  • Typically uses maximum likelihood estimation instead of ordinary least squares

Core Concepts (cont.)

• SEM in fact subsumes:
  • Factor analysis (typically CFA)
  • Canonical correlation (multivariate analysis of correlation)
  • Discriminant analysis (predicting classification into several groups)
  • Regression analysis (linear, logistic, etc.)
  • Is being used with increasing frequency to investigate, e.g.,
    • Communicating health-related research
    • Creating a health index
    • Studying cultural concepts of love

Basic Analyses

• In SEMs, we assess outputs of:
  1. Model fit (χ², etc.)
  2. Model parameters
     • Factor loadings if conducting factor analysis
     • Covariances & variances between factor indicators & factors
     • Often with especial attention given to factor inter-relationships
     • Error variances

Conceiving of SEMs

• SEMs are often conceived of as path diagrams based on the same ideas we used for CFAs:

Same Model without Factor Indicators (Items) or Error Terms

• We can simplify the diagram to present only the factors
  • This does not imply that the unrepresented parameters are not computed
• A double-headed arrow denotes a predicted correlational (or bidirectional) relationship between the factors:

Three Factors

• The power of SEMs begins to show when we consider several factors
• And more complex relationships, e.g.,
  • Factor A affects Factor C
  • Factor B affects Factor C
  • Factor A is unrelated to Factor B

Three Factors with Moderation

• Factor A affects Factor C
• Factor B affects the relationship between A & C

Moderation vs. Mediation

Moderation vs. Mediation (cont.)

• Again, moderators are drawn as affecting the path between the two:

Example of Moderation & Mediation

Yet More Complex Models

Assumptions

Assumptions (cont.)

Assumptions (cont.)

• Sample size
  • Like confirmatory factor analysis, requires large N (>200 or ~20 per factor indicator)
  • Non-normality increases need for larger N
• Ordinal data
  • If monotonic & arguably sample a non-ostensible continuous construct
  • And have a large N (> ~500)
  • Can include via polychoric correlations
    • Or treat as interval
      • When there are several (>4) response options
      • And data are nearly normal

SEM Procedure

• Largely the same as for general(ized) linear models

  • Usually with more parameters

• Therefore, typically:

  1. Use maximum likelihood estimation to try to fit a proposed model to the data’s covariance matrix (may also use variable / factor indicator means)
  2. Review fit indices (e.g., χ², AIC/BIC, plus a few more)
  3. Review final parameters values for insights
  4. Modify model / parameters & test against other theory-driven models

Comparing Models: Modifying Parameters

Modification Indices

• Lagrange multiplier test
  • Computes minimum amount the χ² (of the residual covariance matrix) would decrease if the given parameter were freed
  • Parameters with largest Lagrange multiplier values are most impactful on model
• Still needs to be guided by theory
  • Especially since studies tend to find that models guided by Lagrange multiplier tests generalize poorly to other data

Comparing Models: Modifying Models

• SEMs can test totally different arrangements of the factors
  • Or even different factor loadings
• Through tests of overall model fits
  • Usually via χ², AIC, & BIC

Comparing Models: Comparing Groups

• SEMs can test group differences
  • E.g., whether the same relationships between factors holds for different groups
  • And thus can be sophisticated alternatives to, e.g., (M)AN(C)OVAs
• We can test how well a model fits different groups by placing / removing equality constraints in the model
  • Adding / removing an equality constraints tests how the model performs when assuming the groups are similar / dissimilar on a given parameter

Comparing Models: Comparing Groups (cont.)

• Remember SEMs often include a series of linear regressions
  • These include intercepts for endogenous variables
  • We can compare whether these intercepts between groups
  • This will test the effect of group membership on the (other) model parameters
    • I.e., akin to adding a dummy variable for that group

Two Ways to Test Mediation

Models Tested:
The Unmediaated Model

• In the model below:
  • c is the unmediated path we compare against
  • X is the predictor
  • M is the mediator
  • Y is the outcome

Models Tested:
The Mediated Model

• In the model below:
  • a and b are the components of the mediated pathways
  • c’ is the path from predictor (X) to outcome (Y) accounting for the mediator (M)

Example Using PROCESS

1. Download & install the macro via:
   • Extensions → Utilities → Install Custom Dialog...
2. Open dialogue via:
   • Analyze → Regression → PROCESS v5.0...
   • Y Variable
   • X Variable
   • M Variable(s)
   • Leave Model Number = 4

Overall Strategy

1. Test Path a

• Analyze → General Linear Model → Univariate
  • For Dependent Variable choose the Mediator
  • For Fixed Factor or Covariate choose the Predictor
• Save the unstandardized coefficient and note itse p-value
  • If p < .05, Predictor significantly predicts Mediator

2. Test Paths b and c′

• Analyze → General Linear Model → Univariate
  • For Dependent Variable choose the Outcome
  • For Fixed Factor or Covariate choose the Predictor and Mediator
• The coefficient for the Mediator gives b
• The coefficient for the Predictor gives c′

3. Test Path c

• Run another GLM:
  • For Dependent Variable choose the Outcome
  • For Fixed Factor or Covariate choose the Predictor alone
• The coefficient for Predictor is c

4. Compute the Overall ab Path & 95% CI

1. Multiply a × b
   • E.g., via Transform > Compute Variable
   • Enter a and b manually
2. Rerun test for b and c’ via:
   • Analyze → General Linear Model → Univariate
     • For Dependent Variable choose the Outcome
     • For Fixed Factor or Covariate choose the Predictor and Mediator
     • Click on Bootstrap...
       • For Number of bootstrap samples: 5000
       • For Confidence intervals: BCa or Percentile (95%)

Additional Resources

• Hayes, A. F. (2022). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach (3rd ed.). Guilford Press.