Structural Equation Modeling

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.)

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 sometimes simplify the diagrammed model to present only the factors
    • This does not imply that the unrepresented parameters are not computed
  • A double-headed arrow denotes a predicted correlation (or bidirectional) relationship between the factors:
  • A single-headed arrow denotes a predicted causal relationship between the factors such that Factor A causes changes in Factor B:

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 Mediation

  • Factor A affects Factor C
  • Factor B affects the relationship between A & C
  • I.e., Factor B mediates the relationship between A & C
    • Here, Factor B has no direct effect on Factor C

Mediation vs. Moderation

  • Mediation:
    • When a predictor has no direct effect on an outcome
    • But that predictor affects something else that does have a direct effect on that outcome
    • E.g., hand washing per se doesn’t affect infections
      • Hand washing affects the number of microbes available to infect
  • Moderation:
    • When a predictor has a direct effect on an outcome
    • But the magnitude of the effect is affected by something else
    • E.g., Thoroughness of hand washing affects number of microbes

Mediation vs. Moderation (cont.)

  • Again, mediators are drawn as affecting the path between the two:
  • Moderators are drawn as going through another factor:

Yet More Complex Models

  • And, sure, a model could contain:
    • A direct effect,
    • A mediating effect, and
    • A moderating effect
  • All of which could be tested

Mechanics of SEMs

Assumptions

  • Normality
    • Typically assume multivariate normality
      • N.b., maximum likelihood estimation is rather robust against departures from normality
        • But strongly multivariate non-normal data can create larger χ²s (i.e., greater model misfit)
        • Leading to higher Type 2 errors (false negatives, i.e., a falsely poorly-fit fit model)
          • I.e., parameter estimates per se will likely be reasonable
            • (If not asymptotically unbiased)
          • But standard errors (and thus χ²s) will be large
            • And probably somehow biased

Assumptions (cont.)

Assumptions (cont.)

  • Sample size
    • Like confirmatory factor analysis, requires large N (>200 or ~20 per factor indicator)
    • Non-normality increases need for lager 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, AIB/BIC, plus a few more)
    3. Review final parameters values for insights
    4. Modify model / parameters & test against other theory-driven models

Comparing Models

Comparing Models: Modifying Parameters

  • Simplest comparisons are between models that vary only in parameter estimates
    • E.g., whether to include / remove a relationship between two factors, e.g.:

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

Example of SEMs

Overall Model

Effets of Demographics

Effets of Demographics (cont.)

EFs & Academics

Overall Predicting Academics

The End