• Much like a linear regression model tests relationships between ostensible (exogenous) variables / items

• 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

• Factor analysis (typically CFA)
• Canonical correlation (multivariate analysis of correlation)
• Regression analysis (linear, logistic, etc.)
• Discriminant analysis (predicting classification into several groups)
• And is being used with increasing frequency to investigate, e.g.,
  • Communicasting health-related research
  • Creatting a health index
  • Study cultural concepts of love

1. Model fit (χ², RMSEA, SRMR, CFI, TLI, etc.)
2. Model parameters
   • Factor loadings
   • Covariances & variances between factor indicators & factors
   • Often with especial attention given to factor inter-relationships
   • Error variances

• Not instead unique variance for each factor indicator

• This does not imply that the unpresented parameters are not computed

• Factor A affects Factor C
• Factor B affects Factor C
• Factor A is unrelated to Factor B

• Factor B affects Factor C
• Factor A affects Factor C by first affecting Factor B

• Here, Factor A has no direct effect on Factor C

• 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

• 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

• A direct effect,
• A mediating effect, and
• A moderating effect

• 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
    • Leading to higher Type 2 errors (false negatives, i.e., a falsely poorly-fit fit model)
      • I.e., parameter estimates per se will likley be reasonable
        • (If not asymptotically unbiased)
      • But standard errors (and thus χ²s) will be large
        • And probably somehow biased

• With very non-normal data, can use:
  • Asymptotically distribution free (aka weighted least squares) estimation
  • But the Satorra-Bentler χ² correction (Satorra & Bentler, 2010) is preferred when data are interval / ratio
  • See Finney & DiStefano (2008) for more on robust ML estimation

• Like CFA, requires large N (>200 or ~20 per factor indicator)
• Non-normality increases need for lager N

• 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

Largely the same as for a CFA

• Usually with more parameters
• And usually interested most on relationships between endogenous factors

Therefore, typically:

1. Establish model & initial parameter values based on theory
2. Use maximum likelihood estimation to try to fit a proposed model to the data’s covariance matrix (may also use variable / factor indicator means)
3. Review fit indices (e.g., χ², SRMR, RMSEA, CFI, TLI, plus a few more) and modification indices
4. Review final parameters values for insights
5. Modify model / parameters & test against other theory-driven models

Can compute “effect size” of misfit, e.g., to compensate for larger N

Power is perhaps under-valued in SEMs

• Negative residual variances
• Very large standard errors
• Many non-significant (free) parameters
• Correlations > 1

• Together, these statistics can test the significance of that loading
• By creating a critical ratio (CR, aka Wald test): \[CR=\frac{Factor\;Loading}{Standard\;Error}\]
• Can then use:
  • CR ≥ ±1.96 (|z| = 2) for 2-tailed tests
  • CR ≥ ±1.65 for 1-tailed tests (viz., CR > 0)

• Fixed: Constrained to be a certain value (viz., 0 or 1)
• Freed: Allowed to have its value estimated by the analyses
• Constrained: Allowed to assume only a certain range of values (e.g., non-negative or larger than another parameter)

• Which is a less restrictive model
• More guided by the data than by theory
• So, yeah, quasi-exploratory

• Allow factor indicators to load onto more than one factor (cross-loading)
• Allow factor indicator error terms to correlate
• Perhaps even remove problematic factor indicators

• E.g., whether to include / remove a relationship between two factors, e.g.:

• 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

• Especially since studies tend to find that models guided by Lagrange multiplier tests generalize poorly to other data

• Or even different factor loadings

• Usually via χ², AIC, & BIC

• 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

• Adding / removing an equality constraints tests how the model performs when assuming the groups are similar / dissimilar on a given parameter
• Therefore:
  1. Add an equality constraint, test model fit
  2. Remove an equality constraint, retest model
  3. Compare fits, e.g., via χ² difference

• 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

• Frequency of self-reported environmentally-relevant behaviors
• Divided here into animal items:
  • “I have put up a bird house near my home”
  • “I have asked my parents not to buy products made from animal fur”
• And all other behaviors, e.g.:
  • “I have talked with my parents about how to help with environmental problems”
  • “I turn off the water in the sink while I brush my teeth to conserve water”

• Self-reported empathy of other people
• E.g.:
  • “Other people’s feeling don’t bother me at all”
  • “Seeing a person who has been angered has no effect on my feelings”

• Here divided by comparison stimuli
  • Human comparison (boy or girl)
  • Non-human comparison (food, plant, toy, activity)