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Structural equation modeling (SEM)

Learn about SEM

Model specification

  • Use the SEM Builder or command language
  • SEM Builder uses standard path diagrams
  • Command language is a natural variation on path diagrams
  • Group estimation in linear models as easy as adding group(sex); easily add or relax constraints including adding or omitting paths for some groups but not others

SEM Builder

  • Drag, drop, and connect to create path diagrams
  • Estimate models from path diagrams
  • Display results on the path diagram
  • Save and modify diagrams
  • Tools to create measurement and regression components
  • Set constant and equality constraints by clicking
  • Complete control of how your diagrams look

Classes of models for linear SEM

  • Linear regression
  • Multivariate regression
  • Path analysis
  • Mediation analysis
  • Measurement models
  • Confirmatory factor analysis
  • Multiple indicators and multiple causes (MIMIC) models
  • Latent growth curve models
  • Hierarchical confirmatory factor analysis
  • Correlated uniqueness models
  • Arbitrary structural equation models

Additional classes of models for generalized SEM

  • Generalized linear models
  • Item response theory models
  • Measurement models with binary, count, and ordinal measurements
  • Multilevel CFA models
  • Multilevel mixed-effects models
  • Latent growth curve models with generalized-linear responses
  • Multilevel mediation models
  • Selection models
    • with random intercepts and slopes
    • with binary, count, and ordinal outcomes
  • Endogenous treatment-effect models
  • Any multilevel structural equation models with generalized-linear responses

Linear and generalized-linear responses

  • Models for continuous, binary, count, ordinal, and nominal outcomes
  • Eight distribution families
    • Gaussian
    • Gamma
    • Bernoulli
    • Binomial
    • Poisson
    • Negative binomial
    • Ordinal
    • Multinomial
  • Five links
    • Identity
    • Log
    • Logit
    • Probit
    • Cloglog
  • Support for common regression models: linear, logistic, probit, ordered logit, ordered probit, Poisson, multinomial logistic, tobit, interval measurements, and more

Multilevel models

  • Two-, three-, and higher-level structural equation models
  • Multilevel mixed-effects models
  • Random intercepts and random slopes
  • Crossed and nested random effects

Estimation methods for linear SEM

  • ML—maximum likelihood
  • MLMV—maximum likelihood for missing values; sometimes called FIML
  • ADF—asymptotic distribution free, meaning GMM (generalized method of moments) using ADF weighting matrix

Estimation methods for generalized SEM

  • Maximum likelihood
  • Mean-variance or mode-curvature adaptive Gauss–Hermite quadrature
  • Nonadaptive Gauss–Hermite quadrature
  • Laplace approximation

Standard-error methods

  • OIM—observed information matrix
  • EIM—expected information matrix
  • OPG—outer product of gradients
  • Robust—distribution-free linearized estimator
  • Cluster–robust—robust adjusting for correlation within groups of observations
  • Bootstrap—nonparametric bootstrap and clustered bootstrap
  • Jackknife—delete-one, delete-n, and clustered jackknife

Survey support for linear SEM

  • Sampling weights
  • Stratification and poststratification
  • Clustered sampling at one or more levels

Summary statistics data (SSD)

  • Fit linear SEMs on observed or summary (SSD) data
  • Fit models on covariances or correlations and optionally variances and means
  • SSD may be group specific
  • Easily create and manage SSDs
  • Build SSDs from original (raw) data for distribution or publication
  • Automatic corruption/error checking and repairing
  • Electronic signatures

Starting values

  • Automatic
  • May specify for some or all parameters
  • Grid search available
  • May fit one model, subset or superset, and use fitted values for another model

Identification

  • Automatic normalization (anchoring) constraints provide scale for latent variables; may be overridden

Reliability

  • May specify fraction of variance not due to measurement error

    Direct and indirect effects for linear SEM

    • Confidence intervals
    • Unstandardized or standardized units

    Overall goodness-of-fit statistics for linear SEM

    • Model vs. saturated
    • Baseline vs. saturated
    • RMSEA, root mean squared error of approximation
    • AIC, Akaike's information criterion
    • BIC, Bayesian information criterion
    • CFI, comparative fit index
    • TLI, Tucker–Lewis index, a.k.a. nonnormed fit index
    • SRMR, standardized root mean squared residual
    • CD, coefficient of determination

    Equation-level goodness-of-fit statistics for linear SEM

    • R-squared
    • Equation-level variance decomposition
    • Bentler–Raykov squared multiple-correlation coefficient

    Group-level goodness-of-fit statistics for linear SEM

    • SRMR
    • CD
    • Model vs. saturated chi-squared contribution

    Residual analysis for linear SEM

    • Mean residuals
    • Variance and covariance residuals
    • Raw, normalized, and standardized values available

    Parameter tests

    • Modification indices
    • Wald tests
    • Score tests
    • Likelihood-ratio tests
    • Easy to specify single or joint custom tests for omitted paths, included paths, and relaxing constraints
    • Linear and nonlinear tests of estimated parameters
    • Tests may be specified in standardized or unstandardized parameter units

    Group-level parameter tests for linear SEM

    • Group invariance by parameter class or user specified

    Linear and nonlinear combinations of estimated parameters

    • Confidence intervals
    • Unstandardized or standardized units

    Assess nonrecursive system stability

    Predictions for linear SEM

    • Observed endogenous variables
    • Latent endogenous variables
    • Latent variables (factor scores)
    • Equation-level first derivatives
    • In- and out-of-sample prediction; may estimate on one sample and form predictions in another

    Predictions for generalized SEM

    • Means of observed endogenous variables—probabilities for 0/1 outcomes, mean counts, etc.
    • Linear predictions of observed endogenous variables
    • Latent variables using empirical Bayes means and modes
    • Standard errors of empirical Bayes means and modes
    • Observed endogenous variables with and without predictions of latent variables

    Results

    • May be used with postestimation features
    • May be saved to disk for restoration and use later
    • Displayed in standardized or unstandardized units
    • Optionally display results in Bentler–Weeks form
    • Optionally display results in exponentiated form as odds ratios, incidence rate ratios, and relative risk ratios
    • All results accessible for user-written programs

    Factor variables with generalized SEM

    • Automatically create indicators based on categorical variables
    • Form interactions among discrete and continuous variables
    • Include polynomial terms
    • Perform contrasts of categories/levels

    Marginal analysis

    • Estimated marginal means
    • Marginal and partial effects
    • Average marginal and partial effects
    • Least-squares means
    • Predictive margins
    • Adjusted predictions, means, and effects
    • Contrasts of margins
    • Pairwise comparisons of margins
    • Profile plots
    • Graphs of margins and marginal effects

    Contrasts for generalized SEM

    • Analysis of main effects, simple effects, interaction effects, partial interaction effects, and nested effects
    • Comparisons against reference groups, of adjacent levels, or against the grand mean
    • Orthogonal polynomials
    • Helmert contrasts
    • Custom contrasts
    • ANOVA-style tests
    • Contrasts of nonlinear responses
    • Multiple-comparison adjustments
    • Balanced and unbalanced data
    • Contrasts of means, intercepts, and slopes
    • Graphs of contrasts
    • Interaction plots

    Pairwise comparisons for generalized SEM

    • Compare estimated means, intercepts, and slopes
    • Compare marginal means, intercepts, and slopes
    • Balanced and unbalanced data
    • Nonlinear responses
    • Multiple-comparison adjustments: Bonferroni, Sidak, Scheffe, Tukey HSD, Duncan, and Student–Newman–Keuls adjustments
    • Group comparisons that are significant
    • Graphs of pairwise comparisons

    Explore more about SEM in Stata.

    Additional resources

    See New in Stata 13 for more about what was added in Stata 13.

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