Structural Equation Modeling Using Stata

1 Structural Equation Modeling Using StataPaul D. Allison, Upcoming Seminar: August 16-17, 2018, Stockholm2/3/20171 introduction toStructural Equation ModelingUsing StataStructural Equation ModelsWhat is SEM good for?SEMP review: A Latent Variable SEML atent Variable Model (cont.)CautionsOutlineSoftware for SEMsFavorite TextbookLinear Regression in SEMGSS2014 ExampleLinear Regression with StataFIML for Missing DataFurther ReadingAssumptionsFIML in StataPath Diagram (from Mplus)Path Analysis of Observed VariablesSome Rules and DefinitionsThree Predictor VariablesTwo- Equation SystemWhy combine the two equations ?

2 Calculation of Indirect EffectA More Complex ModelDecomposition of Direct & Indirect EffectsStandardized Coefficients1234567891011121314151617181 9202122232425262728234567891011121314151 61718192021222324252627282/3/20172 Numerical ExamplesMore Complex ExampleDecomposition of EffectsIllness DataSummary DataCovariance MatrixCovariance Matrix for Illness DataIllness Regression in StataStata Results - UnstandardizedCounting Moments & ParametersMplus Results - StandardizedIllness Model with Indirect EffectsModel DiagramPath Diagrams in StataResultsMore Goodness of Fit MeasuresIdentification Status of the ModelImproving the ModelGOF for Improved ModelEstimates for Improved ModelIndirect EffectsIndirect Effects in StataSpecific Indirect Effects in StataPartial CorrelationsPartial Correlations (cont.)

3 Maruyama (1998) DataPartial Correlations in StataPartial Correlation ResultsCausal Ordering 2930313233343536373839404142434445464748 4950515253545556572930313233343536373839 4041424344454647484950515253545556572/3/ 20173 How to DecideNonrecursive SystemsIdentification Problem in Nonrecursive ModelsIdentification Problem (cont.)A Just-Identified ModelReduced Form EquationsSolutions for Structural ParametersSufficient Condition for IdentificationVarieties of IdentificationProblems with Instrumental VariablesExample of a Nonrecursive ModelNonrecursive Example (cont.) Stata Code for Nonrecursive ModelNonrecursive ResultsGOF for NonrecursiveLatent Variable ModelsRoadmap for Latent VariablesClassical Test TheoryRandom Measurement ErrorReliabilityParallel MeasuresTau-Equivalent MeasuresTau-Equivalance: ExampleTau-Equivalence in StataCongeneric TestsThree Congeneric TestsThree Congeneric Measures (cont.)

4 Identification in GeneralStandardized Version585960616263646566676869707172737 4757677787980818283848586585960616263646 5666768697071727374757677787980818283848 5862/3/20174 Digression: Tracing Rule for CorrelationsTracing Rule (cont.)Tracing Rule (cont.)Standardized Version of 3 Congenerics (cont.)Three Congenerics: ExampleThree Congenerics: StataThree Congenerics: ResultsFour Congeneric MeasuresOveridentification with 4 Congeneric MeasuresFour Congeneric Measures with StataEstimates for Four CongenericsGOF for Four CongenericsAlternative Model for 4 Congeneric MeasuresStata for Alternative ModelResults for Alternative ModelHeywood CaseFactor ModelsFactor Models (cont.)

5 Identification (Standardized)Identification (cont.)Two Approaches to Identification ProblemIdentification (Unstandardized)Determining IdentificationNormalizing ConstraintsNormalizing ConstraintsML Estimation of CFA ModelsMultivariate NormalityML DetailsChi-Square Test878889909192939495969798991001011021 0310410510610710810911011111211311411587 8889909192939495969798991001011021031041 051061071081091101111121131141152/3/2017 5 Example: Self-Concept MeasurementSelf Concept Path DiagramSelf Concept. ResultsSelf Concept Results (cont.)Self Concept Results (cont.)Global Goodness of Fit MeasuresOther Global MeasuresOther Global Measures (cont.)

6 Specific Goodness of Fit MeasuresStandardized Residuals for Self-Concept ModelModification IndicesMod Indices for Self-ConceptMod Indices for Self-Concept (cont.)Freeing Up ParametersResults from Freeing 1 ParameterSelected Results (cont.)Correlated ErrorsTwo Correlated ErrorsA Five-Indicator ModelA Two-Factor ModelExample: Self-Concept DataSelected ResultsThe General Structural Equation ModelGSS2014 Example: Stata CodeGSS2014 Example: GOF ResultsGSS2014: Standardized ResultsGSS2014: Standardized ResultsFarm Manager Example (Rock et al. 1977)Farm Managers Path Diagram116117118119120121122123124125126 1271281291301311321331341351361371381391 4014114214314411611711811912012112212312 4125126127128129130131132133134135136137 1381391401411421431442/3/20176 Farm Managers: Stata CodeFarm Managers.

7 Selected ResultsSelected Results (cont.)A Tau-Equivalent ModelParallel ModelIdentification in SEM ModelsAn Identified SEMWhat to Do If Endogenous Variables Aren t NormalExample: NLSY DataML Results for NLSY DataBoth Variables Highly SkewedSatorra-Bentler Robust SE sWeighted Least SquaresWeighted Least SquaresWLS ResultsMultiple Group AnalysisSubjective Class ExampleReading in the Data in StataSubjective Class ModelsStata Code for 2-Group ModelsStata Code (cont.)Tests for Comparing the GroupsModel 2 ResultsModel 2 Results (cont.)Wald & Score Tests Comparing GroupsOutput from estat ginvariantMore output from estat ginvariantInteractions and Non-LinearitiesOrdinal and Binary Data145146147148149150151152153154155156 1571581591601611621631641651661671681691 7017117217314514614714814915015115215315 4155156157158159160161162163164165166167 1681691701711721732/3/20177 Special CorrelationsSpecial CorrelationsSpecialized ModelsCFA Model with Categorical IndicatorsSelected ResultsOther Capabilities of gsemCautions About SEMsExamples I Don t LikeExamples I LikeSEMs and CausalityExemplary ArticleSome Recommendations.

8 Lest you forget1741751761771781791801811821831841 85174175176177178179180181182183184185 introduction toStructural Equation ModelingUsing StataPaul D. Allison, InstructorFebruary 2017 Paul AllisonStructural Equation ModelsThe classic SEM includes many common linear models used in the behavioral sciences: Multiple regression ANOVA Path analysis Multivariate ANOVA and regression Factor analysis Canonical correlation Non-recursive simultaneous equations Seemingly unrelated regressions Dynamic panel data models2 What is SEM good for? Modeling complex causal mechanisms. Studying mediation (direct and indirect effects).

9 Correcting for measurement error in predictor variables. Avoiding multicollinearityfor predictor variables that are measuring the same thing. Analysis with instrumental variables. Modeling reciprocal relationships (2-way causation). Handling missing data (by maximum likelihood). Scale construction and development. Analyzing longitudinal data. Providing a very general Modeling framework to handle all sorts of different problems in a unified of psychometrics and econometrics Simultaneous Equation models, possibly with reciprocal (nonrecursive) relationships Latent (unobserved) variables with multiple indicators.

10 Latent variables are the most distinguishing feature of SEM. For example:4 XYx1x2y1y2e1e2e3e4uabfcdX and Y are unobserved variables, x1, x2, y1, and y2 are observed indicators, e1-e4 and u are random errors. a, b, c, d, and f are correlation coefficients. Preview: A Latent Variable SEM5 Latent Variable Model (cont.)6 If we know the six correlations among the observed variables, simple hand calculations can produce estimates of athrough f. We can also test the fit of the model. Why is it desirable to estimate models like this? Most variables are measured with at least some error.