Structural Equation Modeling: An Overview

1. Structural Equation Modeling: An Overview P. Paxton

2. What are Structural Equation Models? • Also known as: • Covariance structure models • Latent variable models • “LISREL” models • Structural Equations with Latent Variables

3. What are Structural Equation Models? • Special cases: • ANOVA • Multiple regression • Path analysis • Confirmatory Factor Analysis • Recursive and Nonrecursive systems

4. What are Structural Equation Models? • SEM associated with path diagrams intelligence test 2 test 3 test 4 test 5 test 1 δ1 δ2 δ4 δ3 δ5

5. What are Structural Equation Models? Latent variables, factors, constructs Observed variables, measures, indicators, manifest variables Direction of influence, relationship from one variable to another Association not explained within the model

6. What are Structural Equation Models? ε1 ε2 ε3 Depress 1 Depress 2 Depress 3 Family support ζ1 depression ζ2 Physical health Self rated closeness Spousal rating Kids rating δ1 δ2 δ3 Self rating MD rating # visits to MD ε4 ε5 ε6

7. What are Structural Equation Models? • What can you do with these models? • Latent and Observed Variables • Multiple indicators of same concept • Measurement error • Restrictions on model parameters • Tests of model fit

8. What are Structural Equation Models? • What can’t you do? • Prove causation • Prove a model is “correct” All models Models consistent with data Models consistent with reality (Mueller 1997)

9. ε1 ε2 ε7 ε3 ε4 ε8 ε5 ε6 y8 y1 y2 y3 y7 y6 y5 y4 x2 x1 x3 δ1 δ3 δ2 Notation λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11 ζ1 ζ2 β21 η2 η1 γ11 γ21 ξ1 ξ1= industrialization η1= democracy time 1 η2= democracy time 2 x1-x3 = indus. indicators, e.g., energy y1-y4 = democ. indicators time 1 y5-y8 = democ. indicators time 2 λ1 λ2 λ3

10. Notation • η Latent Endogenous Variable • ξ Latent Exogenous Variable • ζ Unexplained Error in Model • x & y Observed Variables • δ & ε Measurement Errors • λ, β, & γ Coefficients

11. Notation • Two components to a SEM • Latent variable model • Relationship between the latent variables • Measurement model • Relationship between the latent and observed variables

12. Notation • Covariance Matrixes of Interest: • Φ • Ψ • Θδ • Θε

13. Example: Trust in Individuals Trust in Individuals ξ1 λ11 λ21 1 people are helpful (x1) people can be trusted (x2) people are Fair (x3) δ1 δ2 δ3

14. Latent Variables • Variables of Interest • Not directly measured • Common • Intelligence • Trust • Democracy • Diseases • Disturbance variables

15. Three Types of SEM • Classic Econometric • Multiple equations • One indicator per latent variable • No measurement error

16. Classic Econometric Citations y3 β32 β43 γ31 Publications y2 β42 β31 Quality rating y4 β41 Size of dept. y1 γ11 γ41 Private x1

17. Classic Econometric Ethnic homogeneity Noncore position industrialization 1980 democracy 1982 democracy 1991 trust 1980 trust 1990 associations 1980 associations 1990

18. Recursive / Nonrecursive • Recursive • Direction of influence one direction • No reciprocal causation • No feedback loops • Disturbances not correlated • Nonrecursive • Either reciprocal causation, feedback loops, or correlated disturbances

19. Recursive x1 y2 y3 y3 x1 y1 x2 y2 x3

20. Nonrecursive x2 y1 x1 y2 y1 x1 x2 y2 x3 y3

21. Confirmatory Factor Analysis • Latent variables • Measurement error • No causal relationship between latent variables x = vector of observed indicators Λx = matrix of factor loadings ξ = vector of latent variables δ = vector of measurement errors

22. Confirmatory Factor Analysis Trust in Individuals ξ1 λ11 λ21 1 people are helpful (x1) people can be trusted (x2) people are Fair (x3) δ1 δ2 δ3

23. General Model • Includes latent variable model • Relationship between the latent variables • And measurement model • Relationship between latent variables and observed variables

24. General Model • Latent Variable Model η = vector of latent endogenous variables ξ = vector of latent exogenous variables ζ = vector of disturbances Β = coefficient matrix for η on η effects Γ =coefficient matrix for ξ on η effects

25. General Model • Measurement Model x = indicators of ξ Λx = factor loadings of ξ on x y = indicators of η Λy = factor loadings of η on y δ = measurement error for x ε = measurement error for y

26. ε1 ε2 ε7 ε3 ε4 ε8 ε5 ε6 y8 y1 y2 y3 y7 y6 y5 y4 x2 x1 x3 δ1 δ3 δ2 General SEM λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11 ζ1 ζ2 β21 η2 η1 γ11 γ21 ξ1 ξ1= industrialization η1= democracy time 1 η2= democracy time 2 x1-x3 = indus. indicator, e.g., energy y1-y4 = democ. indicators time 1 y5-y8 = democ. indicators time 2 λ1 λ2 λ3

27. Six Steps to Modeling • Specification • Implied Covariance Matrix • Identification • Estimation • Model Fit • Respecification

28. Specification • Theorize your model • What observed variables? • How many observed variables? • What latent variables? • How many latent variables? • Relationship between latent variables? • Relationship between latent variables and observed variables? • Correlated errors of measurement?

29. Identification • Are there unique values for parameters? • Property of model, not data • 10 = x + y • 2, 8 • -1, 11 • 4, 6 • x = y

30. Identification • Underidentified • Just identified • Overidentified

31. Identification • Rules for Identification • By type of model • Classic econometric • e.g., recursive rule • Confirmatory factor analysis • e.g., three indicator rule • General Model • e.g., two-step rule

32. Identification • Identified? Yes, by 3-indicator rule. Trust in Individuals ξ1 λ11 λ21 1 people are helpful (x1) people can be trusted (x2) people are Fair (x3) δ1 δ2 δ3

33. Model Fit • Component Fit • Use Substantive Experience • Are signs correct? • Any nonsensical results? • R2s for individual equations • Negative error variances? • Standard errors seem reasonable?

34. Model Fit • How well does our model fit the data? • The Test Statistic (Χ2) • T=(N-1)F • df=½(p+q)(p+q+1) - # of parameters • p = number of y’s • q = number of x’s • Σ=Σ(θ) • Statistical power

35. Model Fit • Many goodness-of-fit statistics • Tb = chi-square test statistic for baseline model • Tm = chi-square test statistic for hypothesized model • dfb = degrees of freedom for baseline model • dfm = degrees of freedom for hypothesized model

36. Model Fit Χ2 = 223, df=5, p=.000 IFI = .87 RMSEA = .25 N=801

37. Respecification • Theory! • Dimensionality? • Correct pattern of loadings? • Correlated errors of measurement? • Other paths? • Modification Indexes • Residuals:

38. Respecification Χ2 = 3.8, df=2, p=.15 IFI = 1.0 RMSEA = .03 N=801

39. Useful References • Book from which this talk is drawn: Bollen, Kenneth A. 1989. Structural Equations with Latent Variables. New York: Wiley. • Ed Rigdon’s website: www.gsu.edu/~mkteer/ • Archives of SEMNET listserv: bama.ua.edu/archives/semnet.html