Structural Equation Modeling Using Mplus

1. Structural Equation Modeling Using Mplus Chongming Yang Research Support Center FHSS College

2. Structural? • Structuralism • Components • Relations

3. Objectives • Introduction to SEM • The model • Parameters • Estimation • Model evaluation • Applications • Estimate simple models with Mplus

4. Continuous Dependent Variables Session I

5. Information of Variable • Mean • Variance • Skewedness • Kurtosis

6. Variance & Covariance

7. Covariance Matrix (S) x1 x2 x3 x1 V1 x2 Cov21 V2 x3 Cov31 Cov32 V3

8. Statistical Model • Probabilistic statement about Relations of variables • Imperfect but useful representation of reality

9. Structural Equation Modeling • A system of regression equations for latent variables to estimate and test direct and indirect effects without the influence of measurement errors. • To estimate and test theories about interrelations among observed and latent variables.

10. Latent Variable (Construct / Factor / Trait) • A hypothetical variable • cannot be measured directly • No objective measurement unit • inferred from observable manifestations • Multiple manifestations (indicators) • Normally distributed interval dimension

11. How is Depression Distributed in? • BYU students • Patients for Therapy

12. Normal Distributions

13. Levels of Analyses • Observed • Latent

14. Test Theories • Classical True Score Theory: Observed Score = True score + Error • Item Response Theory • Generalizability (Raykov & Marcoulides, 2006)

15. Graphic Symbols of SEM • Rectangle – observed variable • Oval -- latent variable or error • Single-headed arrow -- causal relation • Double-headed arrow -- correlation

16. Graphic Measurement Model of Latent  X1 1 1 2 X2  2 X3 3 3

17. Equations • Specific equations X1 = 1 + 1 X2 = 2 + 2 X3 = 3 + 3 • Matrix Symbols X =  +  • True Score Theory?

18. Relations of Variances VX1 = 12 + 1 VX2 = 22 + 2 VX3 = 32 + 3  = measurement error / uniqueness

19. Unknown Parameters VX1 = 12+ 1 VX2 = 22+ 2 VX3 = 32 + 3

20. Sample Covariance Matrix (S) x1 x2 x3 x1 V1 x2 Cov21 V2 x3 Cov31 Cov32 V3

21. Variance of  • Variance of  = common covariance of X1 X2 and X3 1 0 0 Variance of  2 3 0

22. Unstandardized Parameterization(scaling) • 1 = 1 (set variance of X1 =1; X1 called reference Indicator) • Variance of  = common variance of X1 X2 and X3 • Squared  = explained variance of X (R2) • Variance of  = unexplained variance--error • Total Variance = Squared  +  Variance

23. Just Identified Model X1 1 1 2 X2  2 X3 3 3

24. Reference Indicator(marker) • Choose conceptually the best • Small variance  non-convergence • Different markers  different parameters estimates and their standard errors • Affect measurement invariance tests • Not affect standardized estimates

25. Standardized Parameterizations(scaling) • Variance of  = 1 = common variance of X1 X2 and X3 • Squared  = explained variance of X (R2) • Variance of  = 1 - 2 • Mean of  = 0 • Mean of  = 0

26. Two Kinds of Parameters • Fixed at 0, 1, or other values • Freely estimated

30. Structural Equation Modelin Matrix Symbols • X = x +  (exogenous) • Y = y +  (endogenous) •  =  +  +  (structural model) Note: Measurement model reflects the true score theory

31. Structural Equation Modelin Matrix Symbols • X = x + x +  (measurement) • Y = y + y +  (measurement) •  = α +  +  +  (structural) Note: SEM with mean structure.

32. Model Implied Covariance Matrix(Σ) Note: This covariance matrix contains unknown parameters in the equations. (I-B) = non-singular

33. Estimations/Fit Functions • Hypothesis:  = S or  - S = 0 • Maximum Likelihood F = log|||| + trace(S-1) - log||S|| - (p+q)

34. Convergence -- Reaching Limit • Minimize F while adjust unknown Parameters through iterative process • Convergence value: F difference between last two iterations • Default convergence = .0001 • Increase to help convergence (0.001 or 0.01) e.g. Analysis: convergence = .01;

35. No Convergence • No unique parameter estimates • Lack of degrees of freedom  under identification • Variance of reference indicator too small • Fixed parameters are left to be freely estimated • Misspecified model

36. Absolute Fit Index 2 = F(N-1) (N = sample size) df = p(p+1)/2 – q P = number of variances, covariances, & means q = number of unknown parameters to be estimated prob = ? (Nonsignificant 2 indicates good fit, Why?)

37. Sample Information x1 x2 x3 x4 … x1 v1 x2 cov21 v2 x3 cov31 cov32 v3 x4 cov41 cov42 cov43 v4 … … Mean1 Mean2 Mean3 Mean4 … Total info = P(P+1)/2 + Means

38. Absolute Fit -- SRMR • Standardized Root Mean Square Residual • SRMR = Difference between observed and implied covariances in standardized metric • Desirable when < .90, but no consensus

39. Relative Fit: Relative to Baseline (Null) Model • All unknown parameters are fixed at 0 • Variables not related (====0) • Model implied covariance  = 0 • Fit to sample covariance matrix S • Obtain 2, df, prob< .0000

40. Relative Fit Indices • CFI = 1- (2-df)/(2b-dfb) • b = baseline model • Comparative Fit Index, desirable => .95; 95% better than b model • TLI = (2b/dfb - 2/df) / (2b/dfb-1) (Tucker-Lewis Index, desirable => .90) • RMSEA = √(2-df)/(n*df) (Root Mean Square of Error Approximation, desirable <=.06 penalize a large model with more unknown parameters)

41. Special Case A

42. Special Cases A • Assumption: x =  y = x +  +   =  + x + 

43. Special Case B

44. Special Cases B Assumption: y =  x = x + x +  y =  +  + 

45. Other Special Cases of SEM • Confirmatory Factor Analysis (measurement model only) • Multiple & Multivariate Regression • ANOVA / MANOVA (multigroup CFA) • ANCOVA • Path Analysis Model (no latent variables) • Simultaneous Econometric Equations… • Growth Curve Modeling • …

46. EFA vs. CFA

47. Multiple Regression

48. ANCOVA

49. Multivariate Normality Assumption Observed data summed up perfectly by covariance matrix S (+ means M), S thus is an estimator of the population covariance 

50. Consequences of Violation Inflated 2 & deflated CFI and TLI reject plausible models Inflated standard errors  attenuate factor loadings and relations of latent variables (structural parameters) (Cause: Sample covariances were underestimated)