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General Structural Equation (LISREL) Models

General Structural Equation (LISREL) Models. Week #2 Class #1. Today’s class:. Standardization A review: the “metric” of latent variables Relationship with reference indicator Variances of latent variables Interpreting coefficients Other metric/scaling issues

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General Structural Equation (LISREL) Models

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  1. General Structural Equation (LISREL) Models Week #2 Class #1

  2. Today’s class: • Standardization • A review: the “metric” of latent variables • Relationship with reference indicator • Variances of latent variables • Interpreting coefficients • Other metric/scaling issues • Variables with very large or very small variances • Quick introduction to programming in SAS and EQS • Structural Equation Models in Matrix form

  3. Standardization • bstandardized bz= b * {SQRT[var(x)] /SQRT[ var(y)]} • In a measurement model, • bz = b * ( SQRT variance of latent variable ) SQRT variance of manifest variable • In a construct equation model • bz = b * (SQRT variance of exogenous LV / SQRT variance of endogenous LV)

  4. Standardization • In a measurement model, • bz = b * ( SQRT variance of latent variable ) sqrt variance of manifest variable • What LISREL calls “standardized” omits the division by the manifest variable • Look for “COMPLETELY STANDARDIZED” solution • Standardized coefficients are useful for comparing the size of effects of different variables in a single model • As with regular regression, not so useful comparing across samples

  5. Standardization • Standardized coefficients are useful for comparing the size of effects of different variables in a single model • As with regular regression, not so useful comparing across samples • Generally not appropriate for dummy variables • Especially inappropriate for dummy variables where # of categories for a categorical variable > 2

  6. Standardization • Generally not appropriate for dummy variables • Especially inappropriate for dummy variables where # of categories for a categorical variable > 2 • Can calculate R-square for the entire variable

  7. R-square for categorical variables Country with 4 categories: USA; Canada; France; Germany (Germ=ref) R2 explained for country: 1. Get R2 for model shown 2. Subtract R2 for model with B1=0, B2=0, B3=0

  8. Variances of latent variables What is the variance of LV1? X1 = 1.0 LV1 + e1 VAR(X1) = COV(LV1,LV1) + 2*COV(LV1,e1) + COV(e1,e1) = VAR(L1) + VAR(e1) VAR(L1) = VAR(X1) – VAR(e1) If VAR(e1) is small, VAR(L1) is ≈ VAR(X1) Otherwise, VAR(L1) < VAR(X1) Alternative: fix VAR(L1) = 1.0

  9. Variances of latent variables In this model, we can no longer fix VAR(LV1) = 1.0 We could fix VAR(d1) = 1.0 If so, VAR(LV1) = b12 VAR(X4) + VAR(d1) (WE could impose a complicated constraint: VAR(d1) = 1 – ( b12 * VAR(X4)) b12, VAR(X4) are model parameters Complex constraints of this sort not possible in AMOS

  10. Variances of latent variables We could fix VAR(d1) = 1.0 If so, VAR(LV1) = b12 VAR(X4) + VAR(d1) (WE could impose a complicated constraint: VAR(d1) = 1 – ( b12 * VAR(X4)) b12, VAR(X4) are model parameters Complex constraints of this sort not possible in AMOS Also: more complicated with multiple exogenous variables, even more complicated if X4-type variables are latent var’s

  11. Variances of latent variables We could fix VAR(d1) = 1.0 If so, VAR(LV1) = b12 VAR(X4) + VAR(d1) (WE could impose a complicated constraint: VAR(d1) = 1 – ( b12 * VAR(X4)) b12, VAR(X4) are model parameters Complex constraints of this sort not possible in AMOS Also: more complicated with multiple exogenous variables, even more complicated if X4-type variables are latent var’s

  12. How do we interpret b1?

  13. X1 = Avg. church attendance per month X2 = 4 point scale, 4=very religious through 1 = not at all X3 = 10 point scale, importance of religion in my life X4 = 10 point scale, 1=abortion never justified 10 = abortion always justified X5 = 4 point scale, Abortion a matter of woman’s choice: 4=agree strongly through 1 = disagree strongly X6 = 5 point scale, Abortion should never be allowed 5=agree strongly through 1 = disagree strongly (3=“neutral”)

  14. X1 = Avg. church attendance per month * X2 = 4 point scale, 4=very religious through 1 = not at all X3 = 10 point scale, importance of religion in my life X4 = 10 point scale, 1=abortion never justified 10 = abortion always justified * X5 = 4 point scale, Abortion a matter of woman’s choice: 4=agree strongly through 1 = disagree strongly X6 = 5 point scale, Abortion should never be allowed 5=agree strongly through 1 = disagree strongly (3=“neutral”)

  15. X1 = Avg. church attendance per month * X4 = 10 point scale, 1=abortion never justified 10 = abortion always justified * Interpretation: 1 unit increase in church attendance per month leads to b1 increase In abortion support along 10-point scale

  16. Interpretation: 1 unit increase in church attendance per month leads to b1 increase In abortion support along 10-point scale Because the variance of the dependent variable is likely to be (much) higher than the variance of the independent variable, we expect large values for b1 (could be >1 if effect is strong). If this were reversed, we would have small values of b1, even if there is a strong effect.

  17. If X1 is measured on a 4-point scale, Variance of AbortAtt might be quite small (e.g., .5 or .3 or less) Variance of age can be quite large (e.g., 300, 400).  this implies that b2 can be very small, even if effect is large (standardized b2 not affected) Could scale down variance of Age (in SPSS: Age = Age/10)

  18. SAS, EQS programming. • The basics: • Specify equations • Specify “variances” (list variables for which variances are model parameters) • Specify “covariances (list covariances among exogenous variables, if any)

  19. SAS, EQS programming. Sample SAS program: proc calis m=ml mod data=matrix1 cov; lineqs v275 = f1 + e1, v276 = b1 f1 + e2, v277 = b2 f1 + e3, v278 = b3 f1 + e4, v279 = b4 f1 + e5, v280 = b5 f1 + e6, v281 = b6 f1 + e7, v282 = b7 f1 + e8, v283 = b8 f1 + e9; std f1 =vf1, e1=ve1, e2=ve2, e3=ve3, e4=ve4, e5=ve5, e6=ve6, e7=ve7, e8=ve8 , e9=ve9; Run; In SAS, all parameters must be named. Can’t just say “variance of f1 is free” – must give it a parameter name

  20. EQS programming Sample program: /title any title /spec cas=342; var=8; ma=cov; an=cov; da=‘c:\data1.cov’; fo=‘(8f10.2)’; /equ v1 = *F1+ e1; v2 = 1.0 F1 + e2; v3 = *F1 + e3; V4 = *F1 + *F2 + e4; V5 = 1.0 F2 + e5; V6 = *F2 + e6; v7 = *F2 + e7; V8 = *F2 + e8; /var f1=*; f2=*; e1 to e8 = *; /cov f1,f2=* /lmtest /end

  21. LV Structural Equation Models in Matrix terms (continued) From last week

  22. Two scalar equations re-written scalar Matrix Contents of matrices

  23. Reproduced covariances (the formula in matrix terms) Θ above – elements of which are called θ[theta] is not the same as θ in Σ(θ). Latter refers to all parameters in a model. Theta above refers to elements in the variance-covariance matrix of errors/exogenous variables.

  24. A simple model: B Continued……..

  25. Reproduced covariances (observed variable model without latent variables)

  26. (proof of inverse: quick aside)

  27. Measurement (“factor”) model

  28. Alternative notation systems for coefficients:

  29. Matrix Form Latent variables only: PHI (Φ) GAMMA ( Γ) PSI (Ψ)

  30. Matrix form: LISREL matrices LISREL calls the LVs ξ (KSI) or η (ETA), depending on whether they are exogenous or endogenous

  31. Matrix form: LISREL matrices The GAMMA (Γ) matrix is for paths from KSI (ξ ) latent variables to ETA (η ) latent variables. KSI variables must be completely exogenous. In the model shown below, ETA-1 (η1 ) is exogenous with respect to ETA-2 (η2 ) but endogenous with respect to KSI-1 (ξ1 ) . • It is therefore endogenous -- an ETA and not a KSI variable.

  32. Matrix form: LISREL matrices Paths from one ETA (latent) variable to another are represented by elements in the BETA ( Β ) matrix. GAMMA 2 x 2 BETA ( 2 x 2)

  33. Matrix form: LISREL matrices BETA will be sub-diagonal if the model is recursive:

  34. Matrix form: LISREL matrices BETA will be sub-diagonal if the model is recursive: Can be re-expressed as:

  35. Matrix form: LISREL matrices BETA will be sub-diagonal if the model is recursive, but above-diagonals will be used if the model is non-recursive:

  36. Matrix form: LISREL matricesMEASUREMENT MODEL MATRICES From above: Two types of latent variables: ETA (η ) endogenous KSI (ξ ) completely exogenous MEASUREMENT MODEL FOR KSI VARIABLES: Manifest variables: X’s Measurement errors: DELTA ( δ) Coefficients in measurement equations: LAMBDA ( λ ) Sample equation: X1 = λ1ξ1+ δ1

  37. Matrix form: LISREL MEASUREMENT MODEL MATRICES Manifest variables: X’s Measurement errors: DELTA ( δ) Coefficients in measurement equations: LAMBDA ( λ ) Sample equation: X1 = λ1ξ1+ δ1 MATRICES: LAMBDA-x THETA-DELTA PHI

  38. Matrix form: LISREL MEASUREMENT MODEL MATRICES A slightly more complex example:

  39. Matrix form: LISREL MEASUREMENT MODEL MATRICES Labeling shown here applies ONLY if this matrix is specified as “diagonal” Otherwise, the elements would be: Theta-delta 1, 3, 6, 10, 15. OR, using double-subscript notation: Theta-delta 1,1 Theta-delta 2,2 Theta-delta 3,3 Etc.

  40. Matrix form: LISREL MEASUREMENT MODEL MATRICES While this numbering is common in some journal articles, the LISREL program itself does not use it. Two subscript notations possible: Single subscript Double subscript

  41. Matrix form: LISREL MEASUREMENT MODEL MATRICES Models with correlated measurement errors:

  42. Matrix form: LISREL MEASUREMENT MODEL MATRICES Measurement models for endogenous latent variables (ETA) are similar: • Manifest variables are Ys • Measurement error terms: EPSILON ( ε ) • Coefficients in measurement equations: LAMBDA (λ) • same as KSI/X side • to differentiate, will sometimes refer to LAMBDAs as Lambda-Y (vs. Lambda-X) • Equations • Y1 = λ1η1+ ε1

  43. Matrix form: LISREL MEASUREMENT MODEL MATRICES Measurement models for endogenous latent variables (ETA) are similar:

  44. LISREL MATRIX FORM An Example:

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