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General Structural Equation (LISREL) Models

General Structural Equation (LISREL) Models. Week 2 #3 LISREL Matrices The LISREL Program. The LISREL matrices. The variables: Manifest: X, Y Latent: Eta η Ksi ξ Error: construct equations: zeta ζ measurement equations delta δ , epsilon ε. The LISREL matrices.

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General Structural Equation (LISREL) Models

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  1. General Structural Equation (LISREL) Models Week 2 #3 LISREL Matrices The LISREL Program

  2. The LISREL matrices The variables: Manifest: X, Y Latent: Eta η Ksi ξ Error: construct equations: zeta ζ measurement equations delta δ, epsilon ε

  3. The LISREL matrices The variables: Manifest: X, Y Latent: Eta η Ksi ξ Error: construct equations: zeta ζ measurement equations delta δ, epsilon ε Coefficient matrices: x=λ ξ + δ Lambda-X Measurement equation for X-variables (exogenous LV’s) Y = λη + ε Lambda –Y Measurement equation for Y-variables (endogenous LV’s) η = γξ + ζ Gamma Construct equation connecting ksi (exogenous), eta (endogenous) LV’s η = β η + γξ + ζ Construct equation connecting eta with eta LV’s

  4. The LISREL matrices The variables: Manifest: X, Y Latent: Eta η Ksi ξ Error: construct equations: zeta ζ measurement equations delta δ, epsilon ε Variance-covariance matrices: PHI ( Φ) Variance covariance matrix of Ksi (ξ) exogenous LVs PSI (Ψ) Variance covariance matrix of Zeta (ζ) error terms (errors associated with eta (η) LVs Theta-delta (Θδ) Variance covariance matrix of δ (measurement) error terms associated with X-variables Theta-epsilon (Θε)Variance covariance matrix of ε (measurement) error terms associated with Y-variables Also: Theta-epsilon-delta

  5. (slides 5-11 from handout for 1st class this week:) 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

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

  7. 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, 2, 5, 9, 15. OR, using double-subscript notation: Theta-delta 1,1 Theta-delta 2,2 Theta-delta 3,3 Etc.

  8. 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

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

  10. 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

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

  12. Class Exercise #1 Provide labels for each of the variables Slides 12-19 not on handout; see handout for yesterday’s class

  13. #2

  14. #1 epsilon ksi eta zeta delta

  15. #2

  16. Lisrel Matrices for examples. No Beta Matrix in this model

  17. Lisrel Matrices for examples.

  18. Lisrel Matrices for examples (example #2)

  19. Lisrel Matrices for examples (example #2)

  20. Special Cases Single-indicator variables This model must be re-expressed as…. (see next slide)

  21. Special Case: single indicators Error terms with 0 variance

  22. Special Case: single indicators LISREL will issue an error message: matrix not positive definite (theta-delta has 0s in diagonal). Can “override” this.

  23. Special Case: single indicators Case where all exogenous construct equation variables are manifest

  24. Special Case: single indicators Case where all exogenous construct equation variables are manifest

  25. Special Case: correlated errors across delta,epsilon Special matrix: Theta delta-epsilon (TH)

  26. Special Case: correlated errors across exogenous,endogenous variables • Simply re-specify the model so that all variables are Y-variables • Ksi variables must be completely exogenous but Eta variables can be either (only small issue: there will still be a construct equation for Eta 1 above  Eta 1 = Zeta 1 (no other exogenous variables).

  27. Exercise: going from matrix contents to diagrams Matrices: LY 8 x 3 BE 3 x 3 1 0 0 Free elements: ly2,1 0 0 BE 2,1 ly3,1 0 LY3,3 BE 3,1 ly4,1 ly4,2 0 ly5,1 ly5,2 0 PS 3 X 3 0 1 0 Free elements: 0 0 1 - PS(3,2), all diagonals 0 0 LY8,3 (other off-diag’s = 0)

  28. Exercise: going from matrix contents to diagrams Matrices: LX is a 4 x 4 identity matrix! TE is a diagonal matrix with 0’s in the diagonal PH 4 x 4 all elements are free (diagonals and off –diagonals TE 8 x 8 • diagonals free • off-diagonals all zero GAMMA 3 x 4 ga1,1 ga1,2 0 0 ga2,1 0 ga2,3 ga2,4 0 ga3,2 ga3,3 ga3,4

  29. 2 key elements in the LISREL program • The MO (modelparameters) statement • Statements used to alter an “initial specification” • FI (fix a parameter initially specified as free) • FR (free a parameter initially specified as fixed) • VA (set a value to a parameter) • Not normally necessary for free parameters, though it can be used to provide start values in cases where program-supplied start values are not very good

  30. 2 key elements in the LISREL program • Statements used to alter an “initial specification” • FI (fix a parameter initially specified as free) • FR (free a parameter initially specified as fixed) • VA (set a value to a parameter) • Not normally necessary for free parameters, though it can be used to provide start values in cases where program-supplied start values are not very good • EQ (equality constraint)

  31. 2 key elements in the LISREL program MO statement: NY = number of Y-variables in model NX = number of X-variables in model NK = number of Ksi-variables in model NE = number of Eta-variables in model LX = initial specification for lambda-X LY = initial specification for lambda-Y BE = initial specification for Beta GA = initial specification for Gamma

  32. 2 key elements in the LISREL program MO statement: LX = initial specification for lambda-X LY = initial specification for lambda-Y BE = initial specification for Beta GA = initial specification for Gamma PH = initial specification for Phi PS = initial speicification for Psi TE = initial specification for Theta-epsilon TD = initial specification for Theta-delta [there is no initial spec. for theta-epsilon-delta]

  33. 2 key elements in the LISREL program MO specifications Example: NX=6 NK =2 LX = FU,FR “full-free” produces a 6 x 2 matrix: lx(1,1) lx(1,2) lx(2,1) lx(2,2) lx(3,1) lx(3,2) lx(4,1) lx(4,2) lx(5,1) lx(5,2) lx(6,1) lx(6,2) - Of course, this will lead to an under-identified model unless some constraints are applied

  34. 2 key elements in the LISREL program MO specifications Example: NX=6 NK =2 LX = FU,FI “full-fixed” produces a 6 x 2 matrix: 0 0 0 0 0 0 0 0 0 0 0 0

  35. MO specifications Example: With 6 X-variables and 2 Y-variables, we want an LX matrix that looks like this: lx(1,1) 0 lx(2,1) 0 lx(3,1) lx(3,2) lx(4,1) lx(4,2) 0 lx(5,2) 0 lx(6,2) MO NX=6 NK=2 LX=FU,FR FI LX(1,2) LX(2,2) LX(5,1) LX(6,1)

  36. MO specifications Example: With 6 X-variables and 2 Y-variables, we want an LX matrix that looks like this: 1 0 lx(2,1) 0 lx(3,1) lx(3,2) 0 lx(4,2) 0 1 0 lx(6,2) MO NX=6 NK=2 LX=FU,FI FR LX(2,1) LX(3,1) LX(3,2) LX(4,2) LX(6,2) VA 1.0 LX(1,1) LX(5,2)

  37. MO specifications Special case: All X-variables are single indicator. We will want LX as follows: Ksi-1 Ksi-2 Ksi-3 X1 1 0 0 X2 0 1 0 X3 0 0 1 And we will want var(delta-1) = var(delta-2) = var(delta-3) = 0 Specification: LX=ID TD=ZE

  38. VARIANCE-COVARIANCE MATRICES Initial specifications for PH, PS, TE, TD Option 1: PH=SY,FR - entire matrix has parameters (no fixed elements) Option 2: PH=SY,FI - entire matrix has fixed elements (no free elements) Option 3: PH=DI Diagonal matrix (implicit: zeroes in off-diagonals)

  39. VARIANCE-COVARIANCE MATRICES Option 3: PH=DI,FR Diagonal matrix (implicit: zeroes in off-diagonals) • In older versions of LISREL, this specification would not yield modification indices for off-diagonal elements • off-diagonals may not be added later on with FR specifications Option 4: PH=SY (parameters in diagonals, zeroes in off-diagonals) • off-diagonals may be added later with FR specifications Option 5: PH=ZE Zero matrix * * would never do this with PH but perhaps with TD

  40. Single Latent variable (CFA) Model Matrices: LX Lambda-X 3 x1 TD Theta delta 3 x 3 PH Phi 1 x 1 Lambda –X 1.0 Lx(2,1) Lx(3,1) PHI Ph(1,1) Theta-delta td(1,1) 0 td(2,2) 0 0 td(3,3)

  41. Single Latent variable (CFA) Model M0 NX=3 NK=1 LX=FU,FR C PH=SY TD=SY FI LX(1,1) VA 1.0 LX(1,1) C = CONTINUE FROM PREVIOUS LINE Lambda –X 1.0 Lx(2,1) Lx(3,1) PHI Ph(1,1) Theta-delta td(1,1) 0 td(2,2) 0 0 td(3,3)

  42. Single Latent variable (CFA) Model – Could Also be programmed as Y-Eta M0 NY=3 NE=1 LY=FU,FR C PS=SY TE=SY FI LY(1,1) VA 1.0 LY(1,1) C = CONTINUE FROM PREVIOUS LINE Lambda –Y 1.0 LY(2,1) LY(3,1) PSI PS(1,1) Theta-epsilon te(1,1) 0 te(2,2) 0 0 te(3,3)

  43. Two latent variable CFA model Lambda-X 6 x 2 1.0 0 LX(2,1) 0 LX(3,1) 0 0 1.0 0 LX(5,2) 0 LX(6,2) Phi 2 x 2 Ph(1,1) Ph(2,1) Ph(2,2) Theta-delta -- expressed as diagonal TD(1) TD(2) TD(3) TD(4) TD(5) TD(6)

  44. Two latent variable CFA model Lambda-X 6 x 2 1.0 0 LX(2,1) 0 LX(3,1) 0 0 1.0 0 LX(5,2) 0 LX(6,2) Phi 2 x 2 Ph(1,1) Ph(2,1) Ph(2,2) Theta-delta -- expressed as diagonal TD(1) TD(2) TD(3) TD(4) TD(5) TD(6) MO NX=6 NK=2 LX=FU,FI PH=SY,FR TD=DI,FR VA 1.0 LX(1,1) LX(4,2) FR LX(2,1) LX(3,1) LX(5,2) LX(6,2)

  45. Two latent variable CFA model Theta-delta -- expressed as symmetric matrix TD(1,1) TD(2,2) TD(3,3) TD(4,4) TD(5,5) TD(6,6) Theta-delta Td(1,1) 0 td(2,2) 0 0 td(3,3) 0 0 0 td(4,4) 0 0 0 0 td(5,5) 0 0 0 0 0 td(6,6) MO NX=6 NK=2 LX=FU,FI PH=SY,FR TD=SY VA 1.0 LX(1,1) LX(4,2) FR LX(2,1) LX(3,1) LX(5,2) LX(6,2)

  46. Two latent variable CFA model – a couple of complications Correlated error: td(5,3) Added path: LX(2,2) MO NX=6 NK=2 LX=FU,FI PH=SY,FR TD=SY VA 1.0 LX(1,1) LX(4,2) FR LX(2,1) LX(3,1) LX(5,2) LX(6,2) FR LX(2,2) FR TD(5,3)

  47. A model with an exogenous latent variable Lambda-y = same as lambda x previous model Psi 2 x 2 symmetric, free Gamma = 2 x 1 Phi 1 x 1 Lambda-x 3 x 1 Theta delta 3 x 3

  48. A model with an exogenous latent variable Gamma 1 x 2 GA(1,1) GA(2,2) Phi 1 x 1 PH(1,1) PSI 2 x 2 PS(1,1) PS(2,1) PS(2,2) Lambda-Y 1.0 0 LY(2,1) LY(2,2) LY(3,1) 0 0 1.0 0 LY(5,2) 0 LY(6,2) Theta-eps: See previous example TD Theta delta – diagonal TD(1) TD(2) TD(3)

  49. A model with an exogenous latent variable MO NX=3 NY=6 NK=1 NE=2 LX=FU,FR LY=FU,FI GA=FU,FR C PS=SY,FR PH=SY,FR TD=DI,FR TE=SY VA 1.0 LY(1,1) LY(4,2) LX(1,1) FR LY(2,1) LY(2,2) LY(3,1) LY(5,2) LY(6,2) LX(2,1) LX(3,1) FR TE(5,3)

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