1 / 54

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.

verity
Download Presentation

General Structural Equation (LISREL) Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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)

More Related