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 ηKsi ξ Error:construct equations: zeta ζ measurement equations delta δ, epsilon ε

3 The LISREL matrices The variables: Manifest:X, YLatent: Eta ηKsi 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 = λ η + ε Lambda –Y Measurement equation for Y-variables (endogenous LV’s) η = γ ξ + ζ Gamma Construct equation connecting ksi (exogenous), eta (endogenous) LV’s η = β η + γ ξ + ζ Construct equation connecting eta with eta LV’s

4 The LISREL matrices The variables: Manifest:X, YLatent: Eta ηKsi 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 (Θ δ ) Variance covariance matrix of δ (measurement) error terms associated with X-variables Theta-epsilon (Θ ε )Variance covariance matrix of ε (measurement) error terms associated with Y-variables Also: Theta-epsilon-delta

5 Matrix form: LISREL M EASUREMENT MODEL MATRICES Manifest variables: X’s Measurement errors: DELTA ( δ ) Coefficients in measurement equations: LAMBDA ( λ ) Sample equation: X 1 = λ 1 ξ 1 + δ 1 MATRICES: LAMBDA-x THETA-DELTAPHI (slides 5-11 from handout for 1 st class this week:)

6 Matrix form: LISREL M EASUREMENT MODEL MATRICES A slightly more complex example:

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

9 Matrix form: LISREL M EASUREMENT MODEL MATRICES Models with correlated measurement errors:

10 Matrix form: LISREL M EASUREMENT 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 Y 1 = λ 1 η 1 + ε 1

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

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

13 #2

14 #1 delta epsilon ksi eta zeta

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 100Free elements: ly2,100BE 2,1 ly3,10LY3,3BE 3,1 ly4,1ly4,20 ly5,1ly5,20PS 3 X 3 010Free elements: 001- PS(3,2), all diagonals 00LY8,3 (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 ga1,1ga1,20 0 ga2,10ga2,3ga2,4 0ga3,2ga3,3 ga3,4

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30 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

31 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)

32 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

33 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]

34 2 key elements in the LISREL program MO specifications Example: NX=6NK =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

35 2 key elements in the LISREL program MO specifications Example: NX=6NK =2 LX = FU,FI“full-fixed” produces a 6 x 2 matrix:000000

36 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) 0lx(5,2) 0lx(6,2) MO NX=6 NK=2 LX=FU,FR FI LX(1,2) LX(2,2) LX(5,1) LX(6,1)

37 MO specifications Example: With 6 X-variables and 2 Y-variables, we want an LX matrix that looks like this: 10 lx(2,1)0 lx(3,1)lx(3,2) 0lx(4,2) 0 1 0lx(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)

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

39 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)

40 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

41 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) 0td(2,2) 00td(3,3)

42 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) Lambda –X 1.0 Lx(2,1) Lx(3,1) PHI Ph(1,1) Theta-delta td(1,1) 0td(2,2) 00td(3,3) C = CONTINUE FROM PREVIOUS LINE

43 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) Lambda –Y 1.0 LY(2,1) LY(3,1) PSI PS(1,1) Theta-epsilon te(1,1) 0te(2,2) 00te(3,3) C = CONTINUE FROM PREVIOUS LINE

44 Two latent variable CFA model Lambda-X 6 x LX(2,1)0 LX(3,1) LX(5,2) 0LX(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)

45 Two latent variable CFA model Lambda-X 6 x LX(2,1)0 LX(3,1) LX(5,2) 0LX(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)

46 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) 0td(2,2) 00td(3,3) 000td(4,4) 0000td(5,5) 00000td(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)

47 Two latent variable CFA model – a couple of complications 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) Correlated error: td(5,3) Added path: LX(2,2)

48 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 1Theta delta 3 x 3

49 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 LY(2,1)LY(2,2) LY(3,1) LY(5,2) 0LY(6,2) Theta delta – diagonal TD(1) TD(2) TD(3) Theta-eps: See previous example TD

50 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)

51 A model with intervening variables (a non-zero BETA matrix) BETA is 4 x 4 GAMMA is 4 x 1 BETA 0000 BE(2,1)000 0BE(3,2)00 0BE(4,2)00 Zeta1, zeta2 not shown Gamma GA(1,1) GA(2,1) 0

52 A model with intervening variables (a non-zero BETA matrix) MO NX=3 NY=13 NE=4 NK=1 LX=FU,FR LY=FU,FI PS=SY PH=SY,FR C TD=SY TE=SY BE=FU,FI GA=FU,FI FR BE(2,1) BE(4,2) BE(3,2) GA(1,1) GA(1,2) …. Plus LY and LX specifications

53 Single-indicator exogenous variables Special features: MO NX=5 NK=5 LX=ID TD=ZE PH=SY,FR –LX is identity matrix MO NX=5 NK=5 FIXEDX –Special specification if all of the variables in X are single-indicator and measured without error –Specify Gamma and Y-variable matrices as usual

54 Class Exercise (if time permits)