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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 = 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 matricesThe variables:Manifest: X, Y Latent: Eta η Ksi ξError: construct equations: zeta ζ measurement equations delta δ, epsilon ε
Variance-covariance matrices:PHI ( Φ) Variance covariance matrix of Ksi (ξ) exogenous LVsPSI (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
Matrix form: LISREL MEASUREMENT MODEL MATRICES
Manifest variables: X’s
Measurement errors: DELTA ( δ)
Coefficients in measurement equations: LAMBDA ( λ )
Sample equation:
X1 = λ1 ξ1+ δ1
MATRICES:
LAMBDA-x THETA-DELTA PHI
(slides 5-11 from handout for 1st class this week:)
6
Matrix form: LISREL MEASUREMENT MODEL MATRICES
A slightly more complex example:
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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 = λ1 η 1+ ε1
11
Matrix form: LISREL MEASUREMENT MODEL MATRICES
Measurement models for endogenous latent variables (ETA) are similar:
12
Class Exercise
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1
1
1
#1
Provide labels for each of the variables
Slides 12-19 not on handout; see handout for yesterday’s class
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#2
1
1
1
1
1
14
#1
delta
epsilon
ksieta
zeta
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#2
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Lisrel Matrices for examples.
No Beta Matrix in this model
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Lisrel Matrices for examples.
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Lisrel Matrices for examples (example #2)
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Lisrel Matrices for examples (example #2)
20
Special CasesSpecial Cases
Single-indicator variables
1
1
1
1
This model must be re-expressed as…. (see next slide)
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Special Case: single indicatorsSpecial Case: single indicators
1
1
1
1
0
0
1
1
Error terms with 0 variance
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Special Case: single indicatorsSpecial Case: single indicators
1
1
1
1
0
0
1
1
LISREL will issue an error message: matrix not positive definite (theta-delta has 0s in diagonal). Can “override” this.
23
Special Case: single indicatorsSpecial Case: single indicators
1
1
Case where all exogenous construct equation variables are manifest
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Special Case: single indicatorsSpecial Case: single indicators
Case where all exogenous construct equation variables are manifest
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Special Case: correlated errors across delta,epsilonSpecial Case: correlated errors across delta,epsilon
Special matrix:
Theta delta-epsilon (TH)
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Special Case: correlated errors across exogenous,endogenous Special Case: correlated errors across exogenous,endogenous variablesvariables
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 31 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 diagonalPH 4 x 4
all elements are free (diagonals and off –diagonalsTE 8 x 8 • diagonals free• off-diagonals all zeroGAMMA 3 x 4
ga1,1 ga1,2 0 0ga2,1 0 ga2,3 ga2,40 ga3,2 ga3,3 ga3,4
29
eta1
y111
y21
y31
eta2
y4
y5
y6
1
1
11
eta3
y7
y8
11
1
zeta1
1
zeta2zeta3
ksi1x11
01
ksi2x2
0
11
ksi3x3
0
11
ksi4x4
0
11
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-XLY = initial specification for lambda-YBE = initial specification for BetaGA = initial specification for GammaPH = initial specification for PhiPS = initial speicification for PsiTE = initial specification for Theta-epsilonTD = initial specification for Theta-delta[there is no initial spec. for theta-epsilon-delta]
34
2 key elements in the LISREL program
MO specificationsExample: 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
35
2 key elements in the LISREL program
MO specificationsExample: NX=6 NK =2
LX = FU,FI “full-fixed”produces a 6 x 2 matrix:
0 00 00 00 00 00 0
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)
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)
37
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)
38
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
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
11
1
1
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)
42
Single Latent variable (CFA) Model
11
1
1
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)
0 td(2,2)
0 0 td(3,3)
C = CONTINUE FROM PREVIOUS LINE
43
Single Latent variable (CFA) Model – Could Also be programmed as Y-Eta
11
1
1
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)
0 te(2,2)
0 0 te(3,3)
C = CONTINUE FROM PREVIOUS LINE
44
Two latent variable CFA model
11
1
1
11
1
1
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)
45
Two latent variable CFA model
11
1
1
11
1
1
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)
46
Two latent variable CFA model
11
1
1
11
1
1
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)
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)
Ksi-1
X111
x21
x31
Ksi-2
x4
x5
x6
11
1
1
Correlated error: td(5,3)
Added path: LX(2,2)
48
A model with an exogenous latent variable
Eta-1
Y111
Y21
Y31
Eta-2
Y4
Y5
Y6
11
1
1
1
1
Ksi-1
x11
1
x2
1
x3
1
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
49
A model with an exogenous latent variable
Eta-1
Y111
Y21
Y31
Eta-2
Y4
Y5
Y6
11
1
1
1
1
Ksi-1
x11
1
x2
1
x3
1
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 delta – diagonal
TD(1) TD(2) TD(3)
Theta-eps:
See previous example TD
50
A model with an exogenous latent variable
Eta-1
Y111
Y21
Y31
Eta-2
Y4
Y5
Y6
11
1
1
1
1
Ksi-1
x11
1
x2
1
x3
1
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)
eta3
1
1 1 1
eta4
11
1
1
1
eta2
1
1 1 1
eta11
111
ksi1
1
1 1 1
BETA is 4 x 4
GAMMA is 4 x 1
BETA
0 0 0 0
BE(2,1) 0 0 0
0 BE(3,2) 0 0
0 BE(4,2) 0 0
Zeta1, zeta2 not shown
GammaGA(1,1)GA(2,1)00
52
A model with intervening variables(a non-zero BETA matrix)
eta3
1
1 1 1
eta4
11
1
1
1
eta2
1
1 1 1
eta11
111
ksi1
1
1 1 1
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
Ksi-1
x31
1x2
1x1
1
ksi-2x5
11
x41
Eta-1
y11
1
y2
1
y3
1
Eta2
y6
1
1
y51
y41
1
1
Class Exercise (if time permits)
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