General Structural
Equation (LISREL)
Models
Week 2 #3
LISREL Matrices
The LISREL Program
1
The LISREL matrices
The variables:
Manifest: X, Y
Latent: Eta η Ksi ξ
Error:
construct equations: zeta ζ
measurement equations
delta δ, epsilon ε
2
The LISREL matrices
The variables:
Manifest: X, Y
Latent: Eta η
Error:
construct equations: zeta ζ
Ksi ξ
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=γξ+ζ
Gamma
(endogenous) LV’s
Construct equation connecting ksi (exogenous), eta
η = β η + γ ξ + ζ Construct equation connecting eta with eta LV’s
3
The LISREL matrices
The variables:
Manifest: X, Y
Latent: Eta η
Error:
construct equations: zeta ζ
Ksi ξ
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
4
(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 = λ1 ξ1+ δ1
MATRICES:
LAMBDA-x
THETA-DELTA
PHI
5
Matrix form: LISREL MEASUREMENT MODEL MATRICES
A slightly more complex example:
6
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.
7
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
8
Matrix form: LISREL MEASUREMENT MODEL MATRICES
Models with correlated measurement errors:
9
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
10
Matrix form: LISREL MEASUREMENT MODEL MATRICES
Measurement models for endogenous latent variables
(ETA) are similar:
11
Class Exercise
1
1
#1
1
Provide labels for
each of the
variables
1
1
Slides 12-19 not on
handout; see handout for
yesterday’s class
12
#2
1
1
1
1
1
13
#1
epsilon
ksi
eta
zeta
delta
14
#2
15
Lisrel Matrices for examples.
No Beta
Matrix in
this model
16
Lisrel Matrices for examples.
17
Lisrel Matrices for examples (example #2)
18
Lisrel Matrices for examples (example #2)
19
Special Cases
1
1
1
1
Single-indicator variables
This model must be re-expressed as…. (see next slide)
20
Special Case: single indicators
1
1
1
0
1
1
0
1
Error terms with 0 variance
21
Special Case: single indicators
1
1
1
0
1
1
0
1
LISREL will issue an error
message: matrix not positive
definite (theta-delta has 0s in
diagonal). Can “override” this.
22
Special Case: single indicators
1
1
Case where all exogenous
construct equation variables
are manifest
23
Special Case: single indicators
Case where all exogenous
construct equation variables
are manifest
24
Special Case: correlated errors across delta,epsilon
Special matrix:
Theta delta-epsilon (TH)
25
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).
26
Exercise: going from matrix contents to diagrams
Matrices:
LY 8 x 3
BE 3 x 3
1
0
0
Free elements:
ly2,1
ly3,1
ly4,1
ly5,1
0
0
0
0
ly4,2
ly5,2
1
0
0
LY3,3
0
0
0
1
BE 2,1
BE 3,1
0
0
LY8,3
(other off-diag’s = 0)
PS 3 X 3
Free elements:
- PS(3,2), all diagonals
27
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
ga2,1
0
ga1,2
0
ga3,2
0
ga2,3
ga3,3
0
ga2,4
ga3,4
28
zeta1
1
0
1
x1
1
ksi1
y1
1
eta1
y2
0
1
x2
1
y3
y4
x3
1
ksi3
eta2
y5
1
0
1
x4
1
1
1
ksi2
0
1
1
ksi4
y6
zeta2
zeta3
1
eta3
y7
y8
1
1
1
1
1
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:
39
zeroes in off-diagonals)
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:
1
1
LX Lambda-X 3 x1
1
TD Theta delta 3 x 3
PH Phi 1 x 1
1
Lambda –X
PHI
1.0
Ph(1,1)
Lx(2,1)
Theta-delta
Lx(3,1)
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
1
1
PH=SY TD=SY
1
FI LX(1,1)
VA 1.0 LX(1,1)
1
Lambda –X
PHI
1.0
Ph(1,1)
C = CONTINUE FROM
PREVIOUS LINE
Lx(2,1)
Theta-delta
Lx(3,1)
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
1
PS=SY TE=SY
1
FI LY(1,1)
1
VA 1.0 LY(1,1)
1
Lambda –Y
PSI
1.0
PS(1,1)
C = CONTINUE FROM
PREVIOUS LINE
LY(2,1)
Theta-epsilon
LY(3,1)
te(1,1)
0
te(2,2)
0
0
te(3,3)
43
Two latent variable CFA model
Lambda-X 6 x 2
1
1.0
0
1
LX(2,1)
0
1
LX(3,1)
0
0
1.0
0
LX(5,2)
0
LX(6,2)
1
1
1
1
1
Phi 2 x 2
Ph(1,1)
Theta-delta -- expressed as diagonal
Ph(2,1) Ph(2,2)
TD(1) TD(2) TD(3) TD(4) TD(5) TD(6)
44
Two latent variable CFA model
1
1
1
Lambda-X 6 x 2
1.0
0
1
LX(2,1)
0
1
LX(3,1)
0
1
0
1.0
0
LX(5,2)
0
LX(6,2)
1
1
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
1
1
TD(1,1) TD(2,2) TD(3,3) TD(4,4) TD(5,5) TD(6,6)
1
1
1
1
1
1
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
1
Ksi-1
X1
x2
x3
1
Ksi-2
x4
x5
x6
1
1
Correlated error: td(5,3)
1
1
Added path: LX(2,2)
1
1
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
1
1
1
1
1
x1
x2
x3
Eta-1
Y1
Y2
Y3
1
1
1
1
Ksi-1
1
1
Eta-2
Y4
Y5
Y6
1
1
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
48
A model with an exogenous latent variable
1
1
1
1
1
x1
x2
x3
Eta-1
Y1
Y2
Y3
1
1
1
Gamma 1 x 2
GA(1,1) GA(2,2)
1
Ksi-1
1
1
Eta-2
Y4
Y5
Y6
Lambda-Y
1.0
0
LY(2,1) LY(2,2)
LY(3,1) 0
1
1
1
Phi 1 x 1
PH(1,1)
PSI 2 x 2
Theta-eps:
PS(1,1)
See previous
example TD
PS(2,1) PS(2,2)
0
1.0
Theta delta – diagonal
0
LY(5,2)
TD(1) TD(2) TD(3)
0
LY(6,2)
49
A model with an exogenous latent variable
1
1
1
1
1
x1
x2
x3
Eta-1
Y1
Y2
Y3
1
1
1
1
Ksi-1
1
1
Eta-2
Y4
Y5
Y6
1
1
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)
50
A model with intervening variables
(a non-zero BETA matrix)
BETA is 4 x 4
1
1
1
1
1
1
eta2
1
BETA
0
eta3
ksi1
1
1
eta4
1
1
Zeta1, zeta2 not shown
0
0
0
BE(2,1) 0
0
0
1
0
BE(3,2) 0
0
1
0
BE(4,2) 0
0
1
eta1
GAMMA is 4 x 1
1
1
1
1
1
1
1
Gamma
GA(1,1)
GA(2,1)
0
0
51
A model with intervening variables
(a non-zero BETA matrix)
1
1
1
1
1
1
1
1
1
eta2
1
1
1
eta3
ksi1
1
1
1
eta1
1
1
eta4
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
52
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
53
Class Exercise (if time permits)
1
x1
1
x2
1
1
Ksi-1
x3
1
1
1
1
1
y1
y2
y3
1
Eta-1
1
x4
1
ksi-2
x5
1
Eta2
1
y4
y5
y6
1
1
1
54