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:)

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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

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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

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Matrix form: LISREL MEASUREMENT MODEL MATRICES

Measurement models for endogenous latent variables (ETA) are similar:

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Class Exercise

1

1

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

15

#2

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Lisrel Matrices for examples.

No Beta Matrix in this model

17

Lisrel Matrices for examples.

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Lisrel Matrices for examples (example #2)

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Lisrel Matrices for examples (example #2)

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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.

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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).

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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)

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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

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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

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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)

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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

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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]

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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

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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)

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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)

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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

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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)

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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

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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)

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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

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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

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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)

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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)

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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)

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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

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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

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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)

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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

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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

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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

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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)