LISREL matrices, LISREL programming

11

LISREL matrices, LISREL matrices, LISREL programmingLISREL programming

ICPSR General Structural ICPSR General Structural EquationsEquations

Week 2 Class #4Week 2 Class #4

2

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 (from previous class notes:)

3

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

BETA 2 x 2

0 BE(1,2)

BE(2,1) 0

PHI 2 X 2

PHI(1,1)

0 PHI(2,2)

GAMMA 2 X 2

GA(1,1) 0

0 GA(2,2)

PSI 2 x 2

PS(1,1)

PS(2,1) PS(2,2)

Class exercise

4

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

LAMBDA-X

1 0

LX(2,1) 0

LX(3,1) LX(3,2)

0 1

0 LX(5,2)

LAMBDA-Y

1 0

LY(2,1) 0

LY(3,1) 0

0 1

0 LY(5,2)

0 LY(6,2)

5

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

MO NY=6 NX=5 NK=2 NE=2 LX=FU,FI LY=FU,FI C

PH=SY BE=FU,FI GA=FU,FI TD=SY TE=SY PS=SY,FR

VA 1.0 LX 1 1 LX 4 2 LY 1 1 LY 4 2

FR LX 2 1 LX 3 1 LX 3 2 LX 5 2 LY 2 1 LY 3 1 LY 5 2 LY 6 2

FR GA 1 1 GA 2 2

FR BE 2 1 BE 1 2

6

Exercise 2:A (PANEL) MODEL WITH CORRELATED ERRORS

ETA1

Y11

1

Y2

1

Y3

1

ETA2

Y4 Y5 Y61

1 1 1

1

7


ETA1

Y11

1

Y2

1

Y3

1

ETA2

Y4 Y5 Y61

1 1 1

1

Beta 2 x 2

0 0

BE(2,1) 0

Not shown: zeta1

PSI 2 x 2

PS(1,1)

0 PS(2,2)Theta-eps

TE(1,1)

0 TE(2,2)

0 0 TE(3,3)

TE(4,1) 0 0 TE(4,4)

0 TE(5,2) 0 0 TE(5,5)

0 0 0 0 0 TE(6,6)

8


ETA1

Y11

1

Y2

1

Y3

1

ETA2

Y4 Y5 Y61

1 1 1

1

MO NY=6 NE=2 LY=FU,FI PS=SY TE=SY BE=FU,FI

VA 1.0 LY 1 1 LY 4 2

FR LY 2 1 LY 3 1 LY 5 2 LY 6 2

FR BE 2 1

FR TE 4 1 TE 5 2

Notes: PS=SY specification free diagonals (PS(1,1) and PS(2,2), fixed off-diagonals [ps(2,1)=0 in this model].

9

Exercise 3

ETA1

Y111

Y21

Y31

ETA2Y41 1

Y51

1

1

KSI1X11

10

Exercise 3

ETA1

Y111

Y21

Y31

ETA2Y41 1

Y51

1

1

KSI1X11

Gamma 2 x 1

GA(1,1)

0

BETA 2 X 2

0 0

BE(2,1) 0

LAMBDA-Y

1 0

LY(2,1) 0

LY(3,1) LY(3,2)

0 1

0 LY(5,2)

LAMBDA-X 1 X 1

1

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Exercise 3

ETA1

Y111

Y21

Y31

ETA2Y41 1

Y51

1

1

KSI1X11

MO NX=1 NY=5 NK=1 NE=2 LX=ID LY=FU,FI C

PS=SY PH=SY TD=ZE TE=SY BE=FU,FI GA=FU,FI

VA 1.0 LY 1 1 LY 4 2

FR LY 2 1 LY 3 1 LY 3 2 LY 5 2

FR GA 1 1 BE 2 1

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Exercise 4

ETA1

Y111

Y21

Y31

Y41

KSI-1

X31

1X2

1

X11

This is a non-standard model.

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Exercise 4

ETA1

Y111

Y21

Y31

Y41

KSI-1

X31

1X2

1

X11

This parameter cannot be estimated in LISREL; must re-express the model (to an equivalent that CAN be estimated)

ETA1

Y111

Y21

Y31

Y41

KSI-1

X31

1X2

1

Eta2

Y5

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RE-EXPRESSED MODEL

LAMBDA – Y

1 0

LY(2,1) 0

LY(3,1) 0

LY(4,1) 0

0 1

BETA

0 BE(1,2)

0 0

eta-1

y111

y21

y31

y41

eta2

ksi-1x3

11x2

1

y50

11 zeta-2

1

zeta-1

1

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RE-EXPRESSED MODEL

ETA1

Y111

Y21

Y31

Y41

KSI-1

X31

1X2

1

Eta2

Y5

MO NY=5 NX=2 NK=1 NE=2 LY=FU,FI LX=FU,FR C

GA=FU,FR PS=SY PH=SY TD=SY TE=SY

VA 1.0 LX 1 1 LY 1 1 LY 5 2

FR LX 2 1 LY 2 1 LY 3 1 LY 4 1

FI TE 5 5 SINGLE INDICATOR, CANNOT ESTIMATE ERROR

Now X1,X2

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Eta-1

Y1

e1

1

1

Y2

e2

1

Y3

e3

1

Re-expressed as:

Eta-1

Y1

e1

1

Y2

e2

lambda-2

Eta-2

beta 2

zeta-2

Y3

1

0

e3e3 variance=0

Same as lambda parameter in previous model

Same variance as e3 in previous model

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The same sort of principle can be used for other purposes too:

Imposing an inequality constraint.

Example: We wish to impose a constraint such that VAR(e3) > 0 (don’t allow negative error variance).

Ksi-1

X1

e1

1

X2

e2

lambda-2

X3

e2

lambda-3

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Lambda 2, lambda 3: same parm’s

Variance of ksi-2 fixed to 1.0

X3 = lambda3 KSI1 + lambda4 KSI2

VAR(X3) = lambda32*VAR(Ksi-1) + lambda42

*VAR(KSI-2) Since…..VAR(ksi-2) = 1.0

[expression lambda42 replaces VAR(e3)

Regardless of estimate of lambda4, variance >0.

Ksi-1

X1

e1

1

1

X2

e2

lambda-2

1

X3

e3

lambda-3

1

Ksi-1

X1

e1

1

1

X2

e2

lambda-2

1

X3

ksi-2

lambda-3

1

0

e3

1

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The LISREL PROGRAM:

MO modelparameters statement

FR free a parameter

FI fix a parameter

VA set a parameter to a value (if the parameter is free, this is the “start value” to override program default estimate; otherwise, it is the value to which a parameter is constrained

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The LISREL PROGRAM:

If reading in a “system” .dsf file created by prelis:

TitleSY= input file if LISREL .dsfDA - dataparametersSE selection of variablesMO – modelparameters … various FI and FR statementsOU – outputparameters (see handout)

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The LISREL PROGRAM:

! Achievement Values Program #1SY='z:\baer\Week2Examples\LISREL\Achieve1.dsf'SEREDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT /MO NY=6 NE=1 LY=FU,FR PS=SY TE=SYFI LY 1 1 VA 1.0 LY 1 1 OU ME=ML SC MI

• SE statement lists variables to be used (always specify Y variables first)

• can change order on SE statement. Here, REDUCE is Y1, NEVHAPP is Y2, etc. LY 1 1 refers to REDUCE.

•OU : ME=ML (maximum likelihood) SC (standardized solution) MI (provide modification indices)

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LISREL Output:

Parameter Specifications

LAMBDA-Y

ETA 1 -------- REDUCE 0 NEVHAPP 1 NEW_GOAL 2 IMPROVE 3 ACHIEVE 4 CONTENT 5

PSI

ETA 1 -------- 6

THETA-EPS

REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT -------- -------- -------- -------- -------- -------- 7 8 9 10 11 12

Reference indicator is “fixed” All fixed parameters represented by 0.

Theta-eps is diagonal

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LISREL OutputLISREL Estimates (Maximum Likelihood)

LAMBDA-Y

ETA 1 -------- REDUCE 1.00 NEVHAPP 2.14 (0.37) 5.72 NEW_GOAL -2.76 (0.46) -6.00 IMPROVE -4.23 (0.70) -6.01 ACHIEVE -2.64 (0.45) -5.87 CONTENT 2.66 (0.46) 5.78

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LISREL Output

Covariance Matrix of ETA

ETA 1 -------- 0.01

PSI

ETA 1 -------- 0.01 (0.00) 3.08

THETA-EPS

REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT -------- -------- -------- -------- -------- -------- 0.53 0.38 0.19 0.21 0.36 0.50 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) 38.84 36.44 28.79 18.92 34.53 35.92

Squared Multiple Correlations for Y - Variables

REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT -------- -------- -------- -------- -------- -------- 0.02 0.11 0.29 0.46 0.17 0.13

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LISREL Output Modification Indices and Expected Change

No Non-Zero Modification Indices for LAMBDA-Y

No Non-Zero Modification Indices for PSI

Modification Indices for THETA-EPS

REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT -------- -------- -------- -------- -------- -------- REDUCE - - NEVHAPP 323.45 - - NEW_GOAL 24.46 4.29 - - IMPROVE 92.13 52.90 87.29 - - ACHIEVE 19.12 48.71 0.97 33.31 - - CONTENT 170.74 243.43 58.94 21.28 1.82 - -

Expected Change for THETA-EPS


Completely Standardized Expected Change for THETA-EPS


Maximum Modification Index is 323.45 for Element ( 2, 1) of THETA-EPS

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Lisrel program input

If reading in a covariance matrix generated by PRELIS instead of a .dsf file:

DA NO=# cases NI=# of input var’s MA=CM{MA = type of matrix to be analyzed; default = CM, or a covariance matrix}

CM FI=‘c:\file1.cov’input file specification(cov)

SE2 3 6 9 8 7 / Selection: corresponds to order in which variables

located on input covariance matrix (3rd variable on the matrix is now Y2).

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Another LISREL example:

! Achievement Values Program #8: Adding One Extra Measurement Model PathSY='z:\baer\Week2Examples\LISREL\Achieve1.dsf'SEREDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT GENDER AGE EDUC INCOME/MO NX=4 NK=4 NY=6 NE=2 LX=ID PH=SY,FR TD=ZE LY=FU,FI C PS=SY,FR TE=SY GA=FU,FRFI LY 2 1 FI LY 3 2 VA 1.0 LY 2 1 LY 3 2FR LY 1 1 LY 6 1 LY 4 2 LY 5 2 FR LY 1 2 PDOU ME=ML SE TV SC MI

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(from output listing)Parameter Specifications

LAMBDA-Y

ETA 1 ETA 2 -------- -------- REDUCE 1 2 NEVHAPP 0 0 NEW_GOAL 0 0 IMPROVE 0 3 ACHIEVE 0 4 CONTENT 5 0

GAMMA

GENDER AGE EDUC INCOME -------- -------- -------- -------- ETA 1 6 7 8 9 ETA 2 10 11 12 13

PHI

GENDER AGE EDUC INCOME -------- -------- -------- -------- GENDER 14 AGE 15 16 EDUC 17 18 19 INCOME 20 21 22 23

PSI

ETA 1 ETA 2 -------- -------- ETA 1 24 ETA 2 25 26

THETA-EPS

REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT -------- -------- -------- -------- -------- -------- 27 28 29 30 31 32

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(output)LISREL Estimates (Maximum Likelihood)

LAMBDA-Y

ETA 1 ETA 2 -------- -------- REDUCE 1.13 0.65 (0.07) (0.08) 17.32 8.53 NEVHAPP 1.00 - - NEW_GOAL - - 1.00 IMPROVE - - 1.85 (0.12) 16.00 ACHIEVE - - 0.99 (0.06) 15.95 CONTENT 1.16 - - (0.06) 19.84

GAMMA

GENDER AGE EDUC INCOME -------- -------- -------- -------- ETA 1 0.02 -0.01 0.03 0.01 (0.02) (0.00) (0.00) (0.00) 1.14 -10.40 10.04 5.67 ETA 2 0.07 0.00 0.01 0.00 (0.01) (0.00) (0.00) (0.00) • 4.90 4.81 4.19 -0.79

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Covariance Matrix of ETA and KSI

ETA 1 ETA 2 GENDER AGE EDUC INCOME

-------- -------- -------- -------- -------- -------- ETA 1 0.15 ETA 2 -0.04 0.07 GENDER -0.01 0.02 0.25 AGE -2.25 0.37 -0.08 269.69 EDUC 0.53 0.06 -0.07 -18.55 13.75 INCOME 0.47 -0.08 -0.98 -15.71 5.55 20.57

Squared Multiple Correlations for Structural Equations

ETA 1 ETA 2 -------- -------- 0.22 0.03

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(LISREL output)

Modification Indices and Expected Change

Modification Indices for LAMBDA-Y

ETA 1 ETA 2 -------- -------- REDUCE - - - - NEVHAPP - - 3.55 NEW_GOAL 4.90 - - IMPROVE 0.84 - - ACHIEVE 2.18 - - CONTENT - - 3.55

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Completely Standardized Solution

LAMBDA-Y

ETA 1 ETA 2 -------- -------- REDUCE 0.59 0.24 NEVHAPP 0.59 - - NEW_GOAL - - 0.52 IMPROVE - - 0.79 ACHIEVE - - 0.41 CONTENT 0.59 - -

GAMMA

GENDER AGE EDUC INCOME -------- -------- -------- -------- ETA 1 0.03 -0.25 0.25 0.15 ETA 2 0.12 0.11 0.10 -0.02

(could have used LA (labels) statement to provide labels for these latent variables)

33

Reproduced covariances in matrix formFirst examples are for SEM models that are “manifest variable only” – no latent variables.

34

Manifest variables only

35


36


Previous example had no paths connecting endogenous y-variables (no “Beta” matrix). A bit more complicated with these included:

37


With Beta matrix:

38


39


40


41


42

Latent variables included

Measurement model only

43

Latent variables included

44

δ

45

46

47

(last slide)

LISREL matrices, LISREL programming

Documents

syfi ly

fr cga

lambda parameter

fi cps

fi cph

panel model

handoutthe lisrel program

constrainedthe lisrel