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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
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Exercise 2:A (PANEL) MODEL WITH CORRELATED ERRORS
ETA1
Y11
1
Y2
1
Y3
1
ETA2
Y4 Y5 Y61
1 1 1
1
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Exercise 2:A (PANEL) MODEL WITH CORRELATED ERRORS
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)
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Exercise 2:A (PANEL) MODEL WITH CORRELATED ERRORS
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].
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Exercise 3
ETA1
Y111
Y21
Y31
ETA2Y41 1
Y51
1
1
KSI1X11
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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
REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT -------- -------- -------- -------- -------- -------- REDUCE - - NEVHAPP 0.15 - - NEW_GOAL 0.03 0.01 - - IMPROVE 0.08 0.06 0.10 - - ACHIEVE 0.04 0.05 0.01 0.06 - - CONTENT 0.13 0.14 0.06 0.05 0.01 - -
Completely Standardized Expected Change for THETA-EPS
REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT -------- -------- -------- -------- -------- -------- REDUCE - - NEVHAPP 0.32 - - NEW_GOAL 0.09 0.04 - - IMPROVE 0.18 0.15 0.29 - - ACHIEVE 0.08 0.12 0.02 0.14 - - CONTENT 0.23 0.27 0.14 0.10 0.02 - -
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)
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Reproduced covariances in matrix formFirst examples are for SEM models that are “manifest variable only” – no latent variables.
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Manifest variables only
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Manifest variables only
36
Manifest variables only
Previous example had no paths connecting endogenous y-variables (no “Beta” matrix). A bit more complicated with these included:
37
Manifest variables only
With Beta matrix:
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Manifest variables only
39
Manifest variables only
40
Manifest variables only
41
Manifest variables only
42
Latent variables included
Measurement model only
43
Latent variables included
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δ
45
46
47
(last slide)