Transcript LISREL matrices, LISREL programming

LISREL matrices,
LISREL programming
ICPSR General Structural
Equations
Week 2 Class #4
1
Class Exercise
1
(from previous class notes:)
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
2
Class exercise
1
x1
1
x2
1
Ksi-1
1
x3
1
1
1
1
1
y1
y2
y3
BETA 2 x 2
0
1
Eta-1
BE(1,2)
1
BE(2,1)
0
x4
1
ksi-2
x5
1
Eta2
PHI(1,1)
1
PSI 2 x 2
PS(1,1)
PS(2,1) PS(2,2)
PHI 2 X 2
y4
y5
y6
1
1
1
0
PHI(2,2)
GAMMA 2 X 2
GA(1,1) 0
0
GA(2,2)
3
LAMBDA-X
1
1
x1
1
y1
x2
1
Ksi-1
1
x3
1
1
1
1
y2
y3
1
LX(2,1) 0
1
Eta-1
1
LX(3,1) LX(3,2)
x4
1
ksi-2
x5
1
Eta2
LAMBDA-Y
1
0
LY(2,1) 0
0
0
1
0
LX(5,2)
1
y4
y5
y6
1
1
1
LY(3,1) 0
0
1
0
LY(5,2)
0
LY(6,2)
4
1
1
x1
1
y1
x2
1
1
Ksi-1
x3
1
1
1
1
y2
y3
1
Eta-1
1
x4
1
ksi-2
x5
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
1
Eta2
1
y4
y5
y6
1
1
1
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
1
1
1
1
1
1
Y1
Y2
Y3
Y4
Y5
Y6
1
ETA1
1
ETA2
1
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Exercise 2:A (PANEL) MODEL WITH CORRELATED ERRORS
Beta 2 x 2
0
1
1
1
1
1
1
Y1
Y2
Y3
Y4
Y5
Y6
BE(2,1) 0
1
1
1
ETA2
ETA1
Not shown:
zeta1
0
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
7
TE(6,6)
Exercise 2:A (PANEL) MODEL WITH CORRELATED ERRORS
1
1
1
1
1
1
Y1
Y2
Y3
Y4
Y5
Y6
1
1
ETA1
ETA2
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
1
Y1
1
ETA1
1
Y2
Y3
X1
1
1
1
KSI1
1
Y4
ETA2
1
Y5
1
1
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Exercise 3
BETA 2 X 2
1
Y1
1
ETA1
1
Y2
Y3
0
0
1
BE(2,1) 0
1
KSI1
X1
1
Gamma 2 x 1
GA(1,1)
0
1
Y4
ETA2
1
Y5
1
1
LAMBDA-Y
1
0
LY(2,1) 0
LAMBDA-X 1 X 1
LY(3,1) LY(3,2)
1
0
1
0
LY(5,2)
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Exercise 3
1
Y1
1
ETA1
1
Y2
Y3
X1
1
1
1
KSI1
1
Y4
ETA2
1
Y5
1
1
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
1
X1
1
1
1
X2
X3
1
KSI-1
ETA1
Y1
Y2
Y3
Y4
1
1
1
1
This is a non-standard model.
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Exercise 4
1
X1
1
1
1
X2
X3
1
KSI-1
Y1
Y2
ETA1
Y3
Y4
1
1
1
1
This parameter cannot be estimated in LISREL; must reexpress the model (to an equivalent that CAN be estimated)
Y5
Eta2
1
1
1
X2
X3
1
KSI-1
ETA1
Y1
Y2
Y3
Y4
1
1
1
1
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RE-EXPRESSED MODEL
0
1
y5
zeta-2
1
1
eta2
zeta-1
1 1
1
x2
x3
1
1
eta-1
ksi-1
y1
y2
y3
y4
LAMBDA – Y
BETA
1
0
0
BE(1,2)
LY(2,1) 0
0
0
1
1
1
1
LY(3,1) 0
LY(4,1) 0
0
1
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RE-EXPRESSED MODEL
Y5
Eta2
1
1
1
X2
X3
1
ETA1
KSI-1
Now
X1,X2
Y1
Y2
Y3
Y4
1
1
1
1
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
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e1
e2
e3
1
1
1
Y1
Y2
Y3
Re-expressed as:
0
e3 variance=0
e3
1
Eta-1
e1
e2
Same variance as e3
in previous model
Y3
1
Y1
Y2
1
Eta-2
zeta-2
lambda-2
beta 2
Eta-1
Same as lambda parameter
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).
e1
e2
e2
X1
X2
X3
1
lambda-2
lambda-3
Ksi-1
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e1
e2
1
1
X1
X2
1
e3
0
e1
e2
ksi-2
1
1
1
X1
X2
X3
e3
1
1
X3
lambda-2
lambda-3
1
Ksi-1
lambda-2
lambda-3
Ksi-1
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(KSI2) Since…..VAR(ksi-2) = 1.0
[expression lambda42 replaces VAR(e3)
Regardless of estimate of lambda4, variance >0.
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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:
Title
SY=
input file if LISREL .dsf
DA - dataparameters
SE selection of variables
MO – modelparameters
… various FI and FR statements
OU – outputparameters (see handout)
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The LISREL PROGRAM:
! Achievement Values Program #1
SY='z:\baer\Week2Examples\LISREL\Achieve1.dsf'
SE
REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE
CONTENT /
MO NY=6 NE=1 LY=FU,FR PS=SY TE=SY
FI 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
-------0
1
2
3
4
5
REDUCE
NEVHAPP
NEW_GOAL
IMPROVE
ACHIEVE
CONTENT
Reference indicator is “fixed” All fixed
parameters represented by 0.
PSI
Theta-eps is diagonal
ETA 1
-------6
THETA-EPS
REDUCE
-------7
NEVHAPP
-------8
NEW_GOAL
-------9
IMPROVE
-------10
ACHIEVE
-------11
CONTENT
-------12
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LISREL Output
LISREL Estimates (Maximum Likelihood)
LAMBDA-Y
REDUCE
ETA 1
-------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
-------0.53
(0.01)
38.84
NEVHAPP
-------0.38
(0.01)
36.44
NEW_GOAL
-------0.19
(0.01)
28.79
IMPROVE
-------0.21
(0.01)
18.92
ACHIEVE
-------0.36
(0.01)
34.53
CONTENT
-------0.50
(0.01)
35.92
ACHIEVE
-------0.17
CONTENT
-------0.13
Squared Multiple Correlations for Y - Variables
REDUCE
-------0.02
NEVHAPP
-------0.11
NEW_GOAL
-------0.29
IMPROVE
-------0.46
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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
-------- 323.45
24.46
92.13
19.12
170.74
NEVHAPP
--------
NEW_GOAL
--------
IMPROVE
--------
ACHIEVE
--------
CONTENT
--------
- 4.29
52.90
48.71
243.43
- 87.29
0.97
58.94
- 33.31
21.28
- 1.82
- -
Expected Change for THETA-EPS
REDUCE
NEVHAPP
NEW_GOAL
IMPROVE
ACHIEVE
CONTENT
REDUCE
-------- 0.15
0.03
0.08
0.04
0.13
NEVHAPP
--------
NEW_GOAL
--------
IMPROVE
--------
ACHIEVE
--------
CONTENT
--------
- 0.01
0.06
0.05
0.14
- 0.10
0.01
0.06
- 0.06
0.05
- 0.01
- -
Completely Standardized Expected Change for THETA-EPS
REDUCE
NEVHAPP
NEW_GOAL
IMPROVE
ACHIEVE
CONTENT
REDUCE
-------- 0.32
0.09
0.18
0.08
0.23
NEVHAPP
--------
NEW_GOAL
--------
IMPROVE
--------
ACHIEVE
--------
CONTENT
--------
- 0.04
0.15
0.12
0.27
- 0.29
0.02
0.14
- 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)
SE
236987/
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 Path
SY='z:\baer\Week2Examples\LISREL\Achieve1.dsf'
SE
REDUCE 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,FR
FI LY 2 1
FI LY 3 2
VA 1.0 LY 2 1 LY 3 2
FR LY 1 1 LY 6 1 LY 4 2 LY 5 2
FR LY 1 2
PD
OU ME=ML SE TV SC MI
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(from output listing)
Parameter Specifications
LAMBDA-Y
REDUCE
NEVHAPP
NEW_GOAL
IMPROVE
ACHIEVE
CONTENT
ETA 1
-------1
0
0
0
0
5
ETA 2
-------2
0
0
3
4
0
GAMMA
GENDER
-------6
10
ETA 1
ETA 2
AGE
-------7
11
EDUC
-------8
12
INCOME
-------9
13
AGE
--------
EDUC
--------
INCOME
--------
16
18
21
19
22
23
NEW_GOAL
-------29
IMPROVE
-------30
PHI
GENDER
AGE
EDUC
INCOME
GENDER
-------14
15
17
20
PSI
ETA 1
ETA 2
ETA 1
-------24
25
ETA 2
-------26
THETA-EPS
REDUCE
-------27
NEVHAPP
-------28
ACHIEVE
-------31
CONTENT
-------32
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(output)
LISREL Estimates (Maximum Likelihood)
LAMBDA-Y
REDUCE
ETA 1
-------1.13
(0.07)
17.32
ETA 2
-------0.65
(0.08)
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
- -
GENDER
-------0.02
(0.02)
1.14
AGE
--------0.01
(0.00)
-10.40
EDUC
-------0.03
(0.00)
10.04
INCOME
-------0.01
(0.00)
5.67
0.07
(0.01)
4.90
0.00
(0.00)
4.81
0.01
(0.00)
4.19
0.00
(0.00)
-0.79
GAMMA
ETA 1
ETA 2
•
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Covariance Matrix of ETA and KSI
ETA 1
ETA 2
GENDER
AGE
EDUC
INCOME
ETA 1
-------0.15
-0.04
-0.01
-2.25
0.53
0.47
ETA 2
--------
GENDER
--------
AGE
--------
EDUC
--------
INCOME
--------
0.07
0.02
0.37
0.06
-0.08
0.25
-0.08
-0.07
-0.98
269.69
-18.55
-15.71
13.75
5.55
20.57
Squared Multiple Correlations for Structural Equations
ETA 1
ETA 2
-------- -------0.22
0.03
30
(LISREL output)
Modification Indices and Expected Change
Modification Indices for LAMBDA-Y
REDUCE
NEVHAPP
NEW_GOAL
IMPROVE
ACHIEVE
CONTENT
ETA 1
-------- - 4.90
0.84
2.18
- -
ETA 2
-------- 3.55
- - - 3.55
31
Completely Standardized Solution
LAMBDA-Y
REDUCE
NEVHAPP
NEW_GOAL
IMPROVE
ACHIEVE
CONTENT
ETA 1
-------0.59
0.59
- - - 0.59
ETA 2
-------0.24
- 0.52
0.79
0.41
- -
GAMMA
ETA 1
ETA 2
GENDER
-------0.03
0.12
AGE
--------0.25
0.11
EDUC
-------0.25
0.10
INCOME
-------0.15
-0.02
(could have used LA (labels) statement to provide
labels for these latent variables)
32
Reproduced covariances in matrix form
First examples are for SEM models that are
“manifest variable only” – no latent variables.
33
Manifest variables only
34
Manifest variables only
35
Manifest variables only
Previous example had no paths connecting endogenous yvariables (no “Beta” matrix). A bit more complicated with
these included:
36
Manifest variables only
With Beta matrix:
37
Manifest variables only
38
Manifest variables only
39
Manifest variables only
40
Manifest variables only
41
Latent variables included
Measurement
model only
42
Latent variables
included
43
δ
44
45
46
(last slide)
47