General Structural
Equations (LISREL)
Week 1 #4
1
Today:


Quick look at more AMOS examples…
Extending the work with AMOS:
1.
2.
3.


2
Moving from factor model to causal model (construct equations
among latent variables)
adding single-indicator exogenous variables (assume no
measurement error)
adding single-indicator exogenous variables with assumed
measurement error
Equality constraints in structural equation models
Dummy exogenous variables in structural equation
models

SEM equivalents to contrasts

Block tests for dummy variables

AMOS example
Also Today:




3
Model fit (an overview)
The SIMPLIS program (part of
LISREL)
Moving from Standard Stats
packages into SEM software
Conceptualizing SEM models in
Matrix terms (some basics)
The SIMPLIS interface for LISREL




Works in scalar, not matrix, terms
Fairly easy to use
Sometimes, output is provided in regular
LISREL matrix form (can be a bit confusing)
Requires a lower-triangular covariance matrix
(most stats packages produce “square”
matrices) OR a special “.dsf” file (both can be
created by the PRELIS program which
accompanies LISREL).
4
Two examples of SIMPLIS programs
Example 1
SIMPLIS Example for Religion Sexual Morality Data
System file from file f:\Classes\ICPSR2005\Week1Examples\ReligSexMoralSIMPLIS\ReligSex1.dsf
Latent Variables
Relig Sexmor
Relationships:
V9 V175 V176 = Relig
V147 = 1*Relig
V304 V305 V307 V309 = Sexmor
V308 = 1*Sexmor
End of problem
5
SIMPLIS Example for Religion Sexual Morality Data
Output
Covariance Matrix
V9
V147
V175
V176
V304
-------- -------- -------- -------- -------- -------V9
0.82
V147
1.34
6.50
V175
0.31
0.75
0.48
V176
-1.64
-3.49
-1.09
6.77
V304
0.40
1.06
0.29
-1.52
2.90
V305
0.46
0.98
0.27
-1.45
1.34
V307
0.79
1.85
0.46
-2.57
1.70
V308
0.65
1.51
0.37
-1.93
1.59
V309
1.11
2.39
0.58
-3.12
1.61
V305
3.52
1.69
1.61
1.83
Covariance Matrix
V307
V308
V309
-------- -------- -------V307
7.26
V308
3.13
4.61
V309
4.02
2.83
7.76
6
LISREL Estimates (Maximum Likelihood)
Measurement Equations
Output
V9 = 0.44*Relig, Errorvar.= 0.28 , Rý = 0.66
(0.018)
(0.015)
25.20
18.41
V147 = 1.00*Relig, Errorvar.= 3.73 , Rý = 0.43
(0.16)
23.93
V175 = 0.27*Relig, Errorvar.= 0.27 , Rý = 0.44
(0.013)
(0.011)
21.54
23.78
V176 =
- 1.35*Relig, Errorvar.= 1.74 , Rý = 0.74
(0.052)
(0.12)
-25.96
14.68
7
V304 = 0.63*Sexmor, Errorvar.= 1.96 , Rý = 0.33
(0.033)
(0.082)
19.06
24.02
V305 = 0.66*Sexmor, Errorvar.= 2.49 , Rý = 0.29
(0.036)
(0.10)
18.11
24.46
V307 = 1.25*Sexmor, Errorvar.= 3.57 , Rý = 0.51
(0.054)
(0.17)
23.14
20.54
V308 = 1.00*Sexmor, Errorvar.= 2.23 , Rý = 0.52
(0.11)
20.33
V309 = 1.26*Sexmor, Errorvar.= 3.96 , Rý = 0.49
(0.055)
(0.19)
22.81
21.01
8
Covariance Matrix of Independent
Variables
Relig
Sexmor
Relig
-------2.77
(0.21)
13.18
Sexmor
--------
1.59
(0.11)
14.25
2.38
(0.17)
14.32
9
Degrees of Freedom = 26
Minimum Fit Function Chi-Square = 213.09 (P = 0.0)
Normal Theory Weighted Least Squares Chi-Square = 217.29 (P = 0.0)
Estimated Non-centrality Parameter (NCP) = 191.29
90 Percent Confidence Interval for NCP = (147.95 ; 242.10)
Minimum Fit Function Value = 0.15
Population Discrepancy Function Value (F0) = 0.13
90 Percent Confidence Interval for F0 = (0.10 ; 0.17)
Root Mean Square Error of Approximation (RMSEA) = 0.071
90 Percent Confidence Interval for RMSEA = (0.063 ; 0.080)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.00
10
Normed Fit Index (NFI) = 0.97
Non-Normed Fit Index (NNFI) = 0.97
Parsimony Normed Fit Index (PNFI) = 0.70
Comparative Fit Index (CFI) = 0.98
Incremental Fit Index (IFI) = 0.98
Relative Fit Index (RFI) = 0.96
Critical N (CN) = 312.87
Root Mean Square Residual (RMR) = 0.15
Standardized RMR = 0.035
Goodness of Fit Index (GFI) = 0.97
Adjusted Goodness of Fit Index (AGFI) = 0.94
Parsimony Goodness of Fit Index (PGFI) = 0.56
11
Another SIMPLIS Example
(Same 2 latent variables with singleindicator exogenous variables added)
SIMPLIS Example for Religion Sexual Morality Data
Observed variables: V9 V147 V175 V176 V304 V305 V307 V308 V309
V310 V355 V356 SEX OCC1 OCC2 OCC3 OCC4 OCC5
Covariance matrix from file e:\ICPSR2005\RSM1.COV
Sample size = 1457
Latent Variables: Relig Sexmor
Relationships:
V9 V175 V176 = Relig
V147 = 1*Relig
V304 V305 V307 V309 = Sexmor
V308 = 1*Sexmor
Equations:
Relig = V355 V356 SEX
Sexmor = V355 V356 SEX
Let the error covariance of Relig and Sexmor be free
Let the error covariance of V175 and V176 be free
Options MI ND=3 SC
End of problem
12
Output
Covariance Matrix
V307
V308
V309
V355
V356
SEX
V307
-------7.264
3.132
4.023
-7.317
1.447
-0.101
(portion)
V308
--------
V309
--------
V355
--------
V356
--------
SEX
--------
4.606
2.832
-5.385
0.656
0.123
7.758
-4.860
1.455
0.019
305.580
-8.744
-0.107
4.869
0.090
0.250
13
Error Covariance for V176 and V175 = -0.210
(0.0327)
-6.417
Structural Equations
Relig = - 0.0139*V355 + 0.0801*V356 + 0.443*SEX, Errorvar.= 2.735 , R2 = 0.0575
(0.00281)
(0.0222)
(0.0958)
(0.204)
-4.962
3.607
4.626
13.425
Sexmor = - 0.0148*V355 + 0.155*V356 + 0.0413*SEX, Errorvar.= 2.089 , R2 = 0.0973
(0.00255) (0.0204) (0.0860)
(0.149)
-5.795
7.587
0.480
13.995
Error Covariance for Sexmor and Relig = 1.454
(0.104)
13.954
14
Standardized
• In Simplis: OPTIONS SC
Completely Standardized Solution
LAMBDA-Y
V9
V147
V175
V176
V304
V305
V307
V308
V309
Relig
-------0.848
0.668
0.598
-0.821
- - - - - -
Sexmor
-------- - - - 0.560
0.540
0.721
0.709
0.706
15
Standardized
GAMMA
Relig
Sexmor
V355
--------0.143
-0.170
V356
-------0.104
0.225
SEX
-------0.130
0.014
16
Moving from Stat Package System files to
SEM Software
AMOS (reads
Directly from
SPSS
system files)
SPSS
SYSTEM
FILE
AMOS
Use PRELIS
SPSS SYSTEM
FILE
SAS, Stata, etc.
SYSTEM
FILE
A ‘DSF’ file
created by
PRELIS
A raw
covarianc
ematrix
(lower
triangle)
created by
PRELIS
LISREL reads
DSF files
LISREL reads
lower
triangular
matrices
LISREL
17
Fit of a model


How far apart are Σ and S?
Test of significance for H0: Σ=S
chi-square test



Test is a simple function of N:


Note: “Independence chi-square” is a different test! It
tests H0: S=0
Χ2 = F*(N-1)
“Perfect fit” (non-significant chi-square) much
easier to obtain in small samples
18
Fit of a model


Search for “fit indices” that are not a function of N
Desirable properties of fit indices:





Not a direct, linear function of N
Not affected by N (expect wider sampling distribution with
smaller Ns.. this might imply that some types of fit indices
yield “better” values for the same model in larger samples
Easily interpretable metric (e.g., 0  1)
Consistent across estimation methods
Not affected by metric of variables (e.g., same results
whether variables standardized or not)
19
Fit of a model

Desirable properties of fit indices (more):

Do not reward data dredging (vs. construction of
parsimonious models)


So-called “parsimony” measures include a penalty function for
adding parameters to a model
Commonly-used fit measures:




Joreskog & Sorbom’s GFI (affected by N though)
Bentler’s Normed Fit Index (and NNFI)
Incremental, Comparative fit indices
Root Mean Square Error of Approximation (RMSEA)
(for this index, low values are good)
20
Improving the fit of a model: diagnostics

Residuals:



Matrix of differences between sigma and S
Would need to standardize before we could
determine where a model should be improved
A residual is not necessarily connected to one
single parameter:

A high residual might imply any one of 3 or 4 parameters
could/should be added to the model
21
Improving the fit of a model: diagnostics

Modification indices




Based on 2nd order derivative matrix
Estimate the improvement in model fit if a
particular parameter is added
Metric: chi-square (difference)
Any value greater than 3.84 is “significant” at
p<.05 BUT criteria other than straight significance
can/have been employed

Reason: otherwise, sensitive to N; in large samples will
never get parsimonious model, etc.
22
Modification Indices
• In AMOS, click “modification indices”
under output options
• In SIMPLIS, Options MI
Modification Indices and Expected Change (SIMPLIS model
discussed ealrier)
The Modification Indices Suggest to Add the
Path to from
Decrease in Chi-Square
New Estimate
V9
Sexmor
8.4
-0.06
V176
Sexmor
11.6
-0.18
V307
Relig
14.2
-0.21
V309
Relig
34.0
0.33
23
Important note on modification indices


It is not always the case that the parameter
with the highest MI should be added to a
model
Some MIs will not make substantive sense
(e.g., in a causal model, an MI suggesting a
path from respondent’s social status to
parent’s social status).
24
Improving the fit of a model: diagnostics

Estimated parameter change values


Estimated value of a parameter that is currently
fixed (if this parameter is “freed” [included in the
model]).
Standardized values can be helpful in determining
whether adding a parameter is substantively
important
25
Equality Constraints in Structural
Equation Models



We can set “equality
constraints” on any two
(or more) parameters in
a model
E.g.: b1=b2
E.g.: VAR(e1) =
VAR(e2)
VAR-E1 VAR-E2 VAR-E3
E1
E2
E3
1
1
1
1
b1 b2
26
Equality Constraints in Structural
Equation Models


We can set “equality
constraints” on any two
(or more) parameters in
a model
In AMOS we do this by
giving parameters
names, and then using
the same name in the
locations where we
want to impose equality
constraints
VAR-E1 VAR-E2 VAR-E3
E1
E2
E3
1
1
1
1
b1 b1
27
Equality Constraints in Structural
Equation Models
VAR-E1 VAR-E2 VAR-E3


We can set “equality
constraints” on any two
(or more) parameters in
a model
In SIMPLIS, we do this
by adding statements:
E1
E2
E3
1
1
1
1
b1 b1
Let the path from Relig to V176 be equal to the
path from Relig to V167.
28
Equality Constraints in Structural Equation Models

The b1=b2 constraint
may not make sense if
the metric of the 2
latent variables is not
the same (makes most
sense if variances are
the same – would work
if the variables were
standardized]
VAR-E1 VAR-E2 VAR-E3
E1
E2
E3
1
1
1
1
b1 b1
29
Equality Constraints in Structural Equation Models


In this model, we could
test b1=b2, b2=b3,
b1=b3 or b1=b2=b3 by
setting the parameter
names to be the same
Equality constraints
only make sense if
variances of the 3
exogenous manifest
variables are the same,
though
b1
b2
b3
1
1
1
1
30
Equality Constraints in Structural Equation Models





Formal tests:
Model 1 b1, b2 estimated
separately
Model 2 b1=b2 (i.e., labels
“b1” in each of 2 locations)
Model 2 has 1 more degree
of freedom than model 1
A df=1 test for the equality
constraint is obtained by
subtracting the model 1
chi-square from the model
2 chi-square
b1
b2
b3
1
1
1
1
31
Dummy Variables in Structural Equation
Models
• Dummy variables can be included in
structural equation models if they are
completely exogenous
1
1
SEX 1/0
1
1
Sex: 0/1 variable
32
Dummy Variables in Structural Equation
Models
• Dummy variables can be included in
structural equation models if they are
completely exogenous
1
1
SEX 1/0
1
1
Educ
Age
33
Dummy Variables in Structural Equation
Models
• Dummy variables cannot be included in
structural equation models as indicators of
latent constructs
VOTED = 0/1 voted/did
not vote last election
TRUST = 5 pt. trust in
government item
POL COR = 5 pt.
agree/disagree
politicians corrupt
e1
e2
e3
1
1
1
VOTED Trust Pol. Cor
1
 This model is NOT
appropriate
Pol Partic
34
Dummy Variables in Structural Equation
Models
• Dummy variables can be included in
structural equation models if they are
completely exogenous
• For categorical independent variables with
more than 2 categories, sets of dummy
variables can be included (just like in
regression models)
35
Dummy Variables in Structural Equation
Models
•For categorical independent variables with
more than 2 categories, sets of dummy
variables can be included (just like in
regression models)
• Design matrix as with Regression (could
use effects or indicator coding; example
below uses indicator coding):
D1
D2
D3
Catholic
1
0
0
Protestant
0
1
0
Jewish
0
0
1
Atheist
0
0
0
36
DUMMY VARIABLES
D1
1
D2
1
1
D3
1
(add curved arrow D1 D2 )
37
DUMMY VARIABLES
D1
b1
1
D2
b2
1
1
b3
D3
1
Test H0 for entire religion variable: estimate model with
parameters b1, b2 and b3 all set to 0
38
(add curved arrow D1 D2 )
DUMMY VARIABLES
Test H0 for entire religion variable: estimate model with
parameters b1, b2 and b3 all set to 0
39
DUMMY VARIABLES
• Test religion variable : b1=b2=b3=0
Model 1 (3 separate parameters) vs. Model
2 (all parameters = 0) df=3 test
•Test Protestant (category 1) vs. Atheist (reference
group):
• Model 1 (3 separate parameters)
• Model 2 (fix b1=0) df=1
•OR: look at t-test for b1 parameter
40
DUMMY VARIABLES
•Test Protestant (category 1) vs. Catholic
(category2):
• Model 1 (3 separate parameters)
• Model 2 (fix b1=b2) df=1
41
LV Structural Equation Models
in Matrix terms
Thus far, our work has involved “scalar” equations.
• one equation at a time
•Specify a model (e.g, with software) by writing these
equations out, one line per equation
42
Matrix form
We can represent the previous 2 equations in
matrix form:
Matrix Form
(single, double subscript)
43
There are other matrices in this model
Variance-covariance matrix of
error terms (e’s)
44
(other matrices, continued)
Variance covariance matrix of exogenous
(manifest) variables
45
Two scalar equations re-written
scalar
Matrix
Contents of
matrices
46
More generic form (combines all exogenous variables into
single matrix)
More generic:
Where E1 Ξ X1, E2 Ξ X2 and E3 Ξ X3
47
More generic form:
All exogenous variables part of a single
variance-covariance matrix
48