General Structural Equation
(LISREL) Models
Week #2 Class #2
1
Today’s class



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Latent variable structural equations in
matrix form (from yesterday)
Fit measures
SEM assumptions
What to write up
LISREL matrices
2
1
REDUCE
1
e1
From yesterday’s lab:
1
NEVHAPP e2
1
NEW_GOAL e3
Ach1
1
IMPROVE
1
ACHIEVE
e4
Reference indicator:
REDUCE
e5
1
CONTENT e6
Regression Weights:
------------------REDUCE <---------NEVHAPP <--------NEW_GOAL <-------IMPROVE <--------ACHIEVE <--------CONTENT <---------
Ach1
Ach1
Ach1
Ach1
Ach1
Ach1
Estimate
--------
S.E.
-------
C.R.
-------
1.000
2.142
-2.759
-4.226
-2.642
2.657
0.374
0.460
0.703
0.450
0.460
5.721
-5.995
-6.009
-5.874
5.779
Label
------
3
1
REDUCE
1
e1
From yesterday’s lab:
1
NEVHAPP e2
1
NEW_GOAL e3
Ach1
1
IMPROVE
1
ACHIEVE
Reference indicator:
e4
REDUCE
e5
1
CONTENT e6
Standardized Regression Weights:
-------------------------------REDUCE <---------NEVHAPP <--------NEW_GOAL <-------IMPROVE <--------ACHIEVE <--------CONTENT <---------
Estimate
-------Ach1
Ach1
Ach1
Ach1
Ach1
Ach1
0.138
0.332
-0.541
-0.682
-0.410
0.357
4
From yesterday’s lab:
1
REDUCE
1
1
NEVHAPP
e1
e2
1
NEW_GOAL e3
Ach1
1
IMPROVE
1
ACHIEVE
1
CONTENT
e4
e5
Reference indicator:
REDUCE
e6
5
Regression Weights:
Label
---------------------REDUCE <---------NEVHAPP <--------NEW_GOAL <-------IMPROVE <--------ACHIEVE <--------CONTENT <---------
Estimate
-------Ach1
Ach1
Ach1
Ach1
Ach1
Ach1
1.000
-113.975
215.393
373.497
211.419
-155.262
Standardized Regression Weights:
-------------------------------REDUCE <---------NEVHAPP <--------NEW_GOAL <-------IMPROVE <--------ACHIEVE <--------CONTENT <---------
S.E.
-------
1441.597
2717.178
4711.675
2667.067
1961.974
C.R.
-------
---
-0.079
0.079
0.079
0.079
-0.079
Estimate
-------Ach1
Ach1
Ach1
Ach1
Ach1
Ach1
0.002
-0.223
0.534
0.762
0.415
-0.264
6
Solution:
• Use a different reference indicator
• (Note: REDUCE can be used as a
reference indicator in a 2-factor model,
though other reference indicators might be
better because REDUCE is factorally
complex)
7
1
REDUCE e1
1
NEVHAPP
Content
e2
1
1
NEW_GOAL e3
1
IMPROVE
Achieve
1
1
ACHIEVE
1
CONTENT
e4
e5
e6
When to add, when not to add parameters
8
Modification Indices
Covariances:
Par Change
e1
e1
e6
e5
e5
e4
e4
e3
e2
e2
e2
e2
<-->
<-->
<-->
<-->
<-->
<-->
<-->
<-->
<-->
<-->
<-->
<-->
Ach1
Cont1
Ach1
Cont1
e6
e1
e6
e1
e1
e6
e5
e3
M.I.
63.668
6.692
32.540
4.370
13.033
28.242
24.104
4.500
5.440
5.290
14.681
12.410
0.032
0.016
-0.023
0.012
-0.028
0.036
-0.034
0.012
0.016
-0.016
0.025
-0.017
1
REDUCE e1
1
NEVHAPP
Content
1
1
NEW_GOAL e3
1
IMPROVE
Achieve
1
1
ACHIEVE
1
CONTENT
Discrepancy
125.260 0.000
Degrees of freedom
8
P
0.000
0.000
e2
e4
e5
e6
9
Regression Weights:
Change
REDUCE
REDUCE
REDUCE
REDUCE
CONTENT
CONTENT
CONTENT
ACHIEVE
ACHIEVE
IMPROVE
IMPROVE
NEW_GOAL
NEVHAPP
NEVHAPP
NEVHAPP
<-<-<-<-<-<-<-<-<-<-<-<-<-<-<--
Ach1
ACHIEVE
IMPROVE
NEW_GOAL
Ach1
ACHIEVE
IMPROVE
REDUCE
NEVHAPP
REDUCE
CONTENT
NEVHAPP
REDUCE
ACHIEVE
NEW_GOAL
M.I.
Par
52.853
16.291
50.413
23.780
27.051
24.336
31.694
4.791
11.086
18.169
16.219
6.137
4.031
9.687
9.452
0.406
0.076
0.140
0.117
-0.293
-0.094
-0.112
0.033
0.056
0.058
-0.053
-0.032
0.029
0.050
-0.063
1
REDUCE e1
1
NEVHAPP
Content
e2
1
1
NEW_GOAL e3
1
IMPROVE
Achieve
1
1
ACHIEVE
1
CONTENT
e4
e5
e6
10
.37
1
REDUCE
.17
e1
1.08
.26
.99
Cont1
1
NEVHAPP
e2
.19
1.00
-.05
.66
.07
1
NEW_GOAL
.17
1.03
1.76
e3
IMPROVE
1
e4
.35
Ach1
1.00
ACHIEVE
1
e5
.40
1
CONTENT
e6
Choice to add or not to add parameter from Ach1  REDUCE
a matter of theoretical judgement. (Note changes in other
parameters)
11
Goodness of Fit Measures in
Structural Equation Models
A Good Reference: Bollen and Long,
TESTING STRUCTURAL EQUATION
MODELS, Sage, 1993.
12
Goodness of Fit Measures in
Structural Equation Models

A fit measure expresses the difference
between Σ(θ) and S. Using whatever
metric it employs, it should register
“perfect” whenever Σ(θ) = S exactly.


This occurs trivially when df=0
0 to 1 usually thought of as best metric
(see Tanaka in Bollen & Long, 1993)
13
Goodness of Fit Measures in
Structural Equation Models
Early fit measures:
Model Χ2 :
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Asks the question, is there a statistically
significant difference between S and Σ ?
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If the answer to this question is “no”, we
should definitely NOT try to add parameters
to the model (capitalizing on change)
If the answer to this question is “yes”, we can
cautiously add parameters

Contemporary thinking is that we need some other
14
measure that is not sample-size dependent
Goodness of Fit Measures in
Structural Equation Models
Model Χ2 :
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
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X2 = (N-1) * Fml
Contemporary thinking is that we need some
other measure that is not sample-size
dependent
An issue in fit measures: “sample size
dependency” (not considered a good
thing)

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Chi-square is very much sample size
dependent (a direct function of N)
15
Goodness of Fit Measures in
Structural Equation Models
Model Χ2 :



X2 = (N-1) * Fml
Contemporary thinking is that we need some
other measure that is not sample-size
dependent
An issue in fit measures: “sample size
dependency” (not considered a good
thing)


Chi-square is very much sample size
dependent (a direct function of N)
16
Goodness of Fit Measures in
Structural Equation Models
Problem with Χ2 itself as a measure
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(aside from the fact that it is a direct function of
N):
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Logic of trying to “embrace” the null hypothesis.
Even if chi-square not used, it IS important
as a “cut off” (never add parameters to a
model when chi-square is non-signif.
Many measures are based on Χ2
17
Goodness of Fit Measures in
Structural Equation Models
The “first generation” fit measures:

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Jöreskog and Sörbom’s Goodness of Fit
Index (GFI) [LISREL]
Bentler’s Normed Fit Index (NFI) [EQS]
These have now been supplemented
in most software packages with a wide
variety of fit measures
18
Fit Measures
GFI = 1 – tr[Σ-1S – I]2
tr (Σ-1S)2
Takes on value from 0 to 1
Conventional wisdom: .90 cutoff
GFI tends to yield higher values than other
coefficients
GFI is affected by sample size, since in small
samples, we would expect larger differences
between Σ and S even if the model is correct
(sampling variation is larger)
19
Fit Measures
GFI is an “absolute” fit measure
There are “incremental” fit measures that
compare the model against some baseline.
- one such baseline is the “Independence
Model
- Independence Model: models only the
variances of manifest variables (no
covariances) [=assumpt. all MVs
independent] “Independence Model chisquare” (usually very large)
- Σ will have 0’s in the off-diagonals
20
Fit Measures
NFI = (Χ2b-Χ2m)/ Χ2b Normed Fit Index
(Bentler)
(subscript b = baseline m=model)
Both NFI and GFI will increase as the number
of model parameters increases and are
affected by N (though not as a simple *N or
*N-1 function).
GFI = widely used in earlier literature since it
was the only measure (along with AGFI)
available in LISREL
NFI (along with NNFI) only measure available
in early versions of EQs
21
Fit Measures
Thinking about fit indices:
Desirable properties:
1. Normed (esp. to 0  1)
Some measures only approx: TLI
Arbitrary metric: AIC (Tanaka: AIC could be
normed)
2. Not affected by sample size (GFI, NFI are)
3. “Penalty function” for extra parameters (no
inherent advantage to complex models) –
“Parsimony” indices deal with this
4. Consistent across estimation techniques
(ML, GLS, other methods)
22
Fit Measures
Bollens delta-2
(Χ2b – Χ2m )/ Χ2b – dfm
RMR – root mean residual (only works with
standardized residuals)
SRMR - standardized RMR
Parsimony GFI 2df/p * (p+1) * GFI
AGFI = 1 – [1(q+1) / 2df ] [1 – GFI]
RNI (Relative Noncentrality Index)
= [(Χ2b – dfb) – (X2m- dfm)] / (Χ2b – dfb)
CFI = 1 – max[(X2m- dfm),0] / max[(X2m- dfm), (X2
23
b- dfb),0]
Fit Measures
•
•
Some debate on conventional .90 criterion for
most of these measures
Hu & Bentler, SEM 6(1), 1999 suggest:
Use at least 2 measures
Use criterion of >.95 for 0-1 measure, <.06 for
RMSEA or SRMR
24
SEM Assumptions
Fml estimator:
1. No Kurtosis
2. Covariance matrix analysed *
3. Large sample
4. H0: S = Σ(θ) holds exactly
25
SEM Assumptions
Fml estimator:
1. Consistent
2. Asymptotically efficient
3. Scale invariant
4. Distribution approximately
normal as N increases
26
SEM Assumptions
Fml estimator:
Small Samples
1980s simulations:
- Not accurate N<50
- 100 + highly recommended
- “large sample” usually 200+
- in small samples, chi-square tends ot be
too large
27
Writing up results from
Structural Equation Models
What to Report, What to Omit
28
Writing up results from Structural Equation Models


Reference: Hoyle and Panter chapter in
Hoyle.
Important to note that there is a wide
variety of reporting styles (no one
“standard”).
29
Writing up results from Structural Equation Models

A Diagram
Construct Equation Model

Measurement Equation model
Some simplification may be required.
Adding parameter estimates may clutter (but
for simple models helps with reporting).
Alternatives exist (present matrices).
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30
Reporting Structural Equation Models

“Written explanation justifying each path
and each absence of a path” (Hoyle and
Panter)
(just how much journal space is
available here? )
It might make more sense to try to
identify potential controversies (with
respect to inclusion, exclusion).
31
Controversial paths?
1
y1
1
1
1
1
y2 y3 y4 y5 y6
1
20
LV1
LV3
e1
1
e2
LV2
1
1
1
1
1
LV4
1
1
1
32
What to report and what not to report…..

Present the details of the statistical
model
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

4.
Clear indication of all free parameters
Clear indication of all fixed parameters
It should be possible for the reader to
reproduce the model
Describe the data
1.
Correlations and standard errors (or
covariances) for all variables ??
Round to 3-4 digits and not just 2 if you do this
33
What to report and what not to report…
4. Describing the data (continued)

Distributions of the data




Any variable highly skewed?
Any variable only nominally continuous (i.e., 5-6
discrete values or less)?
Report Mardia’s Kurtosis coefficient (multivariate
statistic)
Dummy exogenous variables, if any
5. Estimation Method
If the estimation method is not ML, report ML
results.
34
What to report and what not to report…
6. Treatment of Missing Data


How big is the problem?
Treatment method used?

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
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Pretend there are no missing data
Listwise deletion
Pairwise deletion
FIML estimation (AMOS, LISREL >=8.5)
Nearest neighbor imputation (LISREL >=8.1)
EM algorithm (covariance matrix imputation )
(LISREL >=8.5)
35
What to report and what not to report…
7. Fit criterion
Hoyle and Panter suggest “.90; justify if
lower”.

Choice of indices also an issue.
There appears to be “little consensus on the
best index” (H & P recommend using
multiple indices in presentations)
Standards:
Bollen’s delta 2 (IFI)
Comparative Fit Index
RMSEA

36
Fit indices

Older measures:
 GFI (Joreskog & Sorbom)
 Bentler’s Normed Fit index
 Model Chi-Square
37
What to report & what not to report….
8. Alternative Models used for Nested Comparisons (if appropriate)
US South
US West
1
1
1
1
U.S. Midwest
U.S. Rust Belt
38
9. Plausible explanation for correlated errors

[“these things were just too darned big to ignore”]
Generally assumed when working with panel model with equivalent
indicators across time:
1
1
1
1
1
1
1
1
39
What to report
10. Interpretation of regression-based
model



Present standardized and unstandardized
coefficients (usually)
Standard errors? (* significance test
indicators?)
R-square for equations


Measurement model too?
(expect higher R-squares)
40
What to report.

Problems and issues

Negative error variances or other reasons
for non-singular parameter covariance
matrices


Convergence difficulties, if any



How dealt with? Does the final model entail any
“improper estimates”?
LISREL: can look at Fml across values of given
parameter, holding other parameters constant
Collinearity among exogenous variables
Factorially complex items
41
What to report & what not to report….

General Model Limitations, Future
Research issues:

Where the number of available indicators
compromised the model




2-indicator variables? (any constraints required?)
Single-indicator variables? (what assumptions
made about error variances?)
Indicators not broadly representative of the
construct being measured?
Where the distribution of data presented
problems

Larger sample sizes can help
42
What to report & what not to report….

General Model Limitations, Future
Research issues:


Missing data (extent of, etc.)
Cause-effect issues, if any (what constraints
went into non-recursive model? How
reasonable are these?)
43
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
44
Matrix form: LISREL MEASUREMENT MODEL MATRICES
A slightly more complex example:
45
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.
46
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
47
Matrix form: LISREL MEASUREMENT MODEL MATRICES
Models with correlated measurement errors:
48
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
49
Matrix form: LISREL MEASUREMENT MODEL MATRICES
Measurement models for endogenous latent variables
(ETA) are similar:
50
LISREL MATRIX FORM
An Example:
51
LISREL MATRIX FORM
An Example:
52
LISREL MATRIX FORM
An Example:
53
LISREL MATRIX FORM
An Example:
+ theta-epsilon, 8 x 8 matrix with parameters in diagonal
and 0s in off diagonals (a “diagonal” matrix)
54
Class Exercise
1
1
#1
1
Provide labels for
each of the
variables
1
1
55
#2
1
1
1
1
1
56
#1
epsilon
ksi
eta
zeta
delta
57
#2
58
Lisrel Matrices for examples.
No Beta
Matrix in
this model
59
Lisrel Matrices for examples.
60
Lisrel Matrices for examples (example #2)
61
Lisrel Matrices for examples (example #2)
62