General Structural Equation (LISREL) Models

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General Structural Equation (LISREL) Models

• Week 2 Class 2

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Todays class

• Latent variable structural equations in matrix
  form (from yesterday)
• Fit measures
• SEM assumptions
• What to write up
• LISREL matrices

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From yesterdays lab
Reference indicator REDUCE
Regression Weights Estimate
S.E. C.R. Label -------------------
-------- ------- -------
------- REDUCE lt---------- Ach1
1.000
NEVHAPP lt--------- Ach1 2.142 0.374
5.721 NEW_GOAL
lt-------- Ach1 -2.759 0.460 -5.995
IMPROVE lt--------- Ach1
-4.226 0.703 -6.009
ACHIEVE lt--------- Ach1 -2.642 0.450
-5.874 CONTENT lt---------
Ach1 2.657 0.460 5.779
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From yesterdays lab
Reference indicator REDUCE
Standardized Regression Weights
Estimate --------------------------------
-------- REDUCE lt---------- Ach1
0.138 NEVHAPP lt--------- Ach1
0.332 NEW_GOAL lt-------- Ach1
-0.541 IMPROVE lt--------- Ach1
-0.682 ACHIEVE lt--------- Ach1
-0.410 CONTENT lt--------- Ach1
0.357
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From yesterdays lab
Reference indicator REDUCE
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• Regression Weights Estimate
  S.E. C.R. Label
• ------------------- --------
  ------- ------- -------
• REDUCE lt---------- Ach1 1.000
  
• NEVHAPP lt--------- Ach1 -113.975
  1441.597 -0.079
• NEW_GOAL lt-------- Ach1 215.393
  2717.178 0.079
• IMPROVE lt--------- Ach1 373.497
  4711.675 0.079
• ACHIEVE lt--------- Ach1 211.419
  2667.067 0.079
• CONTENT lt--------- Ach1 -155.262
  1961.974 -0.079

Standardized Regression Weights
Estimate --------------------------------
-------- REDUCE lt---------- Ach1
0.002 NEVHAPP lt--------- Ach1
-0.223 NEW_GOAL lt-------- Ach1
0.534 IMPROVE lt--------- Ach1
0.762 ACHIEVE lt--------- Ach1
0.415 CONTENT lt--------- Ach1
-0.264
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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)

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When to add, when not to add parameters
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Modification Indices Covariances M.I.
Par Change e1 lt--gt Ach1 63.668 0.032 e1 lt--gt
Cont1 6.692 0.016 e6 lt--gt Ach1 32.540 -0.023 e5 lt-
-gt Cont1 4.370 0.012 e5 lt--gt e6 13.033 -0.028 e4 lt
--gt e1 28.242 0.036 e4 lt--gt e6 24.104 -0.034 e3 lt-
-gt e1 4.500 0.012 e2 lt--gt e1 5.440 0.016 e2 lt--gt e
6 5.290 -0.016 e2 lt--gt e5 14.681 0.025 e2 lt--gt e3
12.410 -0.017
Discrepancy 125.260 0.000 Degrees of
freedom 8 P 0.000 0.000
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Regression Weights M.I. Par Change REDUCE
lt-- Ach1 52.853 0.406 REDUCE lt-- ACHIEVE 16.291 0
.076 REDUCE lt-- IMPROVE 50.413 0.140 REDUCE lt-- NE
W_GOAL 23.780 0.117 CONTENT lt-- Ach1 27.051 -0.29
3 CONTENT lt-- ACHIEVE 24.336 -0.094 CONTENT lt-- IM
PROVE 31.694 -0.112 ACHIEVE lt-- REDUCE 4.791 0.033
ACHIEVE lt-- NEVHAPP 11.086 0.056 IMPROVE lt-- REDU
CE 18.169 0.058 IMPROVE lt-- CONTENT 16.219 -0.053
NEW_GOAL lt-- NEVHAPP 6.137 -0.032 NEVHAPP lt-- REDU
CE 4.031 0.029 NEVHAPP lt-- ACHIEVE 9.687 0.050 NEV
HAPP lt-- NEW_GOAL 9.452 -0.063
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Choice to add or not to add parameter from Ach1 ?
REDUCE a matter of theoretical judgement. (Note
changes in other parameters)
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Goodness of Fit Measures in Structural Equation
Models

• A Good Reference Bollen and Long, TESTING
  STRUCTURAL EQUATION MODELS, Sage, 1993.

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Goodness of Fit Measures in Structural Equation
Models

• A fit measure expresses the difference between
  S(?) and S. Using whatever metric it employs, it
  should register perfect whenever S(?) S
  exactly.
• This occurs trivially when df0
• 0 to 1 usually thought of as best metric (see
  Tanaka in Bollen Long, 1993)

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Goodness of Fit Measures in Structural Equation
Models

• Early fit measures
• Model ?2
• Asks the question, is there a statistically
  significant difference between S and S ?
• 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
  measure that is not sample-size dependent

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Goodness of Fit Measures in Structural Equation
Models

• Model ?2
• 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)

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Goodness of Fit Measures in Structural Equation
Models

• Model ?2
• 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)

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Goodness of Fit Measures in Structural Equation
Models

• Problem with ?2 itself as a measure (aside from
  the fact that it is a direct function of N)
• 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 ?2

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Goodness of Fit Measures in Structural Equation
Models

• The first generation fit measures
• Jöreskog and Sörboms Goodness of Fit Index
  (GFI) LISREL
• Bentlers Normed Fit Index (NFI) EQS
• These have now been supplemented in most software
  packages with a wide variety of fit measures

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Fit Measures

• GFI 1 trS-1S I2
• tr (S-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 S and S even if the model is correct
  (sampling variation is larger)

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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 chi-square (usually very large)
• - S will have 0s in the off-diagonals

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Fit Measures

• NFI (?2b-?2m)/ ?2b Normed Fit Index (Bentler)
• (subscript b baseline mmodel)
• 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

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Fit Measures

• Thinking about fit indices
• Desirable properties
• Normed (esp. to 0 ? 1)
• Some measures only approx TLI
• Arbitrary metric AIC (Tanaka AIC could be
  normed)
• Not affected by sample size (GFI, NFI are)
• Penalty function for extra parameters (no
  inherent advantage to complex models)
  Parsimony indices deal with this
• Consistent across estimation techniques (ML, GLS,
  other methods)

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Fit Measures

• Bollens delta-2
• (?2b ?2m )/ ?2b dfm
• RMR root mean residual (only works with
  standardized residuals)
• SRMR - standardized RMR
• Parsimony GFI 2df/p (p1) GFI
• AGFI 1 1(q1) / 2df 1 GFI
• RNI (Relative Noncentrality Index)
• (?2b dfb) (X2m- dfm) / (?2b dfb)
• CFI 1 max(X2m- dfm),0 / max(X2m- dfm), (X2
  b- dfb),0
• RMSEA sqrt (MAX(X2m- dfm),(n-1),0) / dfm

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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 gt.95 for 0-1 measure, lt.06 for
  RMSEA or SRMR

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SEM Assumptions

Fml estimator 1. No Kurtosis 2. Covariance
matrix analysed 3. Large sample 4. H0 S
S(?) holds exactly
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SEM Assumptions

Fml estimator 1. Consistent 2. Asymptotically
efficient 3. Scale invariant 4. Distribution
approximately normal as N increases
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SEM Assumptions

Fml estimator Small Samples 1980s
simulations - Not accurate Nlt50 - 100 highly
recommended - large sample usually 200 - in
small samples, chi-square tends ot be too large
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Writing up results from Structural Equation Models

• What to Report, What to Omit

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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).

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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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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).

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Controversial paths?
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What to report and what not to report..

• Present the details of the statistical model
• 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
• Correlations and standard errors (or covariances)
  for all variables ??
• Round to 3-4 digits and not just 2 if you do this

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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 Mardias Kurtosis coefficient
  (multivariate statistic)
• Dummy exogenous variables, if any
• 5. Estimation Method
• If the estimation method is not ML, report ML
  results.

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What to report and what not to report

• 6. Treatment of Missing Data
• How big is the problem?
• Treatment method used?
• Pretend there are no missing data
• Listwise deletion
• Pairwise deletion
• FIML estimation (AMOS, LISREL gt8.5)
• Nearest neighbor imputation (LISREL gt8.1)
• EM algorithm (covariance matrix imputation )
  (LISREL gt8.5)

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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
• Bollens delta 2 (IFI)
• Comparative Fit Index
• RMSEA

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Fit indices

• Older measures
• GFI (Joreskog Sorbom)
• Bentlers Normed Fit index
• Model Chi-Square

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What to report what not to report.

• 8. Alternative Models used for Nested Comparisons
  (if appropriate)

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• 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

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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)

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What to report.

• Problems and issues
• Negative error variances or other reasons for
  non-singular parameter covariance matrices
• How dealt with? Does the final model entail any
  improper estimates?
• Convergence difficulties, if any
• LISREL can look at Fml across values of given
  parameter, holding other parameters constant
• Collinearity among exogenous variables
• Factorially complex items

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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

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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?)

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Matrix form LISREL MEASUREMENT MODEL MATRICES
Manifest variables Xs Measurement errors
DELTA ( d) Coefficients in measurement equations
LAMBDA ( ? ) Sample equation X1 ?1 ?1 d1
MATRICES
LAMBDA-x THETA-DELTA PHI
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Matrix form LISREL MEASUREMENT MODEL MATRICES
A slightly more complex example
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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.
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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
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Matrix form LISREL MEASUREMENT MODEL MATRICES
Models with correlated measurement errors
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Matrix form LISREL MEASUREMENT MODEL MATRICES
Measurement models for endogenous latent
variables (ETA) are similar

• Manifest variables are Ys
• Measurement error terms EPSILON ( e )
• Coefficients in measurement equations LAMBDA (?)
• same as KSI/X side
• to differentiate, will sometimes refer to LAMBDAs
  as Lambda-Y (vs. Lambda-X)
• Equations
• Y1 ?1 ? 1 e1

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Matrix form LISREL MEASUREMENT MODEL MATRICES
Measurement models for endogenous latent
variables (ETA) are similar
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LISREL MATRIX FORM

• An Example

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LISREL MATRIX FORM

• An Example

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LISREL MATRIX FORM

• An Example

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LISREL MATRIX FORM

• An Example

theta-epsilon, 8 x 8 matrix with parameters in
diagonal and 0s in off diagonals (a diagonal
matrix)
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Class Exercise
1
Provide labels for each of the variables
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2
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1
epsilon
ksi
eta
zeta
delta
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2
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Lisrel Matrices for examples.
No Beta Matrix in this model
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Lisrel Matrices for examples.
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Lisrel Matrices for examples (example 2)
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Lisrel Matrices for examples (example 2)