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Introduction to Structural Equation Modeling with LISREL.

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**1. **Introduction to Structural Equation Modeling with LISREL. E. Kevin Kelloway, Ph.D.
Professor of Management and Psychology
Senior Research Fellow, CN Centre for Occupational Health and Safety Cover PageCover Page

**2. ** Statement of causal relations
Implies a pattern of covariances/correlations
Necessary (but not sufficient) condition for validity is that the oberved pattern of correlations matches the implied pattern of correlations.
Fundamental hypothesis of all SEM applications
= ?(?)
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**3. **Fishbein & Ajzen’s Theory of Reasoned Action Content Page 1Content Page 1

**4. ** Model Specification
Identification
Estimation
Testing Fit
Respecification Content Page 2Content Page 2

**5. **Model Specification ? = ?(?) Cover PageCover Page

**6. ** Association
Isolation
Causal Direction Content Page 2Content Page 2

**7. ** Causal flow is from left to right (top to bottom).
Curved arrows represent bidirectional relationships (correlations).
Straight arrows represent causal associations
Relationships assumed to be linear
What’s not in the model is just as important as what is in the model
Causal Closure Content Page 2Content Page 2

**8. **Path Diagram Content Page 1Content Page 1

**9. **Factor Analysis Y = t + e Content Page 1Content Page 1

**10. ** Identification X + Y = 10 Solve for X Cover PageCover Page

**11. ** Just Identified (e.g., regression or multiple regression)
Under Identified
Over-Identified
The t rule, given a k X k matrix there are
k X (k-1)/2 elements that can be estimated Content Page 2Content Page 2

**12. ** Direction – recursive models
Assigning value – to parameters (often 0) Content Page 2Content Page 2

**13. ** Estimation Cover PageCover Page

**14. **Iterative estimation to a fitting criterion
ML and GL allow for a fit test (N-1) * minimum of the fitting function is distributed as ?2
Partial vs Full information techniques Content Page 2Content Page 2

**15. ** Model Fit Cover PageCover Page

**16. **Absolute
Comparative
Parsimonious Content Page 2Content Page 2

**17. **Available for ML and GLS
Tests the null that ?=?(?)
Distributed with 1/2(q)(q+1)-k df where q is the number of variables and k is the number of estimated parameters
Power
Logical problem of accepting the null Content Page 2Content Page 2

**18. **Indicate degree of fit along a bounded continuum (normed)
Be independent of sample size
Have known distributional properties
No fit indices (except possibly the RMSEA) meet these criteria
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**19. **RMR & Standardized RMR
RMSEA
GFI
AGFI
?2/df Content Page 2Content Page 2

**20. **Tests of individual parameters
Called t values but are interpreted as Z scores
Problems:
Overall fit but parameters are not significant.
Overall fit but parameters are in opposite direction.
Lack of fit but all parameters as predicted Content Page 2Content Page 2

**21. **Null Model (Independence Model)
Saturated Model
Measures of absolute fit test the distance from the saturated model (i.e., are tests of identifying restrictions).
Measures of comparative fit typically test the distance from the null model.
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**22. **Normed Fit Index (NFI)
Non-Normed Fit Index(NNFI)
Incremental Fit Index(IFI)
Comparative Fit Index(CFI)
Relative Fit Index(RFI)
Expected Cross-Validation Index(ECVI) Content Page 2Content Page 2

**23. **Fit (both absolute and comparative) increases with the number of parameters estimated.
Rewards researcher for estimating trivial paths
Parsimonious fit adjusts for the df in the model and penalizes accordingly
Tend to reward the estimation of significant (and only significant) paths
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**24. **Parsimonious Normed Fit Index (PNFI)
Parsimonious Goodness of Fit Index(PGFI)
Akaike Information Criterion(AIC)
Consistent Akaike Information Criterion (CAIC)
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**25. **Compare two (theoretically generated) plausible models of the data
If the models stand in nested sequence (one model is completely contained in the other) then the difference may be tested with a ?2difference test
Subtract the two ?2 values and the result is distribute as ?2 with df equal to the difference in model dfs
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**26. **Compare competing and theoretically plausible models
Identify sources of ambiguity a priori
Using multiple indices/definitions of fit
Recognize that fit does not equate to truth or validity
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**27. ** Model Modification Cover PageCover Page

**28. **Theory trimming (significance tests)
Theory Building (modification indices)
Replication - holdout samples
Simultaneous estimation
“What percentage or researchers would find themselves unable to think up a theoretical justification for freeing a parameter? In the absence of empirical information, I assume that the answer… is near zero” (Steiger, 1990 p. 175)
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**29. ** LISREL The beauty and the horror Cover PageCover Page

**30. **Run in batch (with limited interactivity)
Written in the SIMPLIS language
Three tasks
Specify the data
Specify the model
Specify the output Content Page 2Content Page 2

**31. ** Example 1: A regression Model Cover PageCover Page

**32. **Janes Safety Data (regression)
Observed Variables Injury Training Tfl Passive
Covariance Matrix
1.13
-.05 .096
-.279 -.092 1.973
.439 .067 -.807 2.406
Sample Size: 129
Equation: Injury = Training Tfl Passive
End of Problem
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**33. **Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Donec enim. Fusce libero nisi, feugiat nec, tincidunt eu, accumsan non, justo. Pellentesque mauris. In sit amet velit et libero sollicitudin volutpat. Donec sodales eros id magna. Ut vel neque eget metus sollicitudin semper.
Phasellus vitae augue sed pede convallis laoreet. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos hymenaeos. Nulla posuere, nibh ut dictum lacinia, ipsum augue dignissim felis, quis volutpat felis diam at enim. Content Page 1Content Page 1

**34. **Relationships
Injury = Training - Passive
Paths
Training – Passive -> Injury
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**35. **Add the words “Path Diagram” just before the End of Problem Statement
Theory Trimming
Theory Building Content Page 2Content Page 2

**36. **Compare competing and theoretically plausible models
Identify sources of ambiguity a priori
Using multiple indices/definitions of fit
Recognize that fit does not equate to truth or validity
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**37. ** Example 2: A Path Analysis (observed variable) Cover PageCover Page

**38. ** Basic hypotheses
- [a] leadership affects wellbeing
[b] effects are indirect being mediated by
trust and self efficacy
A FULLY MEDIATED Model Content Page 2Content Page 2

**39. **t rule is met
Null B rule (no relationships among the endogenous variables) - e.g., a multiple regression equation;
Recursive rule - Recursive models are identified
Rank and Order conditions - essentially allows for non-recursive models, need a unique predictor for one of the variables in a non-recursive relationship Content Page 2Content Page 2

**40. **Leadership Data Fully mediated model
Observed Variables = Wellbeing Trust Efficacy Leadership
Means
22.3035294 4.9588235 3.9641765 10.4242353
Standard Deviations
3.9405502 .8590221 .6941727 3.1419617
Correlations
1.0000000
-.2361636 1.0000000
-.1746880 .1860385 1.0000000
-.1441248 .4604753 .1907934 1.0000000
sample size = 425
Paths
Trust Efficacy ->Wellbeing
Leadership ->Trust Efficacy
path diagram
end of problem
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**41. **Does the model fit?
Are the paths significant?
Do the data suggest changing the model?
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**42. **
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**43. **Both the fully mediated and the non-mediated are nested within the partially mediated (but are not directly comparable)
Mediation exists if: [a] Fully mediated Fit is not significantly different than Partially mediated Fit and [b] Non-mediated Fit is significantly worse than Partially mediated fit
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**44. **Content Page 2Content Page 2

**45. **Leadership Data Fully mediated model
Observed Variables = Wellbeing Trust Efficacy Leadership
Means
22.3035294 4.9588235 3.9641765 10.4242353
Standard Deviations
3.9405502 .8590221 .6941727 3.1419617
Correlations
1.0000000
-.2361636 1.0000000
-.1746880 .1860385 1.0000000
-.1441248 .4604753 .1907934 1.0000000
sample size = 425
Y variables = Wellbeing Trust Efficacy
path diagram
end of problem
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**46. ** Example 3: Confirmatory Factor Analysis Cover PageCover Page

**47. **Union commitment literature identifies 3 components of union commitment (loyalty, responsibility, willingness)
Does the same structure hold for commitment to other representative groups (student union). Content Page 2Content Page 2

**48. **One factor model is always a reasonable alternative
Orthogonal models are always nested within oblique models (but may be trivial)
If one generates an alternative model by combining factors (i.e., by fixing the interfactor correlation to 1) a nested sequence is obtained
In this case the literature suggests both a 3 factor (loyalty, responsibility, willingness) and a 2 factor (attitudes and behavior) model
Estimate a 1 factor, two factor and three factor model
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**49. **CFA models are recursive
t rule (estimate less parameters than the number of non-redundant elements in the covariance matrix)
3 indicator rule - 3 observed variables for each latent variable
2 indicator rule - 2 observed variables for each latent variable and latent variables are allowed to correlate
Both 3 indicator and 2 indicator rule assume that unique factor loadings (error terms) are uncorrelated
Monte Carlo research supports the use of 3 indicators with sample sizes greater than 200
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**50. **Example_3.spl (Three Factor) Content Page 1Content Page 1

**51. **Example_3.spl (Two Factor) Content Page 1Content Page 1

**52. **Content Page 2Content Page 2

**53. **INTERACTIVE VERSION Content Page 1Content Page 1

**54. ** Example 4: Latent Variable Path Analysis Cover PageCover Page

**55. **CFA and Path Analysis at the same time
Corrects structural parameters for measurement - modeling with “true” as opposed to “observed” scores
Increased complexity - only real advantage is when you care about both questions of measurement and structural relations Content Page 2Content Page 2

**56. **Lack of fit may result from [a] the measurement model, [b] the structural model, or [c] both
Establish the fit of the measurement model (provides a baseline for the full model), then move to testing structural parameters Content Page 2Content Page 2

**57. **Virtually all of the organizational literature treats gossip as a “bad” thing
We hypothesize that gossip can be a good thing
It enhances individual control
It may enhance organizational citizenship behaviors Content Page 2Content Page 2

**58. **Content Page 2Content Page 2

**59. **OCB = Ocb1 ocb2 ocb3 = item parcels each made up by summing 2 items
CTRL = 3 single indicators (items)
GOSSIP = 4 scale scores (toldsup, hearsup, toldcow, hearcow)
Need to assign a scale for each latent variable
(fix a factor loading to 1 – shouldn’t matter which one) Content Page 2Content Page 2

**60. **Content Page 2Content Page 2

**61. **Does the model fit?
Can it be fixed?
(If so, how?) – Identifying the problem, resolving the problem (Hopefully)
Number of indicators
Single indicator latent variables
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**62. **Start with a correlation matrix (Kevin’s preference)
Reading from a file
Import SPSS data Content Page 2Content Page 2

**63. **THANK YOU Content Page 2Content Page 2