Introduction to Structural Equation Modeling with LISREL.

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

2
What is a theory?

• 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
  
• ?(?)

3
Fishbein Ajzens Theory of Reasoned Action
Behavioral Intentions
Beliefs
Attitudes
Behavior
r.4
r.26
Subjective Norms
4
THE SEM PROCESS

• Model Specification
• Identification
• Estimation
• Testing Fit
• Respecification

5
Model Specification ? ?(?)
6
Causality

• Association
• Isolation
• Causal Direction

7
PATH DIAGRAMS

• 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
• Whats not in the model is just as important as
  what is in the model
• Causal Closure

8
Path Diagram
X
Q
Z
Y
9
Factor AnalysisY t e
F1
F2
Y3
Y5
Y6
Y1
Y2
Y4
E4
E5
E6
E2
E3
E1
10
IdentificationX Y 10Solve for X
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Types of Models

• 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

12
Identifying Restrictions

• Direction recursive models
• Assigning value to parameters (often 0)

13
Estimation
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Estimation

• 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

15
Model Fit
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Types of Fit

• Absolute
• Comparative
• Parsimonious

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Absolute fit The ?2 test

• Available for ML and GLS
• Tests the null that ??(?)
• Distributed with 1/2(q)(q1)-k df where q is the
  number of variables and k is the number of
  estimated parameters
• Power
• Logical problem of accepting the null

18
Criteria For Fit Indices (Gerbing Anderson,
1992)

• 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

19
Indices of Absolute Fit

• RMR Standardized RMR
• RMSEA
• GFI
• AGFI
• ?2/df

20
Condition 9 Tests

• 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

21
Comparative Model Fit

• 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.

22
Comparative Fit Indices

• 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)

23
Parsimonious Fit

• 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

24
Indices of Parsimonious Fit

• Parsimonious Normed Fit Index (PNFI)
• Parsimonious Goodness of Fit Index(PGFI)
• Akaike Information Criterion(AIC)
• Consistent Akaike Information Criterion (CAIC)

25
Nested Model Comparisons

• 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

26
Strategy for assessing fit

• 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

27
Model Modification
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Model Modification

• 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)

29
LISRELThe beauty and the horror
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LISREL Files

• Run in batch (with limited interactivity)
• Written in the SIMPLIS language
• Three tasks
• Specify the data
• Specify the model
• Specify the output

31
Example 1 A regression Model
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Example_1 .spl

• 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

33
The following lines were read from file
C\Documents and Settings\Kevin Kelloway\My
Documents\Example_1.spj 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 Sample
Size 129 Janes Safety Data (regression)

Covariance Matrix Injury
Training Tfl Passive
-------- -------- -------- --------
Injury 1.13 Training -0.05
0.10 Tfl -0.28 -0.09 1.97
Passive 0.44 0.07 -0.81
2.41 Janes Safety Data (regression)
Number of
Iterations 0 LISREL Estimates (Maximum
Likelihood)
Structural Equations Injury - 0.74Training
- 0.11Tfl 0.17Passive, Errorvar. 0.99 , R²
0.12 (0.29)
(0.069) (0.062)
(0.13) -2.51
-1.55 2.70
7.91 Goodness of Fit
Statistics Degrees
of Freedom 0 Minimum Fit
Function Chi-Square 0.0 (P 1.00)
Normal Theory Weighted Least Squares Chi-Square
0.00 (P 1.00) The Model is
Saturated, the Fit is Perfect ! Parameter
Estimates (B weights (not beta) from regression
Standard Errors t values ( gt 2.00 is
significant)

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34
Alternative ways of specifying the model

• Relationships
• Injury Training - Passive
• Paths
• Training Passive -gt Injury

35
Graphical Interface

• Add the words Path Diagram just before the End
  of Problem Statement
• Theory Trimming
• Theory Building

36
Strategy for assessing fit

• 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

37
Example 2 A Path Analysis(observed variable)
38
Model Specification
Trust
Leadership
Wellbeing
Efficacy

• Basic hypotheses
• - a leadership affects wellbeing
• b effects are indirect being mediated by
• trust and self efficacy
• A FULLY MEDIATED Model

39
Model Identification

• 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

40

• Example_2.spl (Output Model Data)

• 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 -gtWellbeing
• Leadership -gtTrust Efficacy
• path diagram
• end of problem

41
Run Example_2.spl

• Does the model fit?
• Are the paths significant?
• Do the data suggest changing the model?

42
Generate Nested Models
The partially mediated
Trust
Wellbeing
Leadership
Efficacy
The Non-mediated
Trust
Wellbeing
Leadership
Efficacy
43
NESTING SEQUENCE
Trust
Wellbeing
Leadership
Efficacy

• 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

44
RESULTS
?2 Difference
45
INTERACTIVE MODEL BUILDING

• 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

46
Example 3 Confirmatory Factor Analysis
47
Model Development

• 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).

48
Alternative Models

• 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

49
Identification in CFA

• 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

50
Example_3.spl (Three Factor)
student union commitment observed variables
loyal1 loyal2 loyal3 loyal4 resp1-resp3 will1 -
will3 Means 3.4095563 3.5460751 2.9044369
3.3412969 3.3378840 3.9590444 4.1945392 3.0955631
2.4744027 3.1194539 Standard deviations
.8457250 .8122099 .8862037 .7716516 .9057455
.9094416 1.0066751 .9745463 .9198310
.9910889 correlation matrix 1.0000000
.5208533 1.0000000 .5595972 .5009598
1.0000000 .6142105 .5376375 .6488150
1.0000000 .4580473 .3767890 .4840866
.4371304 1.0000000 .1999868 .1462896
.0886099 .1712668 .3078848 1.0000000
.1756045 .1628230 .0746533 .1038066 .3257948
.6222075 1.0000000 .3304682 .2064215
.3754222 .4300969 .3706724 .2633207 .2532679
1.0000000 .1235800 .1562927 .2700683
.2873694 .2426644 .1010893 .0775160 .5108488
1.0000000 .2110942 .2079855 .3015793
.3360939 .3173118 .2942106 .3576399 .5980011
.5123867 1.0000000 sample size293 latent
variables Loyal Resp Will relationships
loyal1 -loyal4 Loyal resp1 - resp3 Resp
will1 - will3 Will path diagram end of
problem.
51
Example_3.spl (Two Factor)
student union commitment observed variables
loyal1 loyal2 loyal3 loyal4 resp1-resp3 will1 -
will3 Means 3.4095563 3.5460751 2.9044369
3.3412969 3.3378840 3.9590444 4.1945392 3.0955631
2.4744027 3.1194539 Standard deviations
.8457250 .8122099 .8862037 .7716516 .9057455
.9094416 1.0066751 .9745463 .9198310
.9910889 correlation matrix 1.0000000
.5208533 1.0000000 .5595972 .5009598
1.0000000 .6142105 .5376375 .6488150
1.0000000 .4580473 .3767890 .4840866
.4371304 1.0000000 .1999868 .1462896
.0886099 .1712668 .3078848 1.0000000
.1756045 .1628230 .0746533 .1038066 .3257948
.6222075 1.0000000 .3304682 .2064215
.3754222 .4300969 .3706724 .2633207 .2532679
1.0000000 .1235800 .1562927 .2700683
.2873694 .2426644 .1010893 .0775160 .5108488
1.0000000 .2110942 .2079855 .3015793
.3360939 .3173118 .2942106 .3576399 .5980011
.5123867 1.0000000 sample size293 latent
variables Att Behav relationships loyal1
-loyal4 Att resp1 - will3 Behav
path diagram end of problem.
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RESULTS
?2 Difference
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INTERACTIVE VERSION
student union commitment observed variables
loyal1 loyal2 loyal3 loyal4 resp1-resp3 will1 -
will3 Means 3.4095563 3.5460751 2.9044369
3.3412969 3.3378840 3.9590444 4.1945392 3.0955631
2.4744027 3.1194539 Standard deviations
.8457250 .8122099 .8862037 .7716516 .9057455
.9094416 1.0066751 .9745463 .9198310
.9910889 correlation matrix 1.0000000
.5208533 1.0000000 .5595972 .5009598
1.0000000 .6142105 .5376375 .6488150
1.0000000 .4580473 .3767890 .4840866
.4371304 1.0000000 .1999868 .1462896
.0886099 .1712668 .3078848 1.0000000
.1756045 .1628230 .0746533 .1038066 .3257948
.6222075 1.0000000 .3304682 .2064215
.3754222 .4300969 .3706724 .2633207 .2532679
1.0000000 .1235800 .1562927 .2700683
.2873694 .2426644 .1010893 .0775160 .5108488
1.0000000 .2110942 .2079855 .3015793
.3360939 .3173118 .2942106 .3576399 .5980011
.5123867 1.0000000 sample size293 latent
variables Loyal Resp WIll path diagram end of
problem.
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Example 4 Latent Variable Path Analysis
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Latent Variable Modeling

• 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

56
Two Stage Modeling (Anderson Gerbing, 1988)

• 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

57
Generating Multiple Indicators

• 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

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A Structural Model
Organizational Citizenship Behaviors
GOSSIP
CONTROL
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A measurement model

• 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 shouldnt matter
  which one)

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The Full Model
Organizational Citizenship Behaviors
GOSSIP
CONTROL
Toldsup HearSup Toldcow
HearCow
Item1 Item2 Item3
ocb2
ocb3
Ocb1
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Estimate example_4.spl

• 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

62
Getting Data into LISREL

• Start with a correlation matrix (Kevins
  preference)
• Reading from a file
• Import SPSS data

63

• THANK YOU