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ROB CRIBBIE
Q U A N T I T A T I V E M E T H O D S P R O G R A M – D E P A R T M E N T O F P S Y C H O L O G Y
C O O R D I N A T O R - S T A T I S T I C A L C O N S U L T I N G S E R V I C E
C O U R S E M A T E R I A L S A V A I L A B L E A T :
W W W . P S Y C H . Y O R K U . C A / C R I B B I E
Introduction to Structural Equation Modeling (SEM)
Day 3: November 22, 2012
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Topics Covered in the First Two Weeks
Establishing and Identifying Models Determining Model Fit Checking Statistical Significance of Parameters Checking the r2 of Outcome Variables Relationship between Regression and Structural
Equation Modeling Path Analysis Confirmatory Factor Analysis Full Structural Equation Models
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What are we going to do today?
Mediation Analysis in SEM Multiple Group Models
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Mediation Analysis in SEM
What is mediation? Mediation implies a causal hypothesis where an
independent variable causes a mediating variable which causes a dependent variable.
A mediating variable is responsible for the relationship
between the predictor and the outcome variables In other words, a mediating variable explains “how” or “why” an
independent variable predicts a dependent variable.
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Simple 3-variable Mediation Model.
X predictor
Y outcome
M mediator
X predictor
Y outcome
c
a b
c’
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Terminology
The effect of X on Y through M is referred to as the mediated effect or the indirect effect. A mediating variable is also commonly called an intervening
variable, an intermediate variable, or a process variable.
Suppression and confounding effects also involve 3-
variable systems and are statistically, but not conceptually, related to mediation
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Mediation versus Moderation
Moderation is conceptually and statistically different from mediation.
Mediation – how, why Moderation – when, under what circumstances
Moderation - The nature of the relationship
between the predictor and the outcome differs depending on the level of the moderator.
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Mediation and SEM
Mediational relationships are very common in SEM models
Outdated methods for testing mediation included a series of regression equations However, with SEM we can test multiple regression equations
simultaneously! This makes testing mediational hypotheses very straightforward in
SEM
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The Indirect Effect
Evaluate the significance of the a*b effect Could use the Sobel test, but it can be problematic and produce
biased results (especially with small samples)
Current recommendation in the mediation literature is to evaluate the significance of the indirect effect using bias-corrected bootstrapped confidence intervals
We can do this in AMOS quite easily
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Bootstrapping
Bootstrapping provides an approach to constructing confidence intervals for the mediated effect. Bootstrapped confidence intervals make no assumption
about the distribution of the mediated effect statistic (a*b).
A large number of bootstrap samples are drawn from the data and the effect (a*b) is estimated from each of these bootstrap samples.
The distribution of these samples forms an empirical sampling distribution of the effect.
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Bias-corrected Bootstrap Confidence Intervals
For a 95% CI:
◦ The lower limit is the bootstrapped estimate of a*b at the 2.5 percentile
◦ The upper limit is the bootstrapped estimate of a*b at the 97.5 percentile ◦ Bias correction increases the likelihood that the population value
of a*b is encompassed within the interval in the expected proportion of cases (e.g., 95%)
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Mediation in SEM
In AMOS, the first thing we need to do is to ask for output on the indirect effects View Analysis Properties Output
Click “Indirect, Direct, & Total Effects”
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Mediation in SEM
To obtain bootstrapped confidence intervals for the indirect effect in AMOS: View Analysis Properties “Bootstrap” tab
Select “perform bootstrap” and enter number of resamples wanted (e.g., 10000)
Select “Bias-corrected confidence intervals” and specify width (default is a 90% CI) 95 % CIs are more common
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Mediation Example
Recall the example from the first week’s exercise Predictors: Autonomy Orientation; Controlled Orientation Outcomes: Burnout; Well-being
We can reframe the problem to try to determine if competence is a mediator: The effect of workplace variables on individual well-being and
burnout is mediated by feelings of competence within the workplace.
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ControlledOrientation
AutonomyOrientation
CompetenceSatisfaction
Well Being
Burnout
e11
e21
e3
1
Mediation Example
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Mediation Results in AMOS
Fit χ² (4) = 5.797, p = .215 CFI = .988 TLI = .971 IFI = .989 RMSEA = .0473
Since the fit of the model is good, we can go on to test whether competence is a significant mediator in the model
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Mediation Results in AMOS
Indirect paths: Controlled Orientation Competence Burnout Controlled Orientation Competence Well-Being Autonomy Orientation Competence Burnout Autonomy Orientation Competence Well-Being
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Mediation Results in AMOS
Autonomy Orient
Controlled Orient Competence
Competence .0000000 .0000000 .0000000
Burnout -.0839494 .0882250 .0000000
Well-Being .8815214 -.9264182 .0000000
Indirect Effects
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Mediation Results in AMOS
Autonomy Orientation
Controlled Orientation Competence
Competence .0000000 .0000000 .0000000
Burnout -.1791253 .0092771 .0000000
Well-Being .1200047 -2.0416531 .0000000
Indirect Effects - Lower Bounds (BC)
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Mediation Results in AMOS
Autonomy Orientation
Controlled Orientation Competence
Competence .0000000 .0000000 .0000000
Burnout -.0071333 .1799369 .0000000
Well-Being 1.9094402 -.1631157 .0000000
Indirect Effects - Upper Bounds (BC)
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Mediation Results in AMOS
Autonomy Orientation
Controlled Orientation Competence
Competence ... ... ...
Burnout .0325969 .0261903 ...
Well-Being .0260995 .0170113 ...
Indirect Effects - Two Tailed Significance (BC)
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Interpreting the Mediation Model
The effects of the predictors (autonomous or controlling orientation) on the outcome (well-being or burnout) are significantly accounted for by the mediator (competence) E.g., the effect of autonomy orientation on burnout is
significantly explained by competence Need to look at the parameter estimates to understand the
effect
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Interpreting the Mediation Model
Estimate S.E. C.R. P
Competence ControlledOrient -.1249 .0645 -1.936 .0528
Competence AutonomyOrient .1189 .0687 1.728 .0838
WellBeing Competence 7.412 1.404 5.276 ***
Burnout Competence -.705 .0944 -7.477 ***
Regression Weights:
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Mediation in SEM
Once you know the basics for testing mediation hypotheses in SEM, easily extended to more complex models involving latent variables The process is identical to what we have just covered and
quite straightforward. Testing mediation models with latent variables is not
possible in simple regression.
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Multiple-Group Models
Multiple Group/Sample Models allow us to examine a well specified model in 2 or more groups (e.g., males vs females) or 2 or more samples (e.g., cross validation)
This allows us to test if our loadings, covariances, etc. are different or not different across groups or samples Note: The model should fit both groups before proceeding with
a multi-sample analysis
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Types of Analyses
Examples: Evaluate a path analysis across samples to determine if the
coefficients differ Evaluate if the factorial structure of an instrument varies across
populations Evaluate a latent variable model across multiple groups to
determine if the loadings differ
Interpretations mirror an interaction effect
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Strategies for Multisample Analysis
Full Model Analysis Interest is in comparing all factor loadings, factor
variances/covariances, and structural model paths across groups
Beginning with the measurement model, subsequently “fix” parameters to be equal (i.e., if they do not differ then fix them to be equal in testing future restrictions)
Parameter Level Analysis In this strategy we are testing whether a specific parameter
differs across groups
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Multi-group Models
How do we know if a parameter (or a set of parameters) is invariant across groups? If the chi-square difference test (comparing the constrained and
unconstrained models) is significant then constraining the parameters to be equal significantly increased the chi-square statistic (relative to the degrees of freedom) In other words, constraining the parameters to be equal reduced
the fit of the model Recall: Δχ2 = χ2 (less constrained) - χ2 (more constrained)
• df(Δχ2) = df[χ2 (less constrained)] - df[χ2 (more constrained)] Thus, this parameter can be said to differ across groups
i.e., the model fits better when each sample takes on unique parameter estimates
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Multi-group models
Alternatively, we can look at the difference in the CFI fit index The χ² difference test is sometimes argued to be excessively
stringent when testing for invariance Cheung and Rensvold (2002) present research arguing that
a CFI difference test is a reasonable alternative If change in CFI value is less than or equal to .01, then the
null hypothesis of invariance should not be rejected i.e., the more constrained model is invariant across groups
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EmotionalExhaustion
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Depersonalization
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PersonalAccomplishment
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Multi-group models - Example
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Multi-group Models - Example
The first thing we need to do is to define our groups. Analyze Manage Groups Type in the first group’s name “Elementary teachers”, click
“new” Type in second group’s name “Secondary teachers”
Now we need to assign the data for each group In this example, the data for each group is in separate files However, can have the data for different groups within the
same file. In this case you need to specify the group variable
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Multi-group Models - Example
Selecting the data files: Click on icon “Select data file(s)”
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Multi-group Models - Example
For group data within same dataset: Again, click “select data files” Click “grouping variable”
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Multi-group Models - Example
Select the grouping variable and click “ok” Next, need to tell AMOS how the variable
distinguishes the groups Click “group value”
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Multi-group Models - Example
Once the groups have been defined, we estimate a baseline for both groups In this first model, both data sets are examined
simultaneously, holding only the pattern of factor loadings invariant.
This model serves two functions: First, it serves as a test of configural invariance
that is, poor fit of this model indicates that either the same factor structure does not hold for the two samples, or that the model is misspecified in one or both samples.
Second, the configural invariance model serves as a baseline model for evaluating, which can be used as a comparison model for other more restrictive models.
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Multi-group Models Example
Once we have a good-fitting configural model, we continue in a series of steps in order to test the invariance of specific parts of the model Usually, in the first step we want to see if the factor loadings
are invariant across groups.
Note: There is a short-cut for conducting multiple group analyses in AMOS, using the ‘multiple-group analysis’ command, although unless all difference-based tests are not significant (i.e., no follow-ups are required) then it does not save much time
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Multi-group Models - Example
To fix parameters to be equal, we label each of the parameters with the same name Right-click on path and select “object properties” Enter label under the “parameters” tab Be sure to check off “all groups”
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EmotionalExhaustion
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ITEM22e10L131
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Multi-group Models - Example
Both the male and female models should look the same
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Multi-group Models - Example
Baseline Model: χ² (330) = 1962.3 CFI = .919
Measurement Invariance Model: χ² (347) = 1992.6 CFI = .918
Δχ² (17) = 30.3 – significant (p < .05) ΔCFI = .001 – not significant We will trust the more stringent χ² difference test
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Multi-group Models - Example
Given that we did not find measurement invariance with the χ² difference test, for each of the dimensions of the MBI, we should look at each factor separately for measurement invariance. Label the paths for one factor at a time to set test for group
invariance on that particular factor. Assess using the Δχ²
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Depersonalization
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PersonalAccomplishment
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Multi-group Models - Example
We start with the emotional exhaustion construct
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Multi-group Models - Example
Baseline Model: χ² (330) = 1962.3
Measurement Invariance Model: χ² (337) = 1963.6
Δχ² (7) = 1.3 – not significant (p > .05) Therefore, the groups are invariant for the emotional
exhaustion factor parameters Retain this model and check for invariance of next
factor.
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EmotionalExhaustion
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1ITEM13e4
L61
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Depersonalization
ITEM22e10L131
ITEM15e11L12
1ITEM11e12
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1ITEM5e14
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1
PersonalAccomplishment
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1
Multi-group Models - Example
Now we check the Depersonalization Construct
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Multi-group Models - Example
Baseline Model: χ² (330) = 1962.3
Measurement Invariance Model: χ² (341) = 1972.2
Δχ² (11) = 9.9 – not significant (p > .05) Therefore, the groups are invariant for the
emotional exhaustion and depersonaliztion factors parameters So, the issue seems to be with the non-invariance of all or
some of the parameters in the personal accomplishment factor
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Multi-group Models - Example
Now that we have isolated the factor associated with the non-invariance, test each of the parameters within that factor separately.
Conduct a Δχ² for each item parameter.
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Multi-group Models - Example
Maintaining the most recent model (with emotional exhaustion and depersonalization factors invariant across the 2 groups), we first test for the invariance of item7
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EmotionalExhaustion
ITEM20e2
L81
ITEM14e3L7
1ITEM13e4
L61ITEM8e5
L51ITEM6e6 L41ITEM3e7 L31ITEM2e8 L21ITEM1e9 1
1
Depersonalization
ITEM22e10
L131
ITEM15e11L121
ITEM11e12L11
ITEM10e13 L101ITEM5e14 1
1
PersonalAccomplishment
ITEM21e15
L201
ITEM19e16
L191
ITEM18e171
ITEM17e181
ITEM9e19L161
ITEM7e20 L151ITEM4e21 1
1
1
Multi-group Models - Example
After testing all parameters, we fix to equality the ones that did not result in a significant χ2 difference test
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Multi-group Models - Example
Final Analysis Comparing Configural Model to Constrained Model
Baseline Model: χ² (330) = 1962.3
Measurement Invariance Model: χ² (345) = 1975.3
Δχ² (15) = 13.0 – not significant (p > .05) Therefore, we would examine the 2 item loadings (Items
17 and 18) for each group to see how/why they differ We might also want to determine if the covariances among the factors
differ across groups using the same strategy
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Multisample Analysis in J>2 Groups
We can also test for parameter differences in situations where there are 3 or more groups In this situation, any time that invariance is rejected “pairwise”
comparisons should be conducted to find out which groups differ
When multiple parameters are tested for invariance this can be very time consuming