Multi-Group Structural Equation
Modeling (MGSEM)
Zhan Furner
Kevin Kim
Ajay Swain
Structural Equation Model
 A combination of factor analysis and multiple
regression
 Allows both confirmatory and exploratory modeling,
meaning they are suited to both theory testing and
theory development.
 Extends the possibility of relationship among the
latent variables and encompasses two components:
• A measurement model (essentially the CFA)
– Are the items grouped according to the theory?
– Assessment of convergent and discriminant validity of measurement
• A structural model
– What are the relationships among the latent constructs?
– Assessment of predictive validity
Family Tree of SEM
T-test
ANOVA
Multi-way
ANOVA
Repeated
Measure
Designs
Growth
Curve
Analysis
Bivariate
Correlation
Multiple
Regression
Factor
Analysis
Path
Analysis
Confirmatory
Factor
Analysis
Exploratory
Factor
Analysis
Structural
Equation
Modeling
Latent
Growth
Curve
Analysis
How SEM works
 You supply two main things
• Formal specification of model
• Observed relationship between variables
– (i.e., a covariance or correlation matrix)
 Model implies a set of covariances
 Software tries to reproduce observed covariance
matrix
• It does this by estimating parameters in the model
 Software produces two main things:
• parameter estimates
• information about how well it did in reproducing the covariance matrix
Steps in SEM analysis
 Step 1: Model specification
• usually done by drawing pictures using SEM software
 Step 2: Parameter estimation
• SEM software performs this step
• Iterative process
• Final result is a set of parameters that produce best fit to data possible
 Step 3: Assessment of fit
• Variety of ways to assess fit
– Chi-square goodness of fit
– RMSEA
– GFI
Multi-Group SEM
 A form of SEM analysis where two or more samples of
respondents are compared using similar models.
 Main question addressed:
• Do values of model parameters vary across groups?
• Does group membership moderate the relations specified in the model?
• Is there an interaction between group membership and exogenous
variables in effect on endogenous variables?
Multi-group SEM
 Between-group constraints are used to assess the similarities
between groups on any model parameters.
 Frequently used to evaluate measurement invariance in social
and behavioral science research.
• Measurement invariance refers to equivalent measurement information across
different groups (factorial invariance)
• Measurement invariance may be defined with varying degrees of stringency,
depending on which parameters are constrained to be equal
Measurement Invariance
 Configural invariance
• No constraint imposed on any parameters across groups
• Same basic factor structure exists in all of the groups
• Confirm appropriate levels of model fit and construct validity
 Weak factorial invariance
• Constraints are set to make factor loadings equal across groups
• Determines cross-group validity beyond basic factor structure
 Third level of invariance
• Equivalence of mean and covariance structures across the groups
• Invariance tests on intercepts for observed measures and factors
Multiple-Group model for Purchasing
Behavior
 Data were collected from two different regions, which will be
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labeled as Region 1 ( N =378) and Region 2 (N=423 ), respectively
Spend02 : Total Purchase amount in 2002
Spend03 : Total Purchase amount in 2003
Proc Tcalis
A Restrictive model
A Model with Unconstrained parameters for the two regions
A Model with Partially constrained parameters for the two regions
Simultaneous test
• H0: G2_interspend02 – G1_interspend02 = 0
• H1: G2-interspend03 – G1_interspend03 = 0
Multiple-Group model for Purchasing
Behavior
Why use Multiple-Group Models?
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Do respondents from different cultures interpret a given
measure in a conceptually similar manner?
Do rating sources define performance in similar ways when
rating the same target on identical performance dimensions?
Are there gender, ethnic, or other individual differences that
preclude responding to instruments in similar ways?
Does the very process of substantive interest (i.e., an
intervention or experimental manipulation) alter the
conceptual frame of reference against which a group responds
to a measure over time?
Example Cases for Use of MGSEM
• Test the hypothesis that workers in an individualistic culture are less
committed than are workers in a collectivist culture
• by comparing group mean scores on an organizational commitment measure
across cultures.
 Longitudinal changes in organizational commitment associated with
organizational practices, interventions, or structure
 The analytical approach to the group comparison is
unimportant (e.g., a test of mean differences vs. a test of homogeneity of
regression, etc.).
 The key issue here is the desire to make meaningful inferences regarding
the status of those groups on the measures and to draw conclusions as to
how this group difference may affect organizational functioning
What SEM is and What SEM isn’t
 WHAT IT IS:
• Tests hypotheses about relationships
• Between variables
• Very flexible
• Comprehensive: subsumes many other techniques
– multiple regression
– confirmatory factor analysis
– path analysis
– ANOVA
 WHAT IT ISN’T:
• Only for correlational studies
• A way to test causal hypotheses from correlational data
SEM vs. Other Approaches
 Similar to standard approaches
• based on linear model
• based on statistical theory; conclusions valid only if assumptions are met
• not a magic test of causality
 Different from standard approaches
• requires formal specification of model
• allows latent variables
• statistical tests and assessment of fit more ambiguous
– can seem like less of a science; more of an art
Advantages of SEM
 Use of confirmatory factor analysis (CFA) to reduce
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measurement error by having multiple indicators per latent
variable
Graphical modeling interface
Tests models overall rather than coefficients individually
Tests models with multiple dependents
Modeling indirect variables
Tests coefficients across multiple between-subjects groups
Can handle difficult data (time series with autocorrelated
error, non-normal, incomplete data)
SEM limitations
 Biggest limitation is sample size
• It needs to be large to get stable estimates of the covariances/correlations
• @ 200 subjects for small to medium sized model
• A minimum of 10 subjects per estimated parameter
• Also affected by effect size and power
 Missing data
• Can be dealt with in the typical ways (e.g. regression, EM algorithm, etc.)
through SPSS and data screening
• Most SEM programs will estimate missing data and run the model
simultaneously
SEM limitations
 Multivariate Normality and No outliers
• Screen for univariate and multivariate outliers
• SEM programs have tests for multi-normality
• SEM programs have corrected estimators when there is a violation
• use of bootstrap estimates of parameters and standard errors
 Linearity
• Violation of the linearity assumption means that estimates of model fit and
standard error are biased (not robust)
 Multicollinearity
• complete multicollinearity prevents a SEM solution
Questions?
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