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Structural Equation Modeling SEM: An Introduction

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**1. **Structural Equation Modeling (SEM): An Introduction Also referred to as
covariance structure analysis, analysis of covariance structures, covariance structure modeling, LISREL modeling, causal modeling
Merges the logic of factor analysis and multiple regression
Goal is to develop a model that explains why observed variables are related
i.e., explain the variance-covariance matrix (?)
this is done through the simultaneous solution of equations representing your model

**2. **SEM Advantages of SEM
reduces measurement error by having multiple indicators of a latent variable
ability to test overall models and individual parameters
ability to statistically compare nested and non-nested models
ability to test models with multiple DVs
ability to model mediator variables (processes)
ability to model error terms
ability to model relations across groups, across time

**3. **SEM 2 major approaches, each based on the researcher’s goals
confirmatory factor analysis (CFA)
structural equation modeling (SEM)
For each major approach, one can take 1 of 3 additional methodological approaches
strictly confirmatory
alternative models
model development

**4. **SEM Variables
observed = measured, manifest, indicators
can be items, subscales, or scales
latent = theoretical constructs
variables that are defined by the observed variables
goal is to model the commonality in the observed variables
sound like factor analysis?
and then look at relations between latent variables
sound like multiple regression?

**5. **Confirmatory Factor Analytic Model a priori measurement model is specified and tested
direct relations between observed and latent variable(s) are modeled

**6. **Structural Model CFA plus a priori structural model is tested
two step process (two-step modeling)
establish the measurement model
test the structural model
direct relations among latent variables are modeled
i.e., regression with latent variables
see figure on next page, too big for here!

**7. **Structural Model continued

**8. **Structural Model continued Types of latent variables (LV)
exogenous: LVs that only “cause” other LVs
endogenous: LVs that are “caused” by other LVs
pure DVs: are only “caused”
mediators: are a “cause” and “caused”

**9. **The Process Model specification
you write equations to specify each
factor loading
MV1 = ?(LV) + e1
e.g., BDI = ?11(Depression) + e1
relation among latent variables
endogenous LV = ?(exogenous LV) + d2
Depression LV = ?21(Coping LV) + d2
these equations imply a model
this model attempts to explain the variance-covariance matrix (?)
i.e., relations among observed variables

**10. **The Process Estimating the model
mathematically this is done with some heavy duty matrix algebra and tracing rules
estimation procedures include
maximum likelihood
generalized least squares
asymptotic distribution free methods
etc.
estimation produces a fit function
tells us how well we have reproduced ? with our target model

**11. **The Process Determination of model fit
done at two levels
overall model fit
individual parameter fit
parameters = factor loadings, factor correlations, structural paths
Overall model fit
referred to as goodness of fit
tells us if the model should be accepted or rejected

**12. **The Process if model is accepted, interpret model parameters
if model is rejected, do not interpret model parameters
Determining goodness of fit (see Hu & Bentler, 1999)
the fit function from estimation is converted to a test statistic and numerous descriptive indices
test statistic
?2 provides a statistical test of fit
?2 = ( fit function ) ( N – 1)
we want to this to be nonsignificant
it never (rarely) is (large N)

**13. **The Process descriptive fit indices
comparative fit index (CFI; Bentler, 1990)
compares target model to a baseline model
baseline model = null or independence model
null model = no factors at all
CFI values > .90 are good, .93 better, .95 great
parsimony adjusted fit indices
adjusts fit by weighting indices of fit by the number of parameters estimated
root mean square error of approximation (RMSEA; Steiger, 1990) is best
values less than .08 are good, .06 are better

**14. **The Process Fit of individual parameters
we have statistical tests for each parameter
referred to as critical ratios (CR)
CR = parameter estimate / standard error
these are distributed as z- (or t-)values
we also typically evaluate standardized values for practical fit
factor loadings > .30 ( or .45 in my world )
factor correlations & structural coefficients depend on your research literature

**15. **The Process What if my model and/or individual parameters do not fit?
report that and stop, or
go to the model modification phase
SEM has what are called modification indices that tell you...
whether or not there are parameters that you can add to improve fit
via the LaGrange Multiplier test
whether or not there are parameters that you can delete without hurting fit
via the Wald test
remember, however, that SEM is THEORY-DRIVEN!!!!!!!!

**16. **Practical Issues Assumptions of CFA
large N: at least 200 but 400 is optimal, with the following restrictions
number of cases to MVs
10 to 1
number of cases to estimated parameters
10 to 1
and variations on these if your data is non-normal
number of cases to estimated parameters
15 to 1 or 20 to 1

**17. **Practical Issues Non-normal data
typically evaluate multivariate non-normality
assessed via Mardia’s coefficient
scaled z-value
Satorra-Bentler scaled ?2 corrects for non-normality in 2 ways
overall ?2 and descriptive fit indices
corrects for the overestimation (or inflation) of ?2
standard errors of parameters
corrects for the underestimation of the standard errors

**18. **Practical Issues Measurement scale of MVs
anything, but some assumptions/adjustments need to be made
must assume that discrete/ordinal/interval data has an underlying normal distribution
and/or use special correlations
all binary variables = tetrachoric R
binary and continuous = either polychoric polyserial correlations
Number of observed variables per LV
want a minimum of three (> 3 is better)
Observed variables do NOT have to indicate a LV to be included in the model

**19. **The Process Summary of SEM
propose a priori model(s)
evaluate univariate and multivariate distribution of data for observed variables
determine overall fit of measurement model
by employing confirmatory factor analysis
determine individual parameter fit of measurement model
modify and re-establish fit if necessary or
compare all a priori models

**20. **The Process Summary of SEM continued
add structural model to measurement model
determine overall fit of SEM
determine individual parameter fit of primarily structural model
including variance accounted for in each endogenous LV
modify and re-establish fit if necessary or
compare a priori models