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Confirmatory Factor Analysis CFA

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**1. **Confirmatory Factor Analysis (CFA) CFA is used when strong theory and/or when a strong empirical base is available
Specify relations a priori
number of factors
relations among factors (i.e., correlated vs. uncorrelated)
variables specified as fixed or free on a respective factor(s)

**2. **Confirmatory Factor Analysis a priori measurement model is specified and tested
factor loadings
direct relations between observed and latent variables are modeled
error terms are the “leftovers”
estimate the variance of these things
factor variances/covariances
typically we are interested in the standardized covariances between factors
factor correlations

**3. **Two-factor correlated CFA model

**4. **The Process Specifying the pattern matrix:
* = parameter we will estimated (free)
0 = parameter noted estimated (fixed)

**5. **The Process Model specification via the Bentler-Weeks model
all variables in a model are categorized as IVs or DVs
DV = variable with unidirectional arrows aiming at it
want to explain the variance in these variables with other variables (IVs)
e.g., our eight observed variables
IV = no unidirectional arrows aiming at it
but there can be an unanalyzed association
mathematically we estimate the error variances from slide 3 as well

**6. **The Process you write an equation for each DV
S1 = ?(Academic Self-Esteem) + e1
S2 = ?(Academic Self-Esteem) + e2
.....
S5 = ?(Relationship Self-Esteem) + e5
S6 = ?(Relationship Self-Esteem) + e6
....
? represents a regression coefficient (factor loading)
e represents error (or residual), this path is not directly estimated (fixed)
predetermined by the factor loading

**7. **The Process core parameters in CFA are these factor loadings and variances/covariances for IVS
notice that the latter are not directly specified in these equations
e.g., covariance between Academic and Relationship Self-Esteem
they are a function of the equations that you see
via some complicated matrix algebra
however, you must specify them to solve the equations
software does this for us

**8. **The Process Estimating the model
using primarily maximum likelihood
estimation produces a fit function
Determination of model fit
done at two levels
overall model fit
individual parameter fit
parameters = generally factor loadings in CFA
but include factor covariances (correlations) if specified

**9. **The Process Overall model fit (Goodness of fit)
tells us if the model should be accepted or rejected
if model is accepted, interpret model parameters
if model is rejected, do not interpret model parameters
Determining goodness of fit
test statistic
?2 provides a statistical test of fit
?2 = ( fit function ) ( N – 1)
we want this to be nonsignificant

**10. **The Process types of descriptive indices
absolute fit indices
indexes the amount of variance/covariance accounted for by a model
goodness of fit index (GFI) and adjusted GFI
want values > .90
root mean square residual (RMSR)
average size of residuals generated by a model
want standardized values < .05 if model is good

**11. **The Process comparative fit indices (CFI)
compare target model to a baseline model
baseline model = null or independence model
null model = specifies no factors
CFI values > .90 are good, .93 better, .95 great
parsimony adjusted fit indices
adjusts fit by weighting values by the number of parameters estimated
root mean square error of approximation (RMSEA) is best
values less than .08 are good, .05 are better

**12. **The Process Fit of individual parameters
we have statistical tests for each factor loading and each factor co(variance)
evaluate the critical ratios (CR)
these are distributed as z-values
What if my model and/or individual parameters do not fit?
report that and stop, or
go to the model modification phase
the LaGrange Multiplier test
the Wald test

**13. **Practical Issues Identification
also needed to mathematically solve the equations
based largely on degrees of freedom (df) for the model
df = nonredundant elements in ? - parameters estimated
elements in ? = # variances & covariances
this equals p (p+1) / 2, where p = # observed variables
parameters estimated
count up factor loadings, factor covariances, and IV variances estimated

**14. **Identification of a one-factor model e.g., 4 MVs ? 4 (4 + 1) /2 = 10 variances/covariances
e.g., 4 factor loadings, 4 error variances
e.g., df = 10 – 8 = 2

**15. **Practical Issues over-identified (the ideal)
positive df = more information than parameters to estimate
can determine overall model fit
under-identified
too many parameters, not enough information
model cannot be estimated
just-identified (df = 0)
parameters to be estimated = amount of information
no overall model fit, but you can interpret parameter estimates

**16. **Practical Issues EQS will present unstandardized factor loadings and factor covariances
remember, the analyses are based on ?
however, we generally interpret standardized solutions
this makes factor loadings range (roughly) between 1 and –1
and makes factor covariances into factor correlations

**17. **The Structural Model Testing the directional relations among latent variables
This is just path analysis with latent variables
Latent variables are developed through confirmatory factor analysis (CFA)
General modeling process is the same as with CFA

**18. ** Structural Model Equations

**19. **Comparing nested models

**20. **Comparing nested models

**21. **Comparing nested models we can statistically compare models 1 and 2
model 1: does not have the direct effect
model 2: does have the direct effect
both models have the mediated or indirect effect
model 1, then, is nested within model 2, and thus they can be statistically compared
we do this using the ?2 difference test (? ?2 )

**22. **?2 Difference Test Statistically compares nested models
nested = lower-order models that contain a subset of the parameters from a target higher-order model
e.g., model 1 is nested within model 2 (target)
??2 = ? 2nested - ? 2target; ?df = dfnested – dftarget
notice that the nested model will always
have worse overall model fit (higher ?2 )
and more degrees of freedom
because we are estimating fewer things
??2 and ?df will always be positive because of this

**23. **?2 Difference Test if ? ?2 is not significant...
there is no difference between models
the simpler or more parsimonious model fits "better"
if ? ?2 is significant...
target model fits better and...

**24. **Comparing nonnested models This is a direct comparison between models that have at least a subset of variables that differ
We typically use other descriptive fit indices for these purposes
Akaike Information Criterion (AIC)
Bayesian Information Criterion (BIC)
Expected Cross-Validation Index (ECVI)
For all of the above indices, the model with the smaller index value is the better-fitting model

**25. **Setting the Scale (Metric) for an Endogenous Latent Variable We need to do this to mathematically solve the equations of the model
Two options:
fix variance of latent variable to 1 (standardize)
fix a factor loading for each LV to 1
For an endogenous LV, you can only use the second option
we want to predict the variance of the endogenous LV
setting this value to 1 does not allow for this possibility