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GRA 6020Multivariate Statistics
Confirmatory Factor Analysis
Ulf H. Olsson
Professor of Statistics
Ulf H. Olsson
CFA
• The covariance matrices:
)(
)(
S
Ulf H. Olsson
CFA and ML
k is the number of manifest variables.
If the observed variables comes from a multivariate normal distribution, then
tkkdfwith
ondistributiaeapproximatwillSFN
)1(2
1
))(,()1( 2
kSStrSF log))(()(log))(,( 1
Ulf H. Olsson
Testing Fit
)(:
)(:0
AH
H
Ulf H. Olsson
Problems with the chi-square test• The chi-square tends to be large in large samples if the
model does not hold• It is based on the assumption that the model holds in the
population• It is assumed that the observed variables comes from a
multivariate normal distribution• => The chi-square test might be to strict, since it is based on
unreasonable assumptions?!
Ulf H. Olsson
Univariate Normal Distribution•
•
Ulf H. Olsson
Bivariate normal distribution
Ulf H. Olsson
J-te order Moments• Skewness• Kurtosis
( ) , 1; ( );jj
Population central moments
E X j E X
X is continuous and random
42 2
2
: 3Kurtosis
31 3/ 2
2
:( )
Skewness
Ulf H. Olsson
Kurtosis
2
4 4 4 24
22
tan :
1( ) 3;
2
( ) 1 1
x
Let X be s dardized and normal
EX X f X dX X e dX
we know that f X dX and EX
Ulf H. Olsson
Standard Normal density functions
2
2
2
1)(
u
eu
)2()1(2
1
2
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)1(2
1),(
vuvu
evu
),....,('
;)2(
1),...,.(
21
)'2
1(
2/12/21
1
n
nn
uuu
euuu
u
uu
Ulf H. Olsson
Alternative test- Testing Close fit
ondistributicentralNon 2
FunctionyDiscrepancPopulation
)0);/((0 ndfFMaxF
EAofestimateanisdf
FRMSEA
0
Ulf H. Olsson
How to Use RMSEA
• Use the 90% Confidence interval for EA• Use The P-value for EA• RMSEA as a descriptive Measure
• RMSEA< 0.05 Good Fit• 0.05 < RMSEA < 0.08 Acceptable Fit• RMSEA > 0.10 Not Acceptable Fit
Ulf H. Olsson
Other Fit Indices
• CN• RMR• GFI• AGFI• Evaluation of Reliability• MI: Modification Indices
Ulf H. Olsson
Nine Psychological Tests
9
8
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4
3
2
1
3
2
1
93
83
73
62
52
42
31
21
11
9
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3
2
1
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Ulf H. Olsson
Alternative Estimators
• Assuming multivariate normality• ML• GLS• ULS• If the model holds, ML and GLS are asymptotically
equivalente
Ulf H. Olsson
Alternative Estimators
))(())'(( 1 sUs
S: sample covariance
θ: parameter vector
σ(θ): model implied covariance
Ulf H. Olsson
Alternative Estimators
1, ghijOLSU
Ulf H. Olsson
Alternative Estimators
ghijghijGLS ssU ,
Ulf H. Olsson
Alternative Estimators
ghijghijMLU
,
Ulf H. Olsson
Assignment 1• The Country/People Image• 12 items was used for the country-people Image:• For each item a 7-point semantic differential scale was used • 1) Role in world politics• 2) Refined taste• 3) Likable• 4) Trustworthy• 5) Industrious• 6) Managing the economy well• 7) Technological level• 8) Ability to handle large and complex projects• 9) Would like closer ties• 10) Educational level• 11) Polluted environment• 12) Athletic People
• Research project: • Use the file CI-week11.psf to find a factor structure for the Construct Country Image. Test and evaluate the model.
Ulf H. Olsson
Assignment 2• Political Efficacy • Campel et al. (1954): Political Efficacy is the feeling that individual political action does
have, or can have, an impact upon political process.• Observed (measured) variables:• NOSAY: People like me have “no say” in what the government does• VOTING: Voting is the only way that people like me can have “any say” about how • the government runs things.• COMPLEX: Sometimes politics and government seem so complicated that a person like
me cannot really understand what is going on• NOCARE: I don’t think that public officials care much about what people like me think• TOUCH: Generally speaking, those we elect to Parliament lose touch with people pretty
quickly• INTEREST: Parties are only interested in people’s votes but not in their opinions
• Use the data set in the file: Efficacy2005.psf and find a suitable factor model