GRA 6020 Multivariate Statistics Confirmatory Factor Analysis
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Transcript of GRA 6020 Multivariate Statistics Confirmatory Factor Analysis
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
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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?!
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Univariate Normal Distribution•
•
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Bivariate normal distribution
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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
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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)(
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eu
)2()1(2
1
2
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)1(2
1),(
vuvu
evu
),....,('
;)2(
1),...,.(
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)'2
1(
2/12/21
1
n
nn
uuu
euuu
u
uu
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Alternative test- Testing Close fit
ondistributicentralNon 2
FunctionyDiscrepancPopulation
)0);/((0 ndfFMaxF
EAofestimateanisdf
FRMSEA
0
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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
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Other Fit Indices
• CN• RMR• GFI• AGFI• Evaluation of Reliability• MI: Modification Indices
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Nine Psychological Tests
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Alternative Estimators
• Assuming multivariate normality• ML• GLS• ULS• If the model holds, ML and GLS are asymptotically
equivalente
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Alternative Estimators
))(())'(( 1 sUs
S: sample covariance
θ: parameter vector
σ(θ): model implied covariance
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Alternative Estimators
1, ghijOLSU
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Alternative Estimators
ghijghijGLS ssU ,
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Alternative Estimators
ghijghijMLU
,
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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.
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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