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

222

)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

7

6

5

4

3

2

1

3

2

1

93

83

73

62

52

42

31

21

11

9

8

7

6

5

4

3

2

1

00

00

00

00

00

00

00

00

00

x

x

x

x

x

x

x

x

x

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